Sample path large deviations for marked point processes in the many sources asymptotic with small buffers: Heavily and lightly loaded systems
James R. Cruise Fraser Daly Bemsibom Toh



    Nuh Gedik
    January 22, 2019
====================



Abstract
Consider a queueing system fed by traffic from N independent and identically distributed marked point processes. We establish several novel sample path large deviations results in the scaled uniform topology for such a system with a small buffer. This includes both the heavily loaded case (the load grows as N→∞) and the previously unexplored lightly loaded case (the load vanishes as N→∞); this latter case requires the introduction of novel speed scalings for such queueing systems. Alongside these sample path large deviations results, we introduce a new framework to explore the range of scalings in the many sources asymptotic for these systems.   




Key words and phrases: Queuing theory, large deviations, marked point processes, uniform topology, heavy traffic




AMS 2010 subject classification: 60F10, 60G55



§ INTRODUCTION


In this paper we prove a number of novel and insightful sample path large deviations (SPLD) results related to the many sources asymptotic regime for queueing networks when fed by traffic from marked point processes.  These results extend previously published results for queueing systems with small buffers, i.e., where the size of the buffer grows sub-linearly in the number of sources <cit.>, and a fixed load, to both heavily loaded systems (the load tends to 1 as the number of sources grows) and lightly loaded systems (the load tends to 0 as the number of sources grows).  Throughout this work, for small buffers we retain the flavour of the previous Poisson convergence result <cit.> but for heavily loaded cases this is mixed with a Gaussian limit, so the system behaves as if fed by traffic from a Brownian source. For lightly loaded systems the traditional large deviations scaling breaks down, so we need to introduce a novel speed scaling for the SPLD.  

To better understand how these results relate to each other, we introduce a novel framework to parametrise the different scalings and allow for key insights into how the different scaling results relate to each other, and, furthermore, important insights as to how to select the scaling of importance when considering practical applications.  

In the many sources asymptotic <cit.> we are interested in a series of queueing systems where the traffic arriving externally into the N^th system is produced by a set of N independent and identically distributed traffic sources each with mean arrival rate λ.  This asymptotic was initially introduced by <cit.> and has been well studied in application areas including core network routers, admission control, wireless networks <cit.>. The particular focus on the small buffer regime is driven by a desire to understand the effect of reducing buffer sizes on core networks when modelling a range of different congestion control protocols, for example TCP, but is also relevant for data centres wanting to make use of optical networking technologies where buffering is difficult.  

In addition, the study of the small buffer regime is useful in better understanding the power which can be obtained through multiplexing, rather than using a single server for each source: if services are aggregated together, how does a smaller than linear growth in buffer size affect system performance?  

The many-sources scaling was introduced by Alan Weiss <cit.> to explore the quality of service in data networks. In the original scaling, both the buffer size and the service rate grow linearly with the number of sources feeding the system. In the N^th system, the buffer size of interest is NB and the service rate is of the form Nλ + NC, where λ>0, C>0. In that work, the traffic processes were limited to Markov on/off sources, which produce traffic at a constant rate when on, and for which transitions between states are governed by a Markov chain in continuous time. Since then, a large body of work has developed around using this asymptotic framework to analyse different queuing systems. Buffet <cit.> and  Buffet and Duffield <cit.> used martingale methods to obtain bounds for the queue length in queues with Markovian arrivals within this framework. This was extended to more general sources by Botvich and Duffield <cit.>, who obtained a rate function for the workload process in a single server queue in both discrete and continuous time. Simonian and Guibert <cit.> obtained bounds on the overflow probabilities in continuous time for queues fed by on/off sources. Courcoubetis and Weber <cit.> considered a discrete-time analogue of this system, for which they characterized the rate function associated with the overflow probability. Likhanov and Mazumdar <cit.> extended the work of Courcoubetis and Weber <cit.> by obtaining exact bounds for buffer overflow probabilities for queues with finite capacity buffers. Since then, different authors have used this framework to investigate different problems in queuing theory, including buffer overflow in multiqueue systems <cit.>, conditional delays in queues <cit.> and information loss across networks <cit.>. Different kinds of arrival processes have also been considered, for example, continuous-time Gaussian processes with stationary increments, including fractional Brownian motion and integrated Gaussian processes <cit.> and on/off flows with heavy-tailed (regularly varying) on periods <cit.>.

There have been several attempts to move beyond the linear scaling on buffer size, and consider systems with small buffers <cit.>, and very large buffers <cit.>. This is usually carried out by examining a second limit in B on the associated rate function for the large deviations principle, either to let B increase to ∞ or decrease to zero. This approach does not provide the richness in scalings that we consider, and also obscures the time-scales upon which the most likely events occur. In addition this approach does not enable us to understand the joint effect of varying load and buffer size together.

The moderate deviations framework for queues with many sources was introduced by Wischik <cit.>, to study heavily loaded systems. The inspiration for this work was an attempt to marry together ideas from functional central limit theorems and large deviations results for the many flows asymptotic. To this end, the scaling considered was such that in the system with N sources the buffer is of size N^(1+γ)/2B and the service rate is Nλ + N^(1+γ)/2C for γ∈ (0,1).  Moderate deviations have also been studied by Puhalskii <cit.>, who focused on scaling the load of the system instead of the number of sources, and obtained logarithmic asymptotes for queue length and waiting time processes in single server queues and open queuing networks in heavy traffic, and by Chang et al. <cit.>, who obtained results for queues with long-range dependent input.

Another relevant scaling in the literature is that of the small buffer scaling introduced by Cao and Ramanan <cit.>. Here the buffer size and load are kept constant as the number of flows is increased. This paper observes the short time-scales at which events occur and makes use of this idea to show that for general point processes the behaviour of queue length is as if it had been fed by Poisson traffic. The use of weak convergence enables the authors to obtain the full distribution in the limit but does not easily allow the extension to general networks and sample path results. In comparison to this, the small buffer results proved by Cruise <cit.> enable the discussion of networks and other service disciplines, but provide only tail asymptotics, since they are obtained using large deviations techniques. The small buffer scaling has been investigated in further detail in <cit.>.

The aforementioned results focused on the scaling of a single server queue. Wischik extended these results to consider sample path results in discrete time for single server queues <cit.>, and for switches <cit.> operating in discrete time under the many-sources asymptotic. Subramanian <cit.> also used this approach in analysing multi-queue systems operating in discrete time under a Max-Weight scheduling algorithm. This sample path approach is more powerful than simply considering the behaviour of the system in one dimension, because it allows us to use tools like the contraction principle <cit.> to make very general statements about how various quantities of interest (such as waiting times) scale as the system scales, without needing to re-do analogous calculations from scratch.

The main contribution of this paper is the development of a number of important sample path large deviations in the scaled uniform topology for generic point process arrivals for the many sources asymptotic with small buffers. Previous studies of systems with small buffers have focused on situations where the load was constant as the number of sources increased. Here we extend this to the important situation of heavy traffic and the novel setting of very lightly loaded systems. These situations are important in providing insight into the behaviour of real systems <cit.> but also in providing a better understanding of resource pooling and the design of future systems.  In all cases we demonstrate the importance of understanding the most likely time-scale for events of interest, which for the small buffers considered in this paper are short. These short time-scales lead to parsimony in the results, with the heavily loaded systems demonstrating a Brownian behaviour for a large class of arrival processes; similarly for lightly loaded systems, the result depends only on a small number of parameters of the system. 

In addition, to our knowledge this is the first paper to properly explore the lightly loaded case, i.e., the situation when the load tends to zero as the number of sources increases. Here we demonstrate a large deviations principle which has an unusual rate: rather than being polynomial in N we have log(N). This scaling provides both theoretical insights into the richness of the different scalings which can be achieved, while also providing qualitative insight into the key features which govern the behaviour of real systems.

The final contribution in this work is the development of a novel framework for exploring a range of scalings for the many sources asymptotic. As part of the introduction of this novel framework, we explore the associated sample path scalings. These scalings provide an insight into the key features which will affect the results obtainable in each case, and also into how the various scalings relate to each other. Beyond the small buffer results discussed in this paper, this framework introduces a large range of important and unexplored scalings which will require different techniques and are beyond the scope of this paper. 

The remainder of the paper is organised as follows: Section <ref> introduces our model and the scaling framework we use throughout the work that follows.  Definitions and assumptions related to the traffic arriving at the system as well as other system parameters are detailed in Section <ref>. Our main results are stated in Section <ref>, and proved in Section <ref>.



§ MODEL AND SCALING FRAMEWORK

We consider a sequence of N independent single-server queues, such that the N^th system is fed by traffic from N independent identically distributed sources. This relates to the many sources asymptotic <cit.>.

Consider the N^th system in this sequence. We introduce a new scaling parameterization, indexed by (α, β). The parameter α is used to control the buffer size scaling, such that in the N^th system the buffer is size N^αB. For  α >1 the buffer grows faster than the number of sources, whereas for α <1 the buffer grows slower than the number of sources. β is used to control the excess service rate above the total arrival rate,  such that in the N^th system the excess service capacity is  N^β C. For β<1 the load increases to one as the number of sources increases; the heavily loaded case. For β > 1 the load tends to zero as N increases giving the lightly loaded scenario. 

The relationship between α and β and the division of the parameter space is shown in Figure <ref>.


 We denote the traffic from source i   (1 ≤ i ≤ N) as  A^(i), where A^(i)_t represents the total amount of traffic emitted by source i in the interval [0, t]. We assume that each A^(i)_t has the same distribution as a simple stationary point process A that satisfies 𝔼[A(0,t)] = λ t for each t>0 for some λ > 0. The stability condition C>0 ensures that the queues do not grow unboundedly. The superposed process A^⊕ N = ∑_i=1^N A^(i) represents the aggregate arrival from all N sources. Arriving jobs that cannot be processed immediately are stored in a buffer, assumed to be infinite. We also assume that each job has the same size, and, without loss of generality, set its processing requirement to be one unit, assuming a service rate of Nλ +N^βC. We use the steady state probability of the unfinished work exceeding a certain level N^αB as a surrogate for the steady state buffer overflow probability in a buffer of size N^α B. Let  denote the stationary unfinished work in this system, when the number of sources is N, the arrival process of each source is distributed according to A, and the processing rate is Nλ + N^βC. Under these conditions,  is given by

    = sup_t ∈ [0, ∞)[A^⊕ N (0,t) - N λ t - N^β Ct ] .


To investigate the sample path scalings, we need to find a scaled process of the arrivals, Ã^N_α, β, such that

    (f_C(Ã^N_α, β) > B) = ( > N^α B) ,

where f_C is the queuing map, defined by

    f_C(x) = sup_t>0 (x(t) - Ct) .

Using the continuous form of Loynes' scheme for stationary queue length <cit.> we obtain

    ( > N^α B) = (sup_t ≥ 0∑_i=1^N A_i(0,t) - (N λ + N^β C) t > N^α B ) .

We can re-arrange this to obtain

    ( > N^α B)     = (sup_t ≥ 0∑_i=1^N A_i(0,t)/N^α - N^1-αλ t - N^β - α Ct > B ) 
        = (sup_t' ≥ 0∑_i=1^N A_i(0,N^α - βt')/N^α - N^1-βλ t' - Ct' > B ) 
        = (f_C ( {∑_i=1^N A_i(0,N^α - βt')/N^α -  N^1-βλ t }_t ≥ 0) > B ),

where t' = tN^β - α. The natural scaled process to consider is therefore

    {Ã^N_α, β}_t>0 = {∑_i=1^N A_i(0,N^α - βt')/N^α -  N^1-βλ t' }_t>0 .




We will examine sample path large deviations principles in the following five cases: 


  * α=β=1: original large deviations asymptotic,

  * 0<α<β = 1: small buffer large deviations asymptotic,

  * 1/2<α=β <1: original moderate deviations  asymptotic,

  * α < β<1, α + β >1: small buffer moderate deviations asymptotic,

  * 0<α<1, β>1: large deviations for light-load.





Figure <ref> illustrates the relationship between the scalings (i)–(iv). Both the scalings in (i) and (iii) have been considered previously for many traffic processes for queue length <cit.> and sample path results in discrete time <cit.>. Here we extend these results to sample paths in continuous time.

It is natural to consider scalings (i) and (iii) together, and (ii), (iv) and (v) together because of the timescale upon which the events of interest occur. In the first two scalings the timescales stay constant as we increase the number of flows. In contrast, in the latter three cases they converge to 0, so overflow events happen quickly, which leads to a simple rate function and insensitivity of results to covariance structure. Separately in scalings (iii) and (iv) we see the heavy loads leading to Gaussian structure, and so the proofs reflect this.




§ TRAFFIC ASSUMPTIONS AND ASSOCIATED SPACES

In this section we introduce the various processes of interest in the queuing systems we study. We begin with the arrivals processes, which we assume to be marked point processes.

In many practical applications, the use of a marked point process as a traffic model is sensible since traffic often arrives as units rather than a continuous stream, for example, packets in the Internet or customers in a shop. The marking allows the study of general systems where arrivals can bring more than one customer. An example is computer protocols where multiple packets can be transmitted together. Alternatively,  the marks can represent service times of customers and queue length can represent residual work load using a deterministic server.

Let 𝕏 be a collection of independent and identically distributed simple stationary point processes <cit.>, with mean λ. Let 𝕏^(i) be the i^th point process and 𝕏^(i)(t_1, t_2) be the number of points in the set (-t_2, -t_1] for the i^th process, so that 𝔼(X^(i)(0,t)) = λ t by stationarity. We associate a collection of stationary, positive, identically distributed random variables 𝕐^(i)∈ℝ^∞ with each process, and let Y^(i)_j be the j^th random variable of the i^th collection. Each collection of random variables is independent of the other collections and of all the point processes, and all the collections are identically distributed. Let Y represent a generic random variable Y_j^(i). We define the marked point process A^(i), representing arrivals from source i, where we associate Y_j^(i) with the j^th point, by

    A^(i)(0,t) = ∑_j=1^X^(i)(0,t) Y_j^(i) ,

and 

    A^(i)(t_1, t_2) = A^(i)(0,t_2) - A^(i)(0,t_1) .

We define the aggregates A^⊕ N and X^⊕ N by A^⊕ N(t_1, t_2) =∑_i=0^N A^(i)(t_1, t_2) and X^⊕ N(t_1, t_2) =∑_i=0^N X^(i)(t_1, t_2), respectively, the aggregate of N marked point processes.



The server is a deterministic server such that the N^th system, which has N input sources, has service rate Nλ + N^β C and buffer size N^α B. We let Q^(N) be the stationary queue length in the N^th system. This is given by 

    = sup_t ∈ [0, ∞)[A^⊕ N (0,t) - N λ t - N^β Ct ] .


In addition, we will need the following definitions:
For
x∈ℝ and t∈ [0,∞), 

Ψ(x,t) = sup_θ∈ℝ [θx - t^-1 log𝔼[ e^θA(0,t)]] ,   Ψ_∞^1(x) = lim inf_t →∞ t Ψ(x,t)/logt .

When considering the moderate deviations results, we will also need tighter bounds on the behaviour of the arrival process at large timescales. To achieve this we will need to consider the following limits:
 
 Ψ_∞,d→t^2  = lim inf_t →∞ lim inf_d→0 t Ψ(λ𝔼(Y) + d ,t)/d^2logt
 ,
 and 
  
   Ψ_∞,t→d^2  = lim inf_d→0 lim inf_t →∞ t Ψ(λ𝔼(Y) + d ,t)/d^2logt .
 

We will also need the moment generating function of Y, which we denote M(θ) =𝔼(e^θ Y), and the log moment generating function of A(0,t), which we denote Λ_t(θ) =log𝔼(e^θ A(0,t)). In addition, we define the set on which the moment generating function is finite, 𝒟_M ={θ∈ℝ: M(θ) < ∞}. Finally, let 𝒟_y be the space of cadlag functions, x: ℝ^+ ↦ℝ^+ for which x(0) = 0 and 

    lim_t →∞x(t)/1+t = y .


While considering sample paths we will need to characterize compact sets on which the measure associated with the process is large. Marked point process naturally live in the space of cadlag functions in which it is difficult to classify compact sets. So we instead often make use of a linear interpolation of the marked point process which lives in the space of continuous functions. We define this as follows:

 
Let τ_N^-(t) be the time of the last point before or at
t of X^⊕ N, and  τ_N^+(t) be the time of the next point after t of X^⊕ N, and let ζ_N(t) be its associated mark.
Then let A̅^⊕ N be the
polygonal approximation of A^⊕ N defined by

A̅^⊕N(0,t)= A^⊕N(0,t)
+t-τ_N^-(t)/τ_N^+(t)-τ_N^-(t)ζ_N(t).

We define the scaled version, Ã̅̃^N_α,β, in the obvious way.


Our results will be proved in the space of cadlag functions with a given long run mean rate y, 𝒟_y, with the topology induced by the scaled uniform norm ·_s; see <cit.>, <cit.> . In addition, in stating the results we utilize the subspace of absolutely continuous functions and the reproducing kernel Hilbert space R_v for a given variance function v <cit.>.  We let 𝒜𝒞 denote the set of absolutely continuous functions on an appropriate space as required.




§ RESULTS
 

We use this section to state out main results and the assumptions under which they hold; proofs are deferred until Section <ref>. We begin with case (i), α=β =1; for work in this regime in the discrete time setting, see Wischik <cit.>.  Our results in this continuous time setting will be established under the following assumptions:
 



  * There exist θ_0 >0 and K<∞ such that

lim_t→0t^-1
𝔼[e^θ_0 A(0,t)1_{X(0,t)>K}]=0.


  * Ψ^1_∞(x) > 0 for all x ≠λ𝔼(Y).

  * Λ_t (θ) < ∞ for all t >0 and θ∈ℝ. 

  * {A̅^⊕ N( ·)/N }_t∈ [0,T) is exponentially tight <cit.> in the space of continuous functions on the interval [0,t) with the scaled uniform norm.


With these assumptions we obtain the following:

Let A be a stationary marked point process.
Under Assumptions <ref>, the sequence  Ã^N_1,1 satisfies a sample path large deviations principle in the space 𝒟_0 with the scaled uniform norm ||x||_s. This has rate N, and good rate function I_1,1(x), where

    I_1,1(x)= sup_j ∈𝒥, T∈ℝ^+Λ^*_j,T(x) ,

where 𝒥 is the collection of all ordered finite subsets
of (0,1] with j_0=0, and

    Λ^*_j,t(x) = sup_θ∈ℝ^|j|{∑_i=1^|j| (x(t j_i) - x(t j_i-1) ) θ_i -Λ_j,t(x) } ,

with

    Λ_j,t(x) = log𝔼( e^∑_i=1^| j |θ_i A(j_i-1 t,  j_i t ) - λ𝔼(Y)∑_i=1^| j |θ_i t(j_i -j_i-1)).



We now turn to results for small buffers which involve short timescales. 
We start by considering scaling (ii), where 0<α<β=1. Results here have previously been established by <cit.>; we state them here (together with an outline of the proof in Section <ref>) to illustrate how this fits into the framework which we introduce.  In this setting we need the following assumptions:




  * There exist θ_0 >0 and K<∞ such that

    lim_t→ 0t^-1𝔼[e^θ_0 A(0,t)1_{X(0,t)>K}]=0 .

 
  * Ψ^1_∞(x) > 0 for all x ≠λ𝔼(Y).

  * M (θ)< ∞ for all θ∈ℝ.


With these we have:

Let A be a stationary marked point process.
Under Assumptions <ref> and given 0<α<1, the sequence  Ã^N_α,1 satisfies a sample path large deviations principle in the space 𝒟_0 with the scaled uniform norm ||x||_s. This has rate N^α, and good rate function I_α,1(x), where

    I_α,1(x)= ∫_0^∞Ω^*(ẋ+λ𝔼(Y)) dt     if x(0)=0 and x∈𝒜𝒞 ,
    ∞   otherwise ,

and

    Ω^*(y)=sup_θ∈ℝ[ θ y - λ(M(θ)-1) ] .



Next we look at the moderate deviations case (iii), as first examined in discrete time by Wischik <cit.>; here 1/2 <α=β<1. We need to modify the assumptions under which we work slightly in our continuous time setting.
 



  * There exist θ_0 >0 and K<∞ such that

lim_t→0t^-1
𝔼[e^θ_0 A(0,t)1_{X(0,t)>K}]=0 .


  * Ψ_∞,d→ t^2 > 0  and Ψ_∞,t→ d^2 > 0.

  * There exists θ^*>0  such that, for all θ∈ [0,θ^*) and t∈ℝ^+, Λ_t (θ) < ∞.

  * For α∈ (1/2,1), {Ã̅̃_α,α(·) }_t∈ [0,T) is exponentially tight in the space of continuous  functions on the interval [0,t) with the scaled uniform norm.


We then obtain the following:

Let A be a stationary marked point process with continuous variance function v.
Under Assumptions <ref>, for 1/2<α<1 the sequence  Ã^N_α,α satisfies a sample path large deviations principle in the space 𝒟_0 with the scaled uniform norm ||x||_s. This has rate N^2 α -1, and good rate function I_α,α(x), where

    I_α,α(x)= 1/2 x^2_R_v   if x∈ R_v ,
    ∞   otherwise ,

where R_v is the reproducing kernel Hilbert space associated  with variance function v. 


Next we present moderate deviations (iv) for these fast timescales where α<β<1 and α+β>1. These scalings lie between the small buffer result and the first moderate deviations result.  In this setting we need the following assumptions:




  * There exist θ_0 >0 and K<∞ such that

lim_t→0t^-1
𝔼[e^θ_0 A(0,t)1_{X(0,t)>K}]=0 .
 
 
  * Ψ_∞,d→ t^2(x) > 0  and Ψ_∞,t→ d^2(x) > 0.

  * M (θ)< ∞ for all θ in a neighbourhood of 0.


The result we obtain here is the following:

Let A be a stationary marked point process.
Under Assumptions <ref> and given α < β<1  such that α +β>1, the sequence  Ã^N_α,β satisfies a sample path large deviations principle in the space 𝒟_0 with the scaled uniform norm ||x||_s. This has rate N^α+β -1, and good rate function I_α, β(x), where

    I_α, β(x)= ∫_0^∞ẋ^2/2 λ𝔼(Y^2) dt     if x(0)=0 and x∈𝒜𝒞 ,
    ∞   otherwise .








The final case we consider is (v), large deviations for fast timescales where 0<α<1 and β > 1. In this setting we need the following assumptions:




  * There exist θ_0 >0 and K<∞ such that

lim_t→0t^-1
𝔼[e^θ_0 A(0,t)1_{X(0,t)>K}]=0 .
 

  * Ψ^1_∞(x) > 0 for all x ≠λ𝔼(Y).

  * M (θ)< ∞ for all θ∈ℝ.


Our result here is the following:

Let A be a stationary marked point process.
Under Assumption <ref> and given 0<α <1  and β>1, the sequence  Ã^N_α,β satisfies a sample path large deviations principle in the space 𝒟_0 with the scaled uniform norm ||x||_s. This has rate N^αlog N, and good rate function I_α, β(x), where

    I_α, β(x) = ∫_0^β - 1ẋ dt     if x(0)=0 and x∈𝒜𝒞 ,
    ∞   otherwise .






To understand the assumptions we impose on the traffic process we examine how they lead to properties of the processes required in the proof. The systems studied in this paper are already stable by virtue of the nature of the scaling. We also require the amount of traffic produced in a small time period to be small, and this is guaranteed by the first of each of the assumptions. In addition we require the probability of a source sending at a rate greater than the service rate of the queue for long periods to be small. Often when considering large deviations limits this is dealt with by examining whether lim_t →∞Ψ(C,t) is non-zero. However we consider a scaled form Ψ_∞^1(x) which is a weaker condition.

It is worth noting that for three of these results (Theorems <ref>, <ref> and <ref>) the associated rate functions are the same as for previously studied stochastic processes. Firstly, in Theorem <ref> the rate function in (<ref>) is that of a stationary Gaussian process with variance function v, and of the generalized Schilder's theorem <cit.>. In comparison to this, we have that in the small buffer large deviations case (Theorem <ref>) the rate function (<ref>) is that of a marked Poisson process with mean rate λ and independent marks with distribution Y.  Finally, for the moderate deviations small buffer case (Theorem <ref>) we find that the rate function (<ref>) is that of a Brownian motion with variance parameter λ𝔼(Y^2). In these latter two cases the rate function only depends on the mean rate of the point process and either the moment generating function of the marks or the second moment of the marks. This allows easy calculations in many circumstances, and these calculations are robust to changes in the underlying point process. This is useful from a modelling perspective, as we only need to estimate the mean rate of the point process and properties of the marks to provide useful estimates of tail probabilities. 

The move from large deviations to moderate deviations requires a strengthening of the assumption on the long run behaviour, but we are able to weaken the assumption on the moment generating functions, as in the limit the second moment dominates. In addition, for the large buffer cases the conditions on the log moment generating functions of the processes (our Assumptions <ref>.3 and <ref>.3) imply the associated condition on the moment generating function of the marks M(θ). Finally, in the large buffer cases we have an extra condition which guarantees the processes are exponentially tight on finite timescales but we are able to drop this in the small buffer setting. This is because the limiting process is relatively insensitive to the original process in the small buffer case.
 






















































































































§ PROOFS OF THEOREMS
 

In this section we prove the results we stated in Section <ref>. These proofs are split into three sections: some preliminary lemmas, a framework for the large deviations results, and a final step combining these. Section <ref> contains a series of lemmas on the marked point processes in the various scalings we consider. This includes two limits for the log moment generating function, and also bounds for behaviour at long timescales in the various scenarios. In Section <ref> we set up a general framework of lemmas for proving our large deviations results. We begin by proving a large deviations principle over a fixed time interval and then extending this to infinite time. Finally, for each of the scalings of interest we show how to apply the previous lemmas to obtain the desired results.



 §.§ Preliminary lemmas




  §.§.§ Lemmas for large deviation results ()


 
Suppose A is  a marked point process  which obeys Assumptions <ref> or <ref>. Given 0=j_0<j_1<j_2< ⋯ <j_n-1<j_n = 1 and t ∈ℝ, define the vector 𝐀^𝐣,t by 𝐀^𝐣,t_i = A(t j_i-1, t j_i). Then there exists θ^* >0 such that, uniformly for all θ∈{θ∈ℝ^n: θ_i ∈ [0,θ^*]},

    lim_t → 01/tlog𝔼[ e^⟨θ, 𝐀^𝐣,t⟩]=λ∑_i=1^n (j_i-j_i-1)(M(θ_i) -1) .



Firstly, we have ℙ(X(0,t)=1) =λ t +o(t) and ℙ(X(0,t) ≥ 2) =o(t). These follow directly from
<cit.>. Now let
p_t(k) = ℙ(X(0,t)=k) for k∈ℤ^+. As we choose
θ^*>0 and K<∞ to be such that the  Assumptions <ref> or <ref> are
satisfied with θ^* ≤θ_0 and M(θ^*)<∞,

    𝔼[ e^⟨θ, 𝐀^𝐣,t⟩] -1 =p_t(0) +p_t(1)𝔼[ e^⟨θ, 𝐀^𝐣,t⟩ | X(0,t)=1 ]
     
    + ∑_k=2^K p_t(k) 𝔼[ e^⟨θ, 𝐀^𝐣,t⟩ | X(0,t)=k ] +𝔼[ e^⟨θ, 𝐀^𝐣,t⟩ 1_X(0,t) >K] -1 .

Since 𝔼[ e^⟨θ, 𝐀^𝐣,t⟩ |
X(0,t)=k ] has no dependence on t (as the Y_i are independent of the point process), we use the previous probability
approximations to get

    𝔼[ e^⟨θ, 𝐀^𝐣,t⟩] -1 
    =-λ t + λ t𝔼[ e^⟨θ, 𝐀^𝐣,t⟩ | X(0,t)=1 ] +𝔼[ e^⟨θ, 𝐀^𝐣,t⟩ 1_X(0,t) >K]+o(t) .

Consider 𝔼[ e^⟨θ, 𝐀^𝐣,t⟩ | X(0,t)=1 ]. By conditioning the sub-interval containing the point, we get

𝔼[ e^⟨θ, 𝐀^𝐣,t⟩ | X(0,t)=1 ] = ∑_i=1^n ℙ(X( t j_i-1,t j_i)=1 | X(0,t)=1) M(θ_i) .

Also,

0≤𝔼[ e^⟨θ, 𝐀^𝐣,t⟩ 1_X(0,t) >K ] ≤𝔼[ e^θ^* A(0,t) 1_X(0,t) >K ] ≤𝔼[ e^θ_0 A(0,t) 1_X(0,t) >K ]  .

Using these and our assumptions, we get

    lim_t→ 01/t( 𝔼[ e^⟨θ, 𝐀^𝐣,t⟩] -1 )
    = -λ+λlim_t→ 0(∑_i=1^n ℙ(X( t j_i-1,t j_i)=1 | X(0,t)=1) M(θ_i) ) ,

and

lim_t→0 1/t ( 𝔼[ e^⟨θ, 𝐀^𝐣,t⟩ ] -1 )^2 =0 .

We now need to find lim_t→ 0ℙ(X( t j_i-1,t j_i)=1 | X(0,t)=1).  Firstly,

ℙ(X( 0,t/2)=1 | X(0,t)=1) = ℙ(X( 0,t/2)=1  , X( t/2,t)=0)/ℙ(X( 0,t)=1) .

Since  and ℙ(X(t/2,t)=1)= λ t /2+o(t/2), we have

lim_t→0 ℙ(X( 0,t/2)=1 | X(0,t)=1) = lim_t→0 λt/2 +o(t)/λt +o(t) =1/2 .

A similar argument gives

    lim_t→ 0ℙ(X( t j_i-1,t j_i)=1 | X(0,t)=1) = j_i -j_i-1 .

Thus,

    lim_t→ 01/t( 𝔼[ e^⟨θ, 𝐀^𝐣,t⟩] -1 ) = -λ+λ(∑_i=1^n (j_i- j_i-1) M(θ_i) ) .

Finally, we have that x-x^2/2 ≤log(1+x) ≤ x for all x≥ 0, which gives the desired result.



Suppose A is a marked point process such that Ψ^1_∞(x)>0 for x>λ𝔼(Y). Given B>0 and d ∈ℝ^+, there exists t_0 <∞ such that

lim sup_N →∞ 1/N^α logℙ(sup_t>(N^αt_0)/N [A^⊕N(0,t)-Nxt]≥N^αB )
≤-d.




We use the scaling F^⊕ N(0,t) = A^⊕ N(0,t/N). This gives 
 
    sup_t>(N^α t_0)/N [A^⊕ N(0,t)-Nxt] =sup_t>N^α  t_0 [F^⊕ N(0,t)-xt] .

Hence

    ℙ(sup_t>(N^α t_0)/N[A^⊕ N(0,t)-Nxt]≥ N^αB)
       =ℙ(sup_t>N^α t_0 [F^⊕ N(0,t)-xt]≥ N^αB) 
    
       ≤ℙ(sup_t>N^α t_0 [F^⊕ N(0,t)-xt]≥ B) .


We choose δ∈ (0,B) such that s= δ x^-1 <1.
For l ∈ℤ^+, let t_l = s l and I_l =
[t_l, t_l+1). Since F^⊕ N(0,t) is non-decreasing
in t and B-δ >0 we get

    ℙ(sup_t∈ [t_l , t_l+1 )  [F^⊕ N(0,t)-xt]≥ B)    ≤ℙ(F^⊕ N(0,t_l+1)> B +xt_l) 
       =ℙ(F^⊕ N(0,t_l+1)> B -δ +xt_l+1) 
       ≤ℙ(F^⊕ N(0,t_l+1)> xt_l+1) .

We can now use the Chernoff bound to see that, for θ≥ 0,

ℙ(F^⊕N(0,t_l)> xt_l) ≤e^-xt_l θ 𝔼[e^θF^⊕N(0,t_l)]
=e^-t_l ( x θ-N t_l^-1 log𝔼[e^θA(0,t_l/N)])  .

Taking the infimum over θ≥ 0 we get ℙ(F^⊕ N(0,t_l)> xt_l) ≤ e^-t_l Ψ(x,N^-1 t_l).

Given t_0∈ (s^-1, ∞), let L_N ∈ℤ^+ be such
that N^α t_0 ∈ [t_L_N ,t_L_N +1).  Let N_0 = inf{ N: Nτ >L_N }.  For τ∈ (0, ∞) and all N>N_0, we have that (<ref>) is at most

    ℙ(sup_t∈ [sL_N , s ⌊ Nτ⌋ )  [F^⊕ N(0,t)-xt]≥ B) +ℙ(sup_t∈ [s ⌊ Nτ⌋ , ∞ )  [F^⊕ N(0,t)-xt]≥ B ) 
    ≤∑_l=L_N+1^⌊ Nτ⌋ e^-t_l Ψ(x,N^-1 t_l) + ∑_l=⌊ Nτ⌋^∞ e^-t_l Ψ(x,N^-1 t_l) .

We find a bound for Ψ(x,t) using
trivial extensions to Lemma 5 and Corollary 6 of 
<cit.>, which state that there exist τ́>1, and β_1,β_2>0 such
that Ψ(x,t) ≥β_1 for t∈[0,τ́] and tΨ(x,t)
/log t ≥β_2 for t∈[τ́,∞). This gives

    ∑_l=L_N+1^⌊ Nτ́⌋ e^-t_l Ψ(x,N^-1 t_l)≤∑_l=L_N+1^⌊ Nτ́⌋ e^-t_l β_1≤∑_l=L_N+1^∞ e^-t_l β_1 .

Using the definition of L_N and t_l, and bounding the sum
by an integral, we have

    (<ref>)  ≤∫_N^α t_0/s ^∞
    e^-β_1 sl dl= β_1 s e^-N^α t_0 β_1 .

For the second sum on the right-hand side of (<ref>), we have that

    ∑_l=⌊ Nτ́⌋^∞ e^-t_l Ψ(x,N^-1 t_l)≤∑_l=⌊ Nτ́⌋^∞ e^-Nβ_2 log(N^-1 t_l)
    =∑_l=⌊ Nτ́⌋^∞(ls/N)^-β_2 N
    ≤ s^-β_2 N N∫_τ́^∞ x^-β_2 N dx 
    = - s^-β_2 N N τ́ ^1-β_2 N/1-β_2 N .

Using (<ref>) and the bounds provided by
(<ref>) and (<ref>), we get

    lim sup_N →∞1/N^αlogℙ(sup_t>(N^α t_0)/N [A^⊕ N(0,t)-Nxt]≥
    N^α B)
    ≤max{-β_1 t_0,lim sup_N →∞1/N^αlog(  s^-β_2 N N τ́ ^1-β_2 N/β_2
    N-1) }.

The second term is -∞, as required.



Suppose A satisfies Assumptions <ref>, and that 0<α<1 and β>1. Then, for any B, d >0, there exists T = T(B, d) ∈ℝ_+ and N = N(B, d) ∈ℤ_+ such that, for any t >T, 

    lim sup_N →∞1/N^αlog Nlogℙ( sup_t>N^α - βT  A^⊕ N (0,t) - N λ t - N^β C t ≥ N^α B ) < -d .





We begin by noting that 

    A^⊕ N (0,t) - N λ t - N^β C t ≥ N^α B  
    A^⊕ N (0,t)/N^α - N^1-αλ t - N^β - α C t ≥ B .

Let t' = N^β - αt, so that

    A^⊕ N (0,t)/N^α - N^1-αλ t - N^β - α C t =A^⊕ N (0,N^α-βt')/N^α - N^1-βλ t' -  Ct' .


The problem then reduces to bounding

    ℙ(sup_t ≥ t_0[Ã^N_α, β(0,t) - Ct ] ≥ B ) .

We choose δ∈ (0, B) such that s = δ C^-1 < 1. For l ∈ℤ^+, let t_l = sl and I_l = [t_l, t_l+1). For fixed N, Ã^N_α, β(0,t) is non-decreasing in t, and B - δ >0, therefore we get

    ℙ(sup_t ∈ [t_l, t_l+1)[Ã^N_α, β(0,t) - Ct ] ≥ B )    ≤ℙ(Ã^N_α, β(0, t_l+1) > B + C t_l) 
       ≤ℙ(Ã^N_α, β(0, t_l+1) > B - δ + Ct_l+1) 
       ≤ℙ(Ã^N_α, β(0, t_l+1) > C t_l+1) .

We use the Chernoff bound to see that, for θ>0, 

    ℙ(Ã^N_α, β(0,t_l) > C t_l )  ≤ e^-t_l C θ𝔼[e^θÃ^N_α, β(0, t_l)]
     
     = e^-t_l ( C θ + λθ N^1-β - N/t_l Λ_α-β(θ/N^α) )
    															 ≤ e^-t_l ( C θ + λθ N^1-β - N/t_l Λ_α-β(θ) ) .

Taking the infimum over θ≥ 0 we get

    ℙ(Ã^N_α, β(0,t_l) > Ct_l) ≤ e^-t_lΨ(C+λ N^1- β, N^α-βt_l) ,

where the last equality is obtained by recognizing that N/t_l = N^α-β+1/N^α-βt_l, and that for m ∈ [1, ∞), 
    sup_θ∈ [0, ∞)[θ x - m/tlog𝔼e^θ t] ≤Ψ(x,t) .


Given t_0 ∈ (s^-1, ∞), let L_N ∈ℤ^+ be such that N^α-βt_0 ∈ [t_L_N, t_L_N+1).  Let N_0 = inf{N:N_τ > L_N}. For τ∈ (0, ∞) and all N>N_0  we have that (<ref>) is at most 


    ( sup_t ∈ [sL_N, s Nτ)[ Ã^N_α, β(0,t) - Ct ] ≥ B )
     + (sup_t ∈ [sNτ, ∞)[ Ã^N_α, β(0,t) - Ct ] ≥ B ) 
    ≤∑_L_N + 1^N τ e^-t_l Ψ(C+λ N^1-β, N^α-βt_l) + ∑_l = N τ^∞ e^-t_l Ψ(C+ λ N^1-β, N^α - βt_l) .


We proceed similarly to the proof of Lemma <ref>.  We bound Ψ(x,t) using extensions to Lemma 5 and Corollary 6 in <cit.>: there exist τ̇ > 1, and β_1, β_2 >0 such that Ψ(x,t) ≥β_1 for t ∈ [0, τ̇) and t Ψ(x,t)/log t ≥β_2 for t ∈ [τ̇, ∞). This gives

    ∑_L_N + 1^N τ̇ e^-t_l Ψ(C+ λ N^1-β, N^α-βt_l)≤∑_l=L_N +1^N τ̇ e^-t_l β_1≤∑_l=L_N +1^∞ e^-t_l β_1 ,

which tends to zero as L_N→∞.
For the second sum on the right-hand side of (<ref>) we have that

    ∑_l = N τ̇^∞ e^-t_l Ψ(C+ λ N^1-β, N^α - βt_l)≤∑_l = N τ̇^∞ e^-β_2 N^β-αlog(N^α - β t_l)
     
     = ∑_l = N τ̇^∞(N^α-β sl )^-β_2 N^β - α 								 
     = s^-β_2 N^β-α N ∫_τ̇^∞ x^-β_2 N^β-α  dx 
     = - s^-β_2 N^β - α + 1τ̇^1 - β_2 N^β - α/1 - β_2 N^β - α ,

which also tends to zero as N→∞. We can therefore choose L_N = L_N(,B) and N =  N(,B) such that for T = sL and n>N, each term on the right hand side of (<ref>) is less than ϵ/2. Hence, (<ref>) is at most ϵ for sufficiently large N, and the required conclusion follows from taking logarithms.




  §.§.§ Lemmas for moderate deviations (


We now need to provide equivalent lemmas for the moderate deviations scalings. We begin with a limit on the log moment generating function.

 
Suppose A is  a marked point process which satisfies Assumptions <ref> or <ref>. Given 0=j_0<j_1<j_2< ⋯ <j_n-1<j_n = 1 and t ∈ℝ,  define the vector 𝐀^𝐣,t by 𝐀^𝐣,t_i = A(t j_i-1, t j_i). Also, let f: ℝ^+ →ℝ^+ be a continuous function for which f(t)→0 as t→0.  Then there exists θ^* >0 such that, uniformly for all θ∈ [0,θ^*]^n,

    lim_t → 01/t (f(t))^2log𝔼[ exp{⟨θ f(t), 𝐀^𝐣,t⟩ - λ𝔼(Y)f(t) ∑_i=1^nθ_i (j_i-j_i-1)t }]
     =1/2λ𝔼(Y^2) ∑_i=1^n (j_i-j_i-1)θ_i^2 .




As in the proof for Lemma <ref> we know ℙ(X(0,t)=1) =λ t +o(t) and
ℙ(X(0,t) ≥ 2) =o(t). In addition, by assumption, using (<ref>) and making use of the expansion of e^x about x=0 we get

    𝔼[ e^⟨θ f(t), 𝐀^𝐣,t⟩] -1= 𝔼[ ∑_i=1^∞  (⟨θ f(t), 𝐀^𝐣,t⟩)^i/i!] .

Now we  consider 𝔼(⟨θ f(t), 𝐀^𝐣,t⟩^i) for i>0.  This is bounded above by 𝔼((θ^* f(t) A(0,t))^i), as maxθ_i ≤θ^*. Using (<ref>) and exchanging the sum and expectation, we have, for ν< θ^*,

    ∑_i=1^∞ν^i 𝔼( A(0,t)^i)/i! = λ t( M(ν)-1) +o(t) .

Since A(0,t)≥ 0 we have 𝔼( A(0,t)^i) = c_i t +o(t) for any i>0, giving  𝔼((θ^* f(t) A(0,t))^i) = c_i (f(t)θ^*)^i t + o(t f(t)^i), which will be used to bound 𝔼(⟨θ f(t), 𝐀^𝐣,t⟩^i). 
We know 
    𝔼(⟨θ f(t), 𝐀^𝐣,t⟩) = λ𝔼(Y)f(t) ∑_i=1^nθ_i (j_i-j_i-1)t .

We now let p_k(t) = ℙ(X(0,t)=k), so get 

    𝔼(⟨θ f(t), 𝐀^𝐣,t⟩^2)
    
     = ∑_i=1^k p_i(t) 𝔼(⟨θ f(t), 𝐀^𝐣,t⟩^2| X(0,t)=i) + 𝔼(⟨θ f(t), 𝐀^𝐣,t⟩^2 1_X(0,t)>K) .

Since we have ℙ(X(0,t)>2) = o(t) we get p_i(t) 𝔼(⟨θ f(t), 𝐀^𝐣,t⟩^2| X(0,t)=i) = o(t f(t)^2) for i>1. 
Putting this all together we get

    𝔼[ e^⟨θ f(t), 𝐀^𝐣,t⟩] -1 =
    λ𝔼(Y)f(t) ∑_i=1^nθ_i (j_i-j_i-1)t 
    
    + 1/2(λ t +o(t)) 𝔼(⟨θ f(t), 𝐀^𝐣,t⟩^2| X(0,t)=1)+ o(t f(t)^2) 
    
    + 1/2𝔼(⟨θ f(t), 𝐀^𝐣,t⟩^2 1_X(0,t)>K) 
    +∑_i=3^∞𝔼(⟨θ f(t), 𝐀^𝐣,t⟩^i) .

Using (<ref>) we get
 
    𝔼(⟨θ f(t), 𝐀^𝐣,t⟩^2| X(0,t)=1) = 1/2𝔼(Y^2) ∑_i=1^n (j_i-j_i-1)(θ_i f(t))^2 .

As x-x^2/2 -c ≤log(1+x) -c ≤ x-c, we use the above to get 

    lim_t → 01/tf(t)^2( 𝔼[ e^⟨θ f(t), 𝐀^𝐣,t⟩] -1 -λ𝔼(Y)f(t) ∑_i=1^nθ_i (j_i-j_i-1)t  )
    =  1/2λ𝔼(Y^2) ∑_i=1^n (j_i-j_i-1)θ_i^2+ lim_t → 01/tf(t)^2𝔼(⟨θ f(t), 𝐀^𝐣,t⟩^2 1_X(0,t)>K) .

The second term goes to zero since, by assumption, we have 

    lim_t→ 0t^-1𝔼[e^θ_0 A(0,t)1_{X(0,t)>K}]=0 .

This gives the desired result.



Let A be a marked point process such that Ψ_∞,d→ t^2(x) > 0  and  Ψ_∞,t→ d^2(x) > 0. Given B>0, d >0 and α, β >0 such that β <1 and α +β>1, there exists t_0 <∞ such that

    lim sup_N →∞1/N^α+β -1logℙ(sup_t>N^α-β t_0 [A^⊕ N(0,t)-N λ𝔼(Y) t- N^βx t]≥ N^α B )
    ≤ -d.




We use the scaling F^⊕ N(0,t) =A^⊕ N(0,t/N),  giving

    ℙ(sup_t>N^α-β t_0  [A^⊕ N(0,t)-N λ𝔼(Y) t- N^βx t]≥ N^α B ) 
     =ℙ(sup_t>N^α-β +1 t_0 [F^⊕ N(0,t)-λ𝔼(Y) t- N^β-1x t]≥ N^αB ) .

We choose N_1 such that N_1^α B > λ𝔼(Y)+N_1^β -1x, then choose δ∈ (0,N_1^α B) such that δ > λ𝔼(Y)+N_1^β -1x. Note that for N>N_1 we have N^α B > N_1^α B and λ𝔼(Y)+N^β -1x <λ𝔼(Y)+N_1^β -1x.

For l ∈ℤ^+, let I_l =
[l, l+1). Since F^⊕ N(0,t) is non-decreasing
in t, and because we have N^α B-δ >0 for N>N_1, we get

    ℙ   (sup_t∈ [l , l+1 )  [F^⊕ N(0,t)-λ𝔼(Y) t- N^β-1x t]≥ N^α B)
       ≤ℙ(F^⊕ N(0,l+1)> N^α B +(λ𝔼(Y) + N^β-1x) l)
       ≤ℙ(F^⊕ N(0,l+1)> N^α B - δ +(λ𝔼(Y) + N^β-1x) (l+1))
       ≤ℙ(F^⊕ N(0,l+1)> (λ𝔼(Y) + N^β-1x) (l+1)) .

For the remainder of this proof we will assume N> N_1.
We can now use the Chernoff bound which, for θ≥ 0, gives

    ℙ(F^⊕ N(0,l)> (λ𝔼(Y) + N^β-1x) l) ≤ e^-(λ𝔼(Y) + N^β-1x) l θ𝔼[e^θ F^⊕ N(0,l)]
    
    =e^-l ( (λ𝔼(Y) + N^β-1x) θ -N l^-1log𝔼[e^θ A(0,l/N)])  .

Taking the infimum over θ≥ 0 we get

    ℙ(F^⊕ N(0,l)>(λ𝔼(Y) + N^β-1x)l) ≤ e^-l Ψ((λ𝔼(Y) + N^β-1x),N^-1 l) .


Given t_0∈ (1, ∞), let L_N ∈ℤ^+ be such
that N^α- β +1 t_0 ∈ [L_N ,L_N +1).  Let N_2 = inf{ N: N
τ >L_N }.  For some
τ∈ (0, ∞) and all N>max (N_1,N_2)  we find that (<ref>) is at most

    ℙ(sup_t∈ [L_N ,  ⌊ Nτ⌋ )  [F^⊕ N(0,t)-(λ𝔼(Y) + N^β-1x)t]≥ N^α B)
     
     +ℙ(sup_t∈ [ ⌊ Nτ⌋ , ∞ )  [F^⊕ N(0,t)-(λ𝔼(Y) + N^β-1x)t]≥ N^α B) 
    ≤∑_l=L_N+1^⌊ Nτ⌋ e^-l Ψ((λ𝔼(Y) + N^β-1x),N^-1 l) 
    + ∑_l=⌊ Nτ⌋^∞ e^-l Ψ((λ𝔼(Y) + N^β-1x),N^-1 l) .

We consider Ψ(λ𝔼(Y) + x,t) to find bounds on the above. We start with the assumption that
Ψ_∞,d→ t^2(x) > 0  and  Ψ_∞,t→ d^2(x) > 0.  There exist D and T_1 such that, for all x<D and t>T_1, 

    tΨ(λ𝔼(Y) + x,t)/x^2 log(t) > δ_1 .

 Also, we have
 
    Ψ(λ𝔼(Y) + N^β-1 x,t)  = sup_θ{θ(λ𝔼(Y) + N^β-1x) -t^-1log𝔼( e^A(0,t)θ)}
    
      = N^2(β -1)sup_θ{θ N^1-βx -(tN^2(β -1))^-1log𝔼( e^A(0,t)θ -θλ𝔼(Y))}.

 Letting θ' = N^1-βθ,
 
    (<ref>)    = N^2(β -1)sup_θ{θ x -(tN^2(β -1))^-1log𝔼( e^A(0,t) N^β -1θ -N^β -1θλ𝔼(Y))} .

From Lemma <ref> there is θ_0 such that, for all θ∈ (0, θ_0) and however we take the joint limit, we get 

    lim_N→∞, t→ 01/tN^2(β -1)log𝔼( e^A(0,t) N^β -1θ -N^β -1θλ𝔼(Y)) = λ𝔼(Y^2) θ^2 .
 
So we have that, for x ≠ 0, 

    lim_N→∞, t→ 0 (N^2(1-β)) Ψ(λ𝔼(Y) + N^β-1 x,t) >0 .

This means that there is T_2<1 and N_3 such that, for all N>N_3, (N^2(1-β)) Ψ(λ𝔼(Y) + N^β-1 x,t) > δ_2.  Using Hölder's inequality and stationarity of A means that for t∈(T_2, τ) we have Ψ(λ𝔼(Y) + N^β-1 d,t)  > N^2(β-1) (δ_2 T_2/t).

Let τ́ =max(e^d+1/δ_1, T_1) and N>max(N_1, N_2, N_3). Using the above results we bound the terms of interest: 

    ∑_l=L_N+1^⌊ Nτ́⌋ e^-l Ψ((λ𝔼(Y) + N^β-1x),N^-1 l)≤∑_l=L_N+1^⌊ Nτ́⌋ e^-l N^2(β-1) (δ_2 T_2/τ́)
    ≤∑_l=L_N+1^∞ e^-l N^2(β-1) (δ_2 T_2/τ́) .

Using the definition of L_N  and bounding the sum
by an integral we have

    ∑_l=L_N+1^∞ e^-l N^2(β-1) (δ_2 T_2/τ́)≤∫_N^α -β +1 t_0 ^∞
    e^-l N^2(β-1) (δ_2 T_2/τ́) dl
    = N^2(β-1) (δ_2 T_2/τ́)e^- t_0 N^α+β-1 (δ_2 T_2/τ́) .

For the second sum in (<ref>) we have

    ∑_l=⌊ Nτ́⌋^∞ e^-l Ψ((λ𝔼(Y) + N^β-1x),N^-1 l)≤∑_l=⌊ Nτ́⌋^∞ e^-N^2β-1δ_1 log(N^-1 l)
     
    =∑_l=⌊ Nτ́⌋^∞(l/N)^-δ_1 N^2β -1≤  N∫_τ́^∞ x^-δ_1 N^2β -1 dx 
    = - N τ́ ^1-δ_1 N^2β -1/1-δ_1 N^2β -1 .


So, using (<ref>) and then the bounds provided by
(<ref>) and (<ref>) we get that

    lim sup_N →∞1/N^α+β -1logℙ(sup_t>(N^α-β +1 t_0) [A^⊕ N(0,t)
    -(Nλ𝔼(Y) +N^β)t]≥
    N^α B)

is bounded above by

    max{-t_0δ_2 T_2/τ́ ,lim sup_N →∞1/N^α+β-1log(  N τ́ ^δ_1 N^2β -1/1-δ_1 N^2β -1-1) } .

The final term is -∞ for α <β, and bounded above by -d, showing the desired result.




  §.§.§ Properties of the processes


We now look more closely at the processes of interest. We begin by checking that these processes belong to the space 𝒟_0.


Let A be a marked point process such that Ψ^1_∞(x) >0 for all x ≠λ𝔼(Y). Then Ã^N_α,β is in 𝒟_0 for all N.


Firstly, by definition we have that Ã^N_α,β is a cadlag function. Secondly, we need to show that Ã^N_α,β /(t+1) → 0 as t →∞. This is equivalent to showing
ℙ(Ã^N_α,β∉𝒟_0) =0. 
We do this by considering, for x>λ𝔼(Y), ℙ(A(0,t)>xt)≤ e^-t Ψ(x,t).  Since Ψ^1_∞ (x) >0 given  x ≠λ𝔼(Y),  there exist T and δ>0 such that, for all t>T, t Ψ(x,t)/log t ≥δ > 0.
This shows that  ℙ(A(0,t)>xt) → 0 as t tends to ∞. A similar argument holds for x<λ𝔼(Y).


For our Definition <ref> of the linear interpolation of a point process to be of use, we need to show the processes are exponentially tight (<cit.>) on the scalings of interest. 

Let α>0 and A be a marked point process. If β≥ 1 and
M(θ) < ∞ for all θ∈ℝ, or if β <1 and 
there exists θ>0 such that M(θ) <∞, then the associated measures of Ã^N_α ,β (the scaled original process) and Ã̅̃^N_α,β (the scaled linear interpolation) are exponentially equivalent in 𝒟_0 with the scaled uniform norm ||· ||_s.


The sets we need to consider are 
    {‖Ã̅̃^N_α,β -Ã^N_α,β‖_s >γ} ,
  
which are obviously measurable. To estimate the measure of these we consider whether this event happens in intervals of the form [i,i+1):

    ℙ( ‖Ã̅̃^N_α,β -Ã^N_α,β‖_s >γ ) ≤∑_i =0^∞ℙ( Ã̅̃^N_α,β -Ã^N_α,β_[i,i+1) > γ (i+1) )  ,

where x_[i,i+1) = sup_t∈[i,i+1)|x(t)|. 
Now, we have that 
    |Ã̅̃^N_α,β(0,t) -Ã^N_α,β(0,t)| < |ζ_N(N^α -βt)|/ N^α ,
 which implies that if , either there is a mark in
(-N^α -β (i+1),-N^α -β i] or ζ_N(N^α -β (i+1)) is larger than N^αγ (i+1).  So, for any γ>0,

    ℙ(| Ã̅̃^N_α,β -Ã^N_α,β|_[i,i+1) > γ (i+1) )  
    ≤∑_k=1^Nℙ( ⋃_j=X^(k)(0,N^α-β(i))^X^(k)(0,N^α-β(i+1)){ Y_j^(i) > γ (i+1) N^α})
    + ℙ(Y_1^(1) > γ (i+1) N^α) ,

where the first term represents marks in (-N^α -β(i+1),-N^α -βi], and the second the first mark before this.  Now, the RHS of (<ref>) is

    Nℙ( ⋃_j=X^(k)(0,N^α-β(i))^X^(k)(0,N^α-β(i+1)){ Y_j^(1) > γ (i+1) N^α})+ℙ(Y_1^(1) > γ (i+1) N^α)
       =N 𝔼( 1_{∪_j=X^(k)(0,N^α-β(i))^X^(k)(0,N^α-β(i+1)){ Y_j^(1) > γ (i+1) N^α}}) +ℙ(Y_1^(1) > γ (i+1) N^α)
       ≤ N 𝔼(X^(1)(N^α-βi,N^α-β(i+1)) 1_{ Y_1^(1) > γ (i+1) N^α})+ℙ(Y_1^(1) > γ (i+1) N^α) 
       =(N^α -β  +1λ +1) ℙ(Y_1^(1) > γ (i+1) N^α)  .

We now apply the Chernoff bound to this to get that, for θ≥ 0,

    ℙ(| Ã̅̃^N_α,β -Ã^N_α,β|_[i,i+1) > γ (i+1) ) ≤ (N^α - β +1λ +1) M(θ) e^-N^αθγ (i+1) .

Summing the terms and bounding the sum by an integral we get 

    ℙ( ‖Ã̅̃^N_α,β -Ã^N_α,β‖_s >γ )    ≤∑_i =0^∞ (N^α - β +1λ +1) M(θ) e^-N^αθγ (i+1)
       ≤ e^-N^αθγ((N^α - β +1λ +1) M(θ))(1+ (N^αθγ)^-1) .

Now, for β <1 we just select θ > 0 such that M(θ) < ∞ and we let N →∞.  If β≥ 1 we know that, since M(θ)<∞ for all θ, we can take  N →∞ and then θ→∞, which gives 

lim sup_N→∞ 1/N^α+ β-1 logℙ(‖Ã̅̃^N_α,β|_[0,T] -Ã^N_α,β|_[0,T] ‖_s>γ)  = -∞ ,

for any γ>0. This shows the two measures are exponentially tight.




 §.§ Framework for the proof of the large deviations principles

In this section we provide a series of lemmas which form the basis of the proofs of our main results; the proofs will be completed in subsequent sections.
We start by proving a finite dimensional large deviations principle, which is then extended into a sample path result in finite time.
These lemmas are based on ideas from the proof of Mogulskii's theorem <cit.> and the extension of this to infinite time by O'Connell and Ganesh <cit.>.  

 
Let 𝒥 be the collection of all ordered finite subsets
of (0,1]. For any  T ∈ℝ^+, j ={
0<s_1<s_2<⋯<s_| j|≤ 1}∈𝒥 and
f: [0,T]×[0,T] →ℝ, let p^T_j (f) denote
the vector (f(0,T s_1),f(0,T s_2), …, f(0,T s_| j
|)) ∈ℝ^| j|. Let A be a marked point process. Given α, β∈ℝ^+ such that either 0<α<β and α+β>1, or 0<α<1 and β>1, we define

    Ω_α, β(θ, j,T) = lim_N→∞1/f(N)log𝔼( e^∑_i=1^| j | N^β -1θ_i A( N^α -β j_i-1 T, N^α -β j_i T )- θ_i T(j_i -j_i-1)N^α -1λ𝔼(Y)),

where

    f(N) = {[                  N^α + β - 2 if  0< α < β and α + β > 1 ,;                     N^αlog N    if  0< α < 1  and β > 1 . ].

We assume the following: 


  * Ω_α, β(θ, j,T) is steep <cit.> in the variable θ for every j and T,

  * 𝒟^∘_Ω_α, β  contains θ =0,

  * Ω_α, β(θ, j,T) is differentiable in  θ on  𝒟^∘_Ω_α, β.
 
The sequence of vectors p^T_j ( Ã^N_α, β) satisfies
 a large deviations principle in  ℝ^| j | with rate
f(N) and good rate function

    I_j,T(𝐳)=
    sup_θ∈ℝ^| j |{∑_i=1^| j | (z_i-z_i-1) θ_i - Ω_α, β(θ, j,T) } .




We have that, for a fixed N, p_j^T( Ã_N ) is the following random vector:

𝐙_N^j =( Ã^N_α, β (0,T s_1), …,  Ã^N_α,β(0,T s_|j |)) .

Let

𝐖_N^j =( Ã^N_α,β(0,T s_1), Ã^N_α,β(T s_1,T s_2),…,Ã^N_α,β(T s_|j |-1 , T s_|j |)) .

We prove a large deviations principle for 𝐖  and  apply the contraction mapping principle to find a large deviations principle for 𝐙. We use the
Gärtner-Ellis theorem <cit.>, employing the log moment generating function of W_N^j,
Ψ_N (θ) = log𝔼[ e^⟨θ, W_N^j ⟩]. We
have

    Ψ_N (θ) 
    
     =Nlog𝔼( exp(∑_i=1^| j |1/N^αθ_i A( N^α -β j_i-1 T, N^α -β j_i T )
      - θ_i T(j_i -j_i-1)N^-βλ𝔼(Y)) ) .

By definition Ψ(θ) = lim_N→∞1/f(N)Ψ_N( θ f(N) )=Ω_α, β(θ, j,T).

We need Ψ to satisfy the conditions of the Gärtner-Ellis
theorem; this is ensured by the assumptions on Ω_α, β.
So the rate function for the large deviations principle for 𝐖_N^j is

    Ψ^* (w) = sup_θ∈ℝ^| j |{⟨ w,θ⟩ - Ψ (θ) }
    = sup_θ∈ℝ^| j |{∑_i=1^| j | w_i θ_i - Ω_α, β(θ, j,T) } .

Since the map 𝐖_N^j ↦𝐙_N^j  is continuous and one
to one, we can find the large deviations principle for 𝐙_N^j by the
contraction principle <cit.>. 


As Ã^N_α,β and Ã̅̃^N_α,β are exponentially equivalent in the cases considered above, the results of Lemma <ref>  hold for Ã̅̃^N_α,β.




 
 Let A be a marked point process.  Given α, β∈ℝ, we define 
 
    Ω_α, β(θ, j,T)
     = lim_N→∞1/f(N)log𝔼( e^∑_i=1^| j | N^β -1θ_i A( N^α -β j_i-1 T, N^α -β j_i T ) - θ_i T(j_i -j_i-1)N^α -1λ𝔼(Y)),

where

    f(N) = {[                  N^α + β - 2 if  0< α < β and α + β > 1 ,;                     N^αlog N    if  0< α < 1  and β > 1 . ].
 
We assume the following: 


  * Ω_α, β(θ, j,T) is steep in the variable θ for every j and T,

  * 𝒟^∘_Ω_α, β  contains θ =0,

  * Ω_α, β(θ, j,T) is differentiable in  θ on  𝒟^∘_Ω_α, β.
 
The sequence of vectors p^T_j ( Ã̅̃^N_α, β) satisfies
a large deviations principle in  ℝ^| j | with rate
N^α +β -1 and good rate function

    I_j,T(𝐳)=
    sup_θ∈ℝ^| j |{∑_i=1^| j | (z_i-z_i-1) θ_i - Ω_α, β(θ, j,T) } .




Next we have a lemma which allows us to turn the previous finite dimensional result into a sample path result.

 
For a given T∈ℝ^+, let 𝒞^T be the space of continuous functions, x:[0,T]↦ℝ for which x(0)=0. Let W^N be a random variable on 𝒞^T such that, given any j∈𝒥 and 0<T'≤ T, we have that p^T'_j (W^N) obeys a large deviations principle with rate f(N) and good rate function I_j,T'. Then W^N obeys a large deviations principle in 𝒞^T with the topology of pointwise convergence with rate f(N) and good rate function 

    I_T(x) = sup_j∈𝒥, 0<T'≤ T I_j,T'(x).

Furthermore, if W^N is exponentially tight with respect to the scaled uniform norm then W^N obeys a large deviations principle in 𝒞^T with the scaled uniform norm and the same rate and rate function.



We firstly define a partial ordering on 𝒥 by i =( s_1, … , s_| i |) ≤ j =( t_1, … , t_| j | ), i,j
∈𝒥, if and only if for each l there exists q(l) such that s_l =t_q(l). For a fixed T we can define the projection
p^T_ij : ℝ^| j |→ℝ^| i |
for i≤ j ∈𝒥 in the natural way. We now define
𝒞̃^T to be the projective limit of {𝒴_j = ℝ^| j |}_j∈𝒥 with respect to the projections p_ij^T for a
fixed T. The spaces 𝒞̃^T and 𝒞^T
may be identified with each other. This can be seen by considering
f ∈𝒞^T, which then corresponds to (p^T_j (f))_j∈𝒥 which belongs to 𝒞̃^T since
p^T_i(f)=p_ij^T(p_j^T(f)) for i≤ j ∈𝒥. Also, for
𝐱∈𝒞̃^T we can see that this corresponds to f ∈𝒞^T, where f(t)= x_k if t ∈ [s_k, s_k+1) for t>0 and f(0)=0. In
addition, the projective topology on 𝒞̃^T is
equivalent to the pointwise convergence topology on
𝒞^T. Therefore we can use the Dawson-Gärtner
theorem <cit.> to find a large deviations principle in 𝒞^T with the
topology of pointwise convergence. The good rate function for this
large deviations principle is

    I_T(f) =sup_j∈𝒥, 0<T'≤ T I_j,T'(f) .


For the second part of the lemma we make use of the inverse contraction principle <cit.>, since W^N is exponentially tight in 𝒞^T with the scaled uniform norm. We use the identity function mapping on 𝒞^T from the topology of pointwise convergence to the uniform topology to give the required result. 


We thus have a large deviations principle for sample paths of [0,T). Now we will extend it to those on [0,∞). Again, we use the Dawson-Gärtner theorem to carry out the extension. 


Let W^N be a stationary random process on the space 𝒞 of continuous functions of ℝ^+. Assume that, given T>0, the random variables W^N |_T, the restriction of W^N to the interval [0,T), obey a large deviations principle on 𝒞^T with the uniform topology, rate f(N) and good rate function I_T. Assume also that, given x > 𝔼(W^N(0,1))=λ and B,d>0, there exists t_d,x>0 such that

    lim sup_N→∞1/f(N)logℙ( sup_t> t_d,x W^N(0,t) -xt > B) <-d,

and a similar inequality holds for x< 𝔼(W^N(0,1)). Then W^N obeys a large deviations principle on 𝒞 with topology induced by the scaled uniform norm || · ||_s with rate f(N) and good rate function I(x) = sup_T I_T(x).



We consider the projections q_st: 𝒞^t →𝒞^s, for s ≤ t ∈ℝ^+, which are the restrictions of the
functions to the interval [0,s). This means that the projective limit space
is 𝒞 with the projective limit topology. So we can
apply the Dawson-Gärtner theorem 
<cit.>, which shows that
Y^N satisfies a large deviations principle in 𝒞 with rate f(N) and good rate
function I(x) =sup_T∈ℝ^+ I_T(x).

We now strengthen the topology from the projection topology to that induced by the scaled uniform norm. We start by proving that W_N is exponentially tight in 𝒞 with the scaled uniform norm. We will define two groups of sets, L_α and K_α. Firstly, we
know that W^N|_T satisfies a large deviations
principle in the space 𝒞^T with the uniform norm, which is a Polish space. Thus, it is exponentially tight in this space,
which means that there is a family of compact sets L_α^T
which have the property that

lim sup_N→∞ 1/f(N) logℙ(W^N|_T ∉L_α^T)<-α .

Let L̅_α^T be the smallest compact set with this
property. Then L̅_α^T_2⊂L̅_α^T_1
if T_2>T_1. To see this, assume for contradiction that there is a
function x which is in L̅_α^T_2 but not in
L̅_α^T_1. Since it is not in L̅_α^T_1
we must have that I^T_2 (x) >α. Therefore it must be the
limit of a sequence of functions in L̅_α^T_2,
otherwise it could be removed and the set would still be compact
and have the required property. This cannot be the case, as the
functions truncated to the interval [0,T_1) must also tend to
the limit x truncated to [0,T_1). This shows that x ∈L̅_α^T_1, the required contradiction. Hence there is no x in L̅_α^T_2 but not in
L̅_α^T_1.

Define
L_α=  ⋂_T∈ℕL̅_α^T, which is
compact in the projection topology and has the following property:

lim sup_N→∞ 1/f(N) logℙ(W^N ∉L_α)<-α .

By assumption there exist t_i such that

lim sup_N →∞ 1/f(N) logℙ(sup_t> t_i [W^N(0,t)-(λ+ϵ_i)t]≥1)
≤-α ,

where ϵ_i= 1/ i. Let t_i^+ be the minimum t
such that this holds. Also, let t_i^- be such that

lim sup_N →∞ 1/f(N) logℙ(sup_t> t_i^- [W^N(0,t)-(λ-ϵ_i)t]≤1)
≤-α .

We define K_α to be the set of continuous functions which have the following property:

    x(t)< (λ + 1/i)t +1   for t_i^+<t<t_i+1^+ ,

and

    x(t)> (λ - 1/i)t -1   for t_i^-<t<t_i+1^- .

Obviously we then have that

    lim sup_N→∞1/f(N)logℙ(W^N∉ K_α)<-α .

We now define M_α = K_α∩ L_α, which has the property that

    lim_α→∞lim sup_N →∞1/f(N)logℙ(W^N ∉ M_α) =-∞ .

Also, we know that L_α is compact with the projection topology. Hence, given a sequence x^(n) in  M_α, we can find a subsequence x^(j) which converges to some x in the projective topology. In order to complete the proof we need to show x ∈ M_α and then that x^(j)→ x in the scaled uniform topology.

Since x^(j)→ x uniformly on compact intervals,

    lim_j →∞sup_t∈ [0,T] |
    x^(j)(t)/1+t - x(t)/1+t | =0   for
    every T>0 .
 
Also, since x^(j)∈ K_α, we have 
    |
    x^(j)(t)/t-λ | ≤1/t
    +ϵ_t  for all t> max( t_1^-, t_1^+) ,
 
where
ϵ_t = 1/min{ i-1; t_i^+>t, t_i^->t}, which tends to 0 as
t→∞.  This shows that 
    |
    x(t)/t-λ | ≤1/t
    +ϵ_t  for all t> max( t_1^-, t_1^+) ,
 
thus giving
x∈ M_α. Finally, given ϵ>0 we choose i such
that 1/i <ϵ and then select T>0 such that T>t_i^+ and
T>t_i^-. Then, for j sufficiently large, we have

    x^(j)-x ≤sup_t≤ T | x^(j)(t)/1+t - x(t)/1+t | +sup_t>T | x^(j)(t)/1+t - x(t)/1+t |≤ 2ϵ.



We now make use of these lemmas to prove the large deviations principles for the different scalings of interest. To do this we need to carry out the following three tasks in each setting:


  * Calculate Ω_α, β(θ, j,T) and check the necessary conditions.

  * Calculate the corresponding rate function.

  * Check that Ã̅̃_α, β restricted to the interval [0,T) is exponentially tight in the space 𝒞^T with the uniform norm.




 §.§ The case 


In the case α=β=1 our proof is relatively brief, as nearly all of the above three points are dealt with directly by the assumptions placed on the process. 

Theorem <ref>.

We start by examining 

    Ω_1,1(θ, j,T)= log𝔼( e^∑_i=1^| j |θ_i A(j_i-1 T,  j_i T )) - λ𝔼(Y)∑_i=1^| j |θ_i T(j_i -j_i-1).
 
This is the finite distributional log moment generating function for the process A, minus a linear function of θ. As long as the log moment generating function satisfies the necessary conditions, Ω_1,1 will also obey them. As it is a log moment generating function we automatically have that Ω_1,1(0, j,T) =0 and that it is differentiable on the finite domain. Then Assumption <ref>.3 gives the necessary steepness condition. 

By Lemma <ref>, the finite distributions of Ã^N_1,1 and Ã̅̃^N_1,1 obey large deviations principles.
Now, we know by Assumption <ref>.4 that Ã̅̃^N_1,1 is  exponentially tight in 𝒞^T. By Lemma <ref> we get the sample path result for finite time which we can extend using Lemma <ref> in conjunction with Lemma <ref> to a sample path large deviations principle on 𝒞 with the topology induced by the scaled uniform norm ||·||_s. The issue is then showing that we can restrict ourselves to 𝒞_0; this is done with Lemma <ref> and Lemma 4.1.5 in <cit.>. Then, as 𝒞_0 is a subspace of 𝒟_0, we can expand the space to this. Finally, the exponential tightness of Ã̅̃_1,1^N and Ã^N_1,1 gives the result.





 §.§ The case 


Consider the case 0< α <β =1. In this section we outline the proof of our Theorem <ref>, which has previously been established by Cruise <cit.>, within our framework.  We use the following two lemmas.  The first gives the log-moment generating function and is proved analogously to <ref> below. The second establishes exponential tightness of the scaled process, and may be proved similarly to Lemma <ref>.


For any 0<α <1, we have that 
    Ω_α, 1(θ, j,T)=T ∑_i=1^| j |  (λ (M(θ_i)-1) -λ𝔼(Y)θ_i )(j_i-j_i-1) .
 Furthermore, we have

    I_α,1(x)= ∫_0^∞Ω^*(ẋ+λ𝔼(Y)) dt     if x(0)=0 and x∈𝒜𝒞,̇
    ∞   otherwise ,

where Ω^*(y)=sup_θ∈ℝ[ θ y - λ(M(θ)-1) ].




For A which obeys Assumptions <ref> and with 0<α<1, we have that Ã̅̃^N_α,1 is exponentially tight on 𝒞^T with the uniform topology.


Using these lemmas we have the large deviations result.
Theorem <ref>.

Firstly, Lemma <ref> show us that we have large deviations principles for the finite distributions of Ã_α,1^N and Ã̅̃_α,1^N. Now we extend this to a sample path result on the interval [0,T) for any 0<T<∞ for Ã̅̃_α,1^N in the space 𝒞^T with the uniform topology using Lemmas <ref> and <ref>. This is then extended to the whole of 𝒞 with the scaled uniform norm by Lemma <ref> and <ref>. Now, for all N we have that Ã̅̃^N_α,1 is in 𝒞_0; by Lemma <ref> we can use Lemma 4.1.5 in <cit.> to restrict to this space. We  expand 𝒞_0 to 𝒟_0 and use exponential tightness to get that Ã^N_α,1 obeys a sample path large deviations principle and Lemma <ref> demonstrates that we have the required rate function.





 §.§ The case 


The proof for the case 1/2< α= β <1 follows along similar lines to the previous results, with the major difference being that we obtain a neat form for Ω_α,α(t) in this setting, which enables a simplified rate function. As mentioned previously, this has the form of a rate function for a Gaussian process which leads us to using the Generalized Schilder's Theorem as proved in <cit.>.


Theorem <ref>.

We start by examining 

    Ω_α, α  (θ, j, T)= lim_N →∞1/N^2(α-1)log𝔼( e^∑_i=1^|j| N^α -1 (θ_i A(j_i-1T, j_i T) -θ_i T(j_i- j_i-1) λ𝔼(Y) ) .

We use the Taylor expansion to consider this for large N, since N^α -1 is small. This is possible because, for small θ, Λ_t(θ) < ∞  for all t. This gives

    (<ref>)= lim_N →∞1/N^2(α-1)( ∑_i=1^|j|∑_l=1^|j|Γ(j_i,j_l) θ_i θ_l N^2(α -1) + o(N^3(α-1)) ) ,

where Γ is the covariance function. This is a quadratic function in θ, which means that it has the necessary conditions for us to apply Lemma <ref> to show that the finite distributions of Ã^N_α,α and Ã̅̃^N_α,α obey large deviations principles. 

Now, we know by Assumption <ref>.4 that Ã̅̃^N_α,α is  exponentially tight in 𝒞^T. So we now apply Lemma <ref> to get the sample path result for finite time, which we can extend using Lemma <ref> in conjunction with Lemma <ref> to a sample path large deviations principle on 𝒞 with the topology induced by the scaled uniform norm ||·||_s. The issue is then showing that we can restrict ourselves to 𝒞_0; this is done with Lemma <ref> and Lemma 4.1.5 in <cit.>. Then, as 𝒞_0 is a subspace of 𝒟_0, we can expand the space to this. Finally, the exponential tightness of Ã̅̃_α,α^N and Ã^N_α,α leaves us only the simplified rate function to establish.

We consider the behaviour of the variance function, as this governs the behaviour of Γ. We note that, since Ψ^2_∞, d→ t>0 and Ψ^2_∞, d→ t>0, we have that v(t)/t^2 → 0 as t→∞ and the variance function is continuous.
We consider a Gaussian process Z with variance function v and mean 0, which we use to find the rate function.  Z/N^1/2 obeys a large deviations principle by the Generalized Schilder's Theorem <cit.>. We consider a projection of the process indexed by T>0 and j a finite partition of [0,1] given by the vector

    (Z(Tj_1)-Z(0), Z(Tj_2)-Z(Tj_1),…, Z(T)-Z(T j_|j|-1))/N^1/2 .

This then obeys a large deviations principle with rate 
    I_j,T(x) = sup_θ (⟨ x,θ⟩ -Ω_α, α) (θ,j,T).
 Since this is the same rate function as for the finite distributions of Ã_α,α^N, the uniqueness of rate functions give the rate function for Ã_α,α^N.




 §.§ The cases ,  and , 


Finally, we consider the cases (iv) and (v): α<β <1, α + β >1 and 0<α< 1, β >1.  Here the proofs again follow a similar style to previous cases. We will state the lemmas as for the previous cases, but omit later proofs for brevity; Lemmas 
<ref> and <ref>
may be obtained by analogous arguments to those used for Lemmas <ref>  and <ref>.  We will, however, give some calculations to motivate the form of the rate function in the lightly loaded case, β>1. 


For a given 0<α<β <1 and α + β >1,  we have that 
    Ω_α, β(θ, j,T)=T ∑_i=1^| j |λ𝔼(Y^2)θ_i^2/2 .
 Furthermore,

    I_α,β(x)= ∫_0^∞ẋ^2/2 λ𝔼(Y^2) dt     if x(0)=0 and x∈𝒜𝒞 ,
    ∞   otherwise .



We have

    Ω_α, β(θ, j,T) = lim_N→∞  N^2-α-βlog𝔼( e^∑_i=1^| j |  N^β-1θ_i A( N^α -β j_i-1 T, N^α -β j_i T )
     - θ_i T(j_i -j_i-1)N^α -1λ𝔼(Y)).

Now we let Δ = T N^α -β, which means Δ→ 0 as N →∞.  Let f(Δ) be such that f(Δ)=N^β-1, so that

    Ω_α, β(θ, j,T)
     = lim_Δ→ 0T/Δ f(Δ)^2log𝔼( e^∑_i=1^| j |  f(Δ)θ_i A( Δ j_i-1, Δ j_i ) - f(Δ)θ_i Δ (j_i -j_i-1)λ𝔼(Y)) .
 
We can then apply Lemma <ref>, yielding

    Ω_α, β(θ, j,T)= 1/2Tλ𝔼(Y^2) ∑_i=1^| j |θ_i^2(j_i-j_i-1) .

So, by Lemma <ref> the rate function is

    I_j,T(𝐳)   =
    sup_θ∈ℝ^| j |{∑_i=1^| j | (z_i-z_i-1) θ_i - 1/2Tλ𝔼(Y^2) ∑_i=1^| j |θ_i^2(j_i-j_i-1) }
       = sup_θ∈ℝ^| j |{∑_i=1^| j | T(j_i-j_i-1)( ((z_i-z_i-1)/T(j_i-j_i-1)) θ_i - 1/2λ𝔼(Y^2) θ_i^2 ) }
       =   ∑_i=1^| j | T(j_i-j_i-1) Ω^*((z_i-z_i-1)/T(j_i-j_i-1)) ,

where 
    Ω^*(y)=sup_θ∈ℝ(θ y - 1/2λ𝔼(Y^2) θ^2)= y^2/2λ𝔼(Y^2).

So, we have that

    I_T(x)    = sup_j∈𝒥, 0<T'≤ T I_j,T'(x) 
        =sup_j∈𝒥, 0<T'≤ T∑_i=1^| j | T'(j_i-j_i-1) Ω^*((x(T'j_i)-x(T'j_i-1))/T'(j_i-j_i-1)) .

We want to show this is equal to 

    I^T(x)= ∫_0^Tẋ^2/2 λ𝔼(Y^2) dt     if x(0)=0 and x∈𝒜𝒞 ,
    ∞   otherwise .

Firstly,  Ω^* is non-negative for all y, as the value at θ =0 is 0, so the supremum  is at j_|j|=1 and T'=T.
In addition,  we have that I_T ≤ I^T by 
Jensen's inequality and convexity of Ω^* 
<cit.>. We need to prove the reverse inequality. Let us
start with  f, which is absolutely continuous, and let g(t)
= d f /dt, which is in L_1([0,T]). Then, for k>1 we can
define τ_k = T/k and

    g^(k)(t)    = τ_k^-1∫_⌊ t/τ_k⌋τ_k^(⌊ t/τ_k⌋ +1)τ_k g(s) ds   for t∈[0,T) , 
    
    g^(k) (T)    = τ_k^-1∫_T-τ_k^T g(s) ds .

Using these definitions we have

    I_T (f) ≥lim inf_k→∞∑_l=1^kτ_k Ω^* [ f(l τ_k )-f ((l-1) τ_k) /τ_k] 
    =lim inf_k→∞∫_0^TΩ^*(g^(k)(t))dt .

In addition to this, by Lebesgue's theorem, lim_k→∞ g^(k)(t) =g(t) almost everywhere in [0,T]. So, by Fatou's lemma and the lower semicontinuity of Ω^*, we have

    lim inf_k→∞∫_0^TΩ^*(g^(k)(t))dt ≥∫_0^Tlim inf_k→∞Ω^*(g^(k)(t))dt 
    ≥∫_0^TΩ^*(g(t))dt = I^T(f),

which together show that  I_T(f) ≥ I^T(f), for
f absolutely continuous.

Now let f be a function from [0,T] to ℝ which is
not absolutely continuous. This implies that there exists
δ>0 and s_1^n<t_1^n ≤⋯≤ s_k_n^n<t_k_n^n such that ∑_l=1^k_n (t_l^n -s_l^n) → 0 but
∑_l=1^k_n |f(t_l^n) -f(s_l^n)| ≥δ.  In addition,
 Ω^* is non-negative, so we get

    I_T(f)    = sup_0<t_1<t_2<⋯ <t_k =T
    θ_1 , … , θ_k ∈ℝ∑_l=1^kθ_l(f(t_l)-f(t_l-1)) -(t_l-t_l-1)1/2λ𝔼(Y^2) θ_l^2 
       ≥sup_0≤ s_1<t_1≤ s_2 <t_2<⋯≤ s_k <t_k =T
    θ_1 , … , θ_k ∈ℝ∑_l=1^kθ_l(f(t_l)-f(s_l)) -(t_l-s_l) 1/2λ𝔼(Y^2) θ_l^2 .

Now let t_l=t_l^n, s_l=s_l^n and θ_l have the same sign as f(t_l^n)-f(s_l^n), with |θ_i|=ρ. This then gives

    I_T(f) ≥lim sup_n →∞{ρ∑_l=1^k_n |f(t_l^n) -f(s_l^n)|  - sup_|θ|=ρ{1/2λ𝔼(Y^2) θ^2 }∑_l=1^k_n (t_l^n -s_l^n) }≥ρδ .

The choice of ρ
is arbitrary, implying that I_T(f)=∞.

Finally, we have that I_T(x) is increasing in T as Ω^* is non-negative. So, 
    I_α,β(x) =lim_T→∞ I_T(x) =  ∫_0^∞ẋ^2/2 λ𝔼(Y^2) dt     if x(0)=0 and x∈𝒜𝒞 ,
    ∞   otherwise .



Let 0<α<β <1 and α + β >1.  For A which satisfies Assumptions <ref> we have that Ã̅̃^N_α,β is exponentially tight on 𝒞^T with the uniform topology.


For a fixed γ>0, let

K_γ(T) = { f ∈𝒜𝒞 : f(0)=0, ∫_0^∞ ẋ^2/2 λ𝔼(Y^2) ≤γ+ 1 } .

We want to show that K̅_γ (T) is compact and that 
    lim_γ→∞lim sup_N →∞1/N^αlogℙ(Ã̅̃^N_α,β
    |_T∉ K_γ (T)) = - ∞.
 We show the second of these
first. We have that ℙ( Ã̅̃^N_α,β
|_T∉ K_γ(T)) =ℙ( Ã̅̃^N_α,β
|_T∉ K_γ(T) ∩𝒜𝒞),
as, by definition, paths of Ã̅̃^N_α,β
|_T are almost surely absolutely continuous. We also have that Ã̅̃^N_α,β
|_T obeys a large deviations principle in this
space, but with the topology of pointwise convergence by the first half of Lemma <ref>. So,

lim sup_N →∞ 1/N^α logℙ(Ã̅̃^N_α,β
|_T ∉K_γ(T)) ≤-inf_f ∈K^c_γ(T)∩𝒜𝒞 I_T(f) ,

where K^c_γ(T) is the complement of K_γ(T) and K̅ is the closure of the set K.
By the definition of K_γ (T) we have I_T(f) > γ for f ∈K^c_γ(T)∩𝒜𝒞. If there exists g ∈K^c_γ(T)∩𝒜𝒞 such that I_T(g) ≤γ, then there exists a sequence f_n ∈ K^c_γ(T)∩𝒜𝒞  whose limit is g; but this cannot be the case because there would have to exist N such that I_T(f_N)< α +
β, and this is untrue by the definition of
K_γ(T). Hence,

lim sup_N →∞ 1/N^α logℙ(Ã̅̃^N_α,β
|_T ∉K_γ(T)) ≤-γ .

This goes to -∞ as γ→∞.

To prove compactness we make use of the Arzelá-Ascoli theorem,
which says that if K_γ (T) is a closed and
bounded set of equicontinuous functions then it is compact.  To check the functions are equicontinuous, if f∈ K_γ (T) then the
continuous function f is differentiable almost everywhere on
[0,T], and, for all 0≤ s ≤ t≤ T,

Ω^* ( (f(t)-f(s)/t-s)^2 ·1/2λ𝔼(Y)) ≤1/t-s ∫_s^t Ω^*(ḟ(t')/2λ𝔼(Y)) dt' ≤γ+1/t-s .














By definition, the functions in the set K_γ (T) have bounded derivatives. As such, in any interval of time δ, the variation of the function is bounded. In addition, we have a bound on the set by setting s=0 and
δ =T.


For a given 0<α<1 and β >1,  we have that 
    Ω_α, β(θ, j,T)= {[                     0 if θ_j < β - 1  ∀ j ,;                     1   if θ_j = β-1  ∀ j ,;                     ∞           otherwise , ].

and

    I_α, β(x) = ∫_0^β - 1ẋ dt     if x(0)=0 and x∈𝒜𝒞 ,
    ∞   otherwise .


To see why Ω_α, β(θ, j,T) takes this form in Lemma <ref>, note that, for 0<α<1 and β > 1,

    Ω_α, β(θ, j, T) = lim_N →∞N^1-α/log Nlog𝔼[e^⟨θlog N, A^j, N^α-βT⟩] .

By replacing t with N^α - βt in Lemma <ref>, we can see that the process A(N^α - βt_j-1, N^α-βt_j) can be approximated by a Poisson process with mean λ N^α - β(t_j - t_j-1).  Therefore,

    Ω_α, β(θ, j, T)     = lim_N →∞N^1-α/log N∏_j 𝔼[e^θ_j log N A(N^α -βt_j-1, N^α - βt_j)]
        =  lim_N →∞N^1-α/log N∑_j N^α - βλ (t_j - t_j-1) (e^θ_j log N -1) 
        =  lim_N →∞N^1 - β/log N∑_jλ (t_j - t_j-1)(N^θ_j - 1) 
        = lim_N →∞∑_j λ (t_j - t_j-1) ( N^1 - β + θ_j/log N) 
        = 
                        0     if θ_j < β-1   ∀ j , 
    
                        1     if θ_j = β-1   ∀ j , 
    ∞    otherwise .


Let 0<α<1 and β >1.  For A which obeys Assumptions <ref> we have that Ã̅̃^N_α,β is exponentially tight on 𝒞^T with the uniform topology.


Using these lemmas we can now prove the large deviations results.
Theorems <ref> and <ref>.
We begin with Theorem <ref>. Lemma <ref>  show us that we have large deviations principles for the finite distributions of Ã^N_α,β and Ã̅̃^N_α,β. Now we extend this to a sample path result on the interval [0,T) for any 0<T<∞ for Ã̅̃^N_α,β in the space 𝒞^T with the uniform topology using Lemmas <ref> and <ref>. This is then extended to the whole of 𝒞 with the scaled uniform norm by Lemma <ref> and <ref>. For all N,  Ã̅̃^N_α,β belongs to 𝒞_0 (Lemma <ref>). We use Lemma 4.1.5 in <cit.> to restrict to this space. We expand 𝒞_0 to 𝒟_0 and use exponential tightness to get that Ã^N_α,β obeys a sample path large deviations principle, and Lemma <ref> demonstrates that we have the required rate function.  This  established Theorem <ref>.  

The proof of Theorem <ref> is similar, with Lemmas <ref>, <ref> and <ref> used in place of Lemmas <ref>,<ref> and <ref> respectively.


unsrt


