Diffusion approximations via Stein's method and time changes
Mikolaj J. Kasprzak




Diffusion approximations via Stein's method


,


Mikołaj J. Kasprzak

University of Oxford

Mikołaj Kasprzak

Department of Statistics

University of Oxford

24-29 St Giles'

Oxford, OX1 3LB

United Kingdom

e1

E-mail: 



We extend the ideas of <cit.> and use Stein's method to obtain a bound on the distance between a scaled time-changed random walk and a time-changed Brownian Motion. We then apply this result to bound the distance between a time-changed compensated scaled Poisson process and a time-changed Brownian Motion. 

This allows us to bound the distance between a process whose dynamics resemble those of the Moran model with mutation and a process whose dynamics resemble those of the Wright-Fisher diffusion with mutation upon noting that the former may be expressed as a difference of two time-changed Poisson processes and the diffusive part of the latter may be expressed as a time-changed Brownian Motion.

The method is applicable to a much wider class of examples satisfying the Stroock-Varadhan theory of diffusion approximation (<cit.>). 


[class=MSC]
[Primary ]60B10
60F17
[; secondary ]60J70, 60J65, 60E05, 60E15



Stein's method
functional convergence
time-changed Brownian Motion
Moran model
Wright-Fisher diffusion





§ INTRODUCTION

In his seminal paper <cit.>, Charles Stein introduced a method for proving normal approximations and obtained a bound on the speed of convergence to the standard normal distribution. He observed that a random variable Z has the standard normal law if and only if EZf(Z)=Ef'(Z) for all smooth functions f. Therefore, if, for a random variable W with mean 0 and variance 1, Ef'(W)-EWf(W) is close to zero for a large class of functions f, then the law of W should be approximately Gaussian. He then proposed that, instead of evaluating |Eh(W)-Eh(Z)| directly for a given function h, one can first find an f=f_h solving the following Stein equation:

    f'(w)-wf(w)=h(w)-Eh(Z)

and then find a bound on |Ef'(W)-EWf(W)|. This approach often turns out to be much easier, due to some bounds on the solutions f_h, which can be derived in terms of the derivatives of h. Since then, Stein's method has been significantly developed and extended to approximations by distributions other than normal.

The aim of Stein's method is to find a bound on the quantity |E_ν_nh-E_μh|, where μ is the target (known) distribution, ν_n is the approximating law and h is chosen from a suitable class of real-valued test functions ℋ. The idea is to find an operator 𝒜 acting on a class of real-valued functions such that 
    (∀ f∈Domain(𝒜)  E_ν𝒜f=0)  ⟺  ν=μ,

where μ is our target distribution. In the next step, for a given function h∈ℋ, a solution f=f_h to the following Stein equation:

    𝒜f=h-E_μh

is sought and its properties studied. Finally, using various mathematical tools (among which the most popular are Taylor's expansions in the continuous case, Malliavin calculus, as described in <cit.>, and coupling methods), a bound is sought for the quantity |E_ν_n𝒜f_h|.

An accessible account of the method can be found, for example, in the surveys <cit.> and <cit.> as well as the books <cit.> and <cit.>, which treat the cases of Poisson and normal approximation, respectively, in detail. <cit.> is a database of information and publications connected to Stein's method.

Approximations by infinite-dimensional laws have not been covered in the Stein's method literature very widely, with the notable exceptions of <cit.>,  <cit.> and recently <cit.>. We will focus on the ideas taken from <cit.>, which provides bounds on the Brownian Motion approximation of a one-dimensional scaled random walk and some other one-dimensional processes including scaled sums of locally dependent random variables and examples from combinatorics. In the sequel, we show that the approach presented in <cit.> can be extended to time-changes of Brownian Motion, including diffusions in the natural scale.

The most important example we apply our theory to is the approximation of the Moran model with mutation by the Wright-Fisher diffusion with mutation. The former, first introduced in <cit.> as an alternative for the Wright-Fisher model (first formally described in <cit.>) is one of the simplest and most important models of the genetic drift, i.e. the change in the frequencies of alleles in a population. It assumes that the population is divided into two allelic types (A and a) and the frequency of each of the alleles is governed by a birth-death process and an independent mutation process. Specifically, in a population of size n, at exponential rate n 2, a pair of genes is sampled uniformly at random. Then one of them is selected at random to die and the other one gives birth to another gene of the same allelic type. In addition, every gene of type a changes its type independently at rate ν_2 and every gene of type A changes its type independently at rate ν_1. The model then looks at the proportion of a-genes in the population.

On the other hand, the Wright-Fisher model is a discrete Markov chain and does not allow for overlapping generations. Specifically, each step represents a generation. In generation k each of the n individuals chooses its parent independently, uniformly at random from the individuals present in generation k-1 and inherits its genetic type. This model also then looks at the proportion of a-genes in the population.

The Moran model turns out to be easier to study mathematically. It may be proved, for instance using the Stroock-Varadhan theory of diffusion approximation (see <cit.>), that it converges weakly to the Wright-Fisher diffusion, which is also a scaling limit of the Wright-Fisher model. We show how to put a bound on the speed of this convergence. The Wright-Fisher diffusion is often used in practice in genetics for inference concerning large populations. It is given by the equation:

    dM(t)=γ(M(t))dt+√(M(t)(1-M(t)))dB_t,

where γ:[0,1]→R encompasses mutation.  For a discussion of probabilistic models in genetics see <cit.>.

In Section <ref> we introduce the space of test functions we will find the bounds for. In Section <ref> we present our main reuslts. Theorem <ref> shows how the approach in <cit.> can be extended to the approximation of a scaled, time-changed random walk by a time-changed Brownian Motion. In Theorem <ref> we apply Theorem <ref> to look at the distance between a time-changed Poisson Process and a time-changed Browian Motion. Theorem <ref> shows how this can be extended to find the speed of convergence of a process whose dynamics resemble those of the Moran model with mutation to one whose dynamics resemble those of the Wright-Fisher diffusion with mutation. The bound obtained therein makes it possible to analyse the impact the mutation rates and the number of individuals have on the quality of the approximation and the interplay between those parameters. Section <ref> provides proofs of the reults presented in Section <ref> and comments on how the proof of Theorem <ref> may be adapted to find the speed of convergence in other examples satisfying the Stroock-Varadhan theory of diffusion approximation (see <cit.>).

In what follows, · will always denote the sup norm and D=D[0,1]=D([0,1],R) will be the Skorokhod space of càdlàg real-valued functions on [0,1].


§ SPACE M

Let us define:

    f_L:=sup_w∈ D[0,1]|f(w)|/1+w^3,

and let L be the Banach space of the continuous functions f:D[0,1]→R such that f_L<∞. We now let M⊂ L consist of the twice Fréchet differentiable functions f, such that:

    D^2f(w+h)-D^2f(w)≤ k_fh,

for some constant k_f, uniformly in w,h∈ D[0,1]. By D^kf we mean the k-th Fréchet derivative of f and the k-linear norm B on L is defined to be B=sup_{ h:h=1} |B[h,...,h]|. Note the following:

For every f∈ M, f_M<∞, where:

    f_M:=   sup_w∈ D[0,1]|f(w)|/1+w^3+sup_w∈ D[0,1]Df(w)/1+w^2+sup_w∈ D[0,1]D^2f(w)/1+w
       +sup_w,h∈ D[0,1]D^2f(w+h)-D^2f(w)/h.



Note that for f∈ M it is possible to find a constant K_f satisfying:

    A)     Df(w)≤Df(w)-Df(0)+Df(0)
    MVT≤   wsup_θ∈[0,1]D^2f(θ w)+Df(0)
    ≤   w[sup_θ∈[0,1](D^2f(θ w)-D^2f(0)+D^2f(0))]+Df(0)
    (<ref>)≤   w[k_fsup_θ∈[0,1]θ w+D^2f(0)]+Df(0)
    ≤   k_fw^2+D^2f(0)(1∨w^2)+Df(0)< K_f(1+w^2);
    B)     D^2f(w)≤D^2f(w)-D^2f(0)+D^2f(0)
    (<ref>)≤   k_fw+D^2f(0)<K_f(1+w);
    C)     |f(w+h)-f(w)-Df(w)[h]-1/2D^2f(w)[h,h]|≤ K_fh^3,

uniformly in w,h∈ D, where the last inequality follows by Taylor's theorem and (<ref>). Therefore:

    f_M=   sup_w∈ D|f(w)|/1+w^3+sup_w∈ DDf(w)/1+w^2+sup_w∈ DD^2f(w)/1+w
       +sup_w,h∈ DD^2f(w+h)-D^2f(w)/h<∞.


We now let M^0⊂ M be the class of functionals g∈ M such that:

    g_M^0:=g_M+sup_w∈ D|g(w)|+sup_w∈ DDg(w)+sup_w∈ DD^2g(w)<∞.

This is Proposition 3.1 of <cit.>:

Suppose that, for each n≥ 1, the random element Y_n of D[0,1] is piecewise constant with intervals of constancy of length at least r_n. Let (Z_n)_n≥ 1 be random elements of D[0,1] converging weakly in D[0,1], with respect to the Skorokhod topology, to a random element Z∈ C([0,1],R). If:

    |Eg(Y_n)-Eg(Z_n)|≤ Cτ_ng_M^0

for each g∈ M^0 and if τ_nlog^2(1/r_n)0, then Y_n⇒ Z in D[0,1] (weakly in the Skorokhod topology).

A similar result holds when Y_n is a continuous-time Markov chain:

Suppose that, for each n≥ 1, the random element Y_n of D[0,1] is a contiuous-time Markov chain with mean holding time 1/λ_n→ 0, identically distributed for each state. Let (Z_n)_n≥ 1 be random elements of D[0,1] converging weakly in D[0,1], with respect to the Skorokhod topology, to a random element Z∈ C([0,1],R). Suppose further that:

    |Eg(Y_n)-Eg(Z_n)|≤ Cτ_ng_M^0

for each g∈ M^0 and that τ_nlog^2((λ_n)^3)0. Then Y_n⇒ Z in D[0,1] (weakly in the Skorokhod topology).

We provide a proof of Proposition <ref> in the Appendix.


§ MAIN RESULTS

Theorem <ref> below is an extension of Theorem 1 in <cit.> to the case of a time-changed scaled random walk: 

Let X_1,X_2,... be i.i.d. with mean 0, variance 1 and finite third moment. Let s:[0,1]→[0,∞) be a strictly increasing, continuous function with s(0)=0. Define:

    Y_n(t)=n^-1/2∑_i=1^⌊ ns(t)⌋ X_i,   t∈[0,1]

and let (Z(t),t∈[0,1])=(B(s(t)),t∈[0,1]), where B is a standard Brownian Motion. Suppose that g∈ M.
Then:

    |Eg(Y_n)-Eg(Z)|≤   g_M30+54· 5^1/3s(1)/√(πlog 2)n^-1/2√(log (2s(1)n))
       +g_Ms(1)(1+(3/2)^3√(2/π)s(1)^3/2)E|X_1|^3n^-1/2
       +g_M2160/√(π)(log 2)^3/2n^-3/2(log (2s(1)n))^3/2.


In Theorem <ref> we do not claim that our bounds are sharp. Our bound in Theorem <ref> is of the same order as the one obtained in the original case in <cit.>. This result can also be extended in a straightforward way to instances in which the time change is random and independent of the step sizes of the random walk. We can obtain this by conditioning on the time change.

Theorem <ref> below treats a time-changed Poisson process and can also be extended to random time changes, independent of the Poisson process of interest, by conditioning.

Suppose that P is a Poisson process with rate 1 and S^(n):[0,1]→[0,∞) is a sequence of increasing continuous functions, such that S^(n)(0)=0. Let S:[0,1]→[0,∞) be also increasing and continuous. Let Z(t)=B(S(t)),t∈[0,1] where B is a standard Brownian Motion and 

    Ỹ_n(t)=P(nS^(n)(t))-nS^(n)(t)/√(n),     t∈[0,1].

Then, for all g∈ M:

    |Eg(Ỹ_n)-Eg(Z)|≤g_M{(2+27√(2)/2√(π)S(1))√(S-S^(n))+27√(2)/2√(π)S-S^(n)^3/2.
       + n^-1/2[30+54· 5^1/3S(1)/√(πlog 2)√(log (2s(1)n)).
       +(1+(3/2)^3√(2/π)S^(n)(1)^3/2)S^(n)(1)(1+2e^-1).+1+log(1+2e^-1)+2log n/loglog(n+2)]
       +n^-19√(S^(n)(1))/2(1+3nS^(n)(1))^1/2[4+16701+128(log n)^3/(loglog(n+3))^3]^1/3
       +.n^-3/2[2160/√(π)(log 2)^3/2(log (2S(1)n))^3/2+8+33402+256(log n)^3/(loglog(n+3))^3]}.



The bound in Theorem <ref> goes to 0 as long as the time changes S^n→ S uniformly.

Theorem <ref> below gives a bound on the speed of convergence of a process whose dynamics are similar to those of the Moran model with mutation to a process whose dynamics are similar to those of the Wright-Fisher diffusion with mutation. In the Moran model with mutation, in a population of size n, each individual carries a particular gene of one of the two forms: A and a. Each individual has exactly one parent and offspring inherit the genetic type of their parent. Now, at exponential rate n 2 a pair of genes is sampled uniformly at random from the population. One of the pair is selected at random to die and the other one splits in two. In addition, every individual of type A changes its type independently at rate ν_2 and every individual of type a changes its type independently at rate ν_1. 

Let X_n(t) be the proportion of type a genes in the population at time t∈[0,1] under the Moran model with mutation rates ν_1,ν_2, as described above. Let (X(t),t∈[0,1]) denote the Wright-Fisher diffusion given by:

    dX(t)=(ν_2-(ν_1+ν_2)X(t))dt+√(X(t)(1-X(t)))dB_t.


Suppose that:

    P[X_n(t+Δ t)-X_n(t)=1/n]=n^2R_1^(n)(t)Δ t
    P[X_n(t+Δ t)-X_n(t)=1/n]=n^2R_-1^(n)(t)Δ t
    .

and:

    M_n(t)=1/nP_1(n^2R_1^(n)(t))-1/nP_-1(n^2R_-1^(n)(t)),

for t∈[0,1], where P_1 and P_-1 are i.i.d. rate 1 Poisson processes, independent of X_n.

Suppose further that 

    M(t)=W(∫_0^tX(t)(1-X(t))ds)+∫_0^t(ν_2-(ν_1+ν_2)X(s))ds,

where W is a standard Brownian Motion, independent of X.
Then, for any g∈ M:

    |Eg(M_n)-Eg(M)|
    ≤   g_M{(18+ν_1^1/2+47ν_1^3/4+31ν_1^3/2+ν_2+3ν_2^2+9ν_2^3).
       ·(1.02· 10^6+425ν_2^1/2+623ν_2+39ν_2^3/2+7ν_2^5/2)
       + (12+3ν_2+3ν_2^2+9ν_2^3)(1.02· 10^6+425ν_1^1/2+623ν_1+39ν_1^3/2+7ν_1^5/2)
       .+7(1/2(1+2ν_2)(ν_1+ν_2)+31(ν_1+ν_2)^3)} n^-1/4
       +g_M2112[(18+ν_1^1/2+47ν_1^3/4+31ν_1^3/2+ν_2+3ν_2^2+9ν_2^3)(log(n^2/4+ν_2n))^3/2.
       +.(12+3ν_2+3ν_2^2+9ν_2^3)(log(n^2/4+ν_1n))^3/2]n^-3.



If ν_1≥ 1 and ν_2≥ 1 then we can write:

    |Eg(M_n)-Eg(M)|
    ≤   g_M[(18+79ν_1^3/2+13ν_2^3)(1.02· 10^6+1094ν_2^5/2).
       +(12+15ν_2^3)(1.02· 10^6+1094ν_2^5/2)
       +.7(31.5ν_1^3+32.5ν_2^3+ν_1ν_2(1+93ν_1+93ν_2))]n^-1/4
       +g_M2112(18+79ν_1^3/2+13ν_2^3)n^-3(log(n^2/4+ν_1n))^3/2
       +g_M2112(12+15ν_2^3)n^-3(log(n^2/4+ν_1n))^3/2.

The approximation gets worse as the mutation rates increase and the number of individuals decreases. Should we want to make the mutation rates depend on n and be of the same order, we will require them to be o(n^1/22) in order for the bound to converge to 0 as n→∞.


Note that the Moran model X_n jumps up by 1/n with intensity n^2R_1^(n)(t) and down by 1/n with intensity n^2R_-1^(n)(t), as defined in (<ref>). Using the ideas from <cit.> we can write X_n in the following form:

    X_n(t)=1/nP̃_1(n^2R_1^(n)(t))-1/nP̃_-1(n^2R_-1^(n)(t))

for some indepenent rate 1 Poisson processes P̃_1 and P̃_-1 which are, however, not independent of R_1^(n) and R_-1^(n). In Theorem <ref> we consider a similar process, given by (<ref>), whose definition uses Poisson processes P_1 and P_-1 independent of R_1^(n) and R_-1^(n).

Similarly, the diffusive part of the Wright-Fisher diffusion may be expressed as a time-changed Brownian Motion W̃(∫_0^tX(t)(1-X(t)ds). However, we consider a process given by (<ref>) with the assumption that W is independent of X.

We will  appeal to Theorem <ref> to obtain the bounds. The time changes we apply in this case are random and therefore we will first condition on them.


The Moran model M_n converges weakly to the Wright-Fisher diffusion M, which can be proved using, for instance, the Stroock-Varadhan theory of diffusion approximation <cit.>. However, our paper does not provide the tools necessary for treating this convergence with Stein's method and obtaining bounds on its rate.

A key idea used in the proof will be the Donnelly-Kurtz look-down construction coming from <cit.> and decribed in Chapter 2.10 of <cit.>. The n-particle look-down process is denoted by a vector (ψ_1(t),...,ψ_n(t)) with each index representing a "level" and each of ψ_i's representing the type of the individual at level i at time t. Individual at level k is equipped with an exponential clock with rate k-1, independent of other individuals, and at the times the clock rings it "looks down" at a level chosen uniformly at random from { 1,...,k-1} and adopts the type of the individual at that level. In addition, the type of each individual evolves according to the mutation process. A comparison of the generators of the Moran model and the look-down process shows that, as long as the two are started from the same initial exchangeable condition, they produce the same distribution of types in the population. In addition, it may be shown that the Wright-Fisher diffusion may be represented as the proportion of type a individuals in the population, in which the types are distributed according to the infinite-particle look-down process. The corresponding Moran model is then the proportion of type a individuals among the ones located on the first n levels. Due to exchangeability, we may then describe the Moran model X_n at a fixed time, as depending on the Wright-Fisher diffusion X in the following way: nX_n(t)∼Binomial(n,X(s)).

Our bound in Theorem <ref> is sufficient to conclude that process M_n converges to process M weakly on compact intervals. This follows from Proposition <ref>. Using the notation therein, in this case, τ_n=n^-1/4 and λ_n=n(n-1)/2.



§ SETTING UP STEIN'S METHOD

Let us first define:

    A_n(t)=n^-1/2∑_i=1^⌊ ns(1)⌋ Z_i1_[i/n,s(1)](s(t))=n^-1/2∑_i=1^⌊ ns(1)⌋ Z_i1_[s^-1(i/n),1](t),

for Z_ii.i.d∼𝒩(0,1).
In a preparation for the proof of Theorem <ref>, we will apply Stein's method to find the distance between A_n and Y_n.


 §.§ The Stein equation

We first note that if U_1,U_2,... are i.i.d. Ornstein-Uhlenbeck processes with stationary law 𝒩(0,1), then defining:

    W_n(t,u)=n^-1/2∑_i=1^⌊ ns(t)⌋ U_i(u),   u≥ 0,t∈[0,1],

we obtain that the law of A_n is stationary for (W_n(·,u))_u≥ 0. Denote the generator of (W_n(·,u))_u≥ 0 by 𝒜_n. By properties of stationary distributions,  E_μ𝒜_nf=0 for all f∈Domain(𝒜_n) if and only if μ=ℒ(A_n). Therefore, we can treat

    𝒜_nf=g-Eg(A_n)

as our Stein equation.

In the next subsection, for any g from a suitable class of functions, we will find an f satisfying equation (<ref>). Then, in the sequel, we will find a bound on |E𝒜_nf(Y_n)|, which will readily give us a bound on |Eg(Y_n)-Eg(A_n)|.


The generator 𝒜_n of the process (W_n(·,u))_u≥ 0 acts on any f∈ M in the following way:

    (𝒜_nf)(w):=-Df(w)[w]+ED^2f(w)[A_n^(2)].


Before we prove this result, we need a lemma:

Letting ℱ_n,u=σ(W_n(·,v),v≤ u), we have:

    W_n(·,u+v)-e^-vW_n(·,u)𝒟=σ(v)A_n(·).


 We first note that for each i≥ 1 we can construct independent standard Brownian Motions B_i such that (X_i(u),u≥ 0)=(e^-uB_i(e^2u),u≥ 0). Then:


    W_n(·,u+v)-e^-vW(·,u)=   n^-1/2∑_k=1^⌊ ns(·)⌋ U_k(u+v)-e^-vn^-1/2∑_k=1^⌊ ns(·)⌋ U_k(u)
    𝒟=   n^-1/2e^-(u+v)∑_k=1^⌊ ns(·)⌋[B_k(e^2(u+v))-B_k(e^2u)]
    𝒟=   n^-1/2σ(v)∑_k=1^⌊ ns(·)⌋Z_k=σ(v)A_n(·).
 


Note that the semigroup of  (W_n(·,u))_u≥ 0, acting on L, is defined by:

    (T_n,uf)(w):=E[f(W_n(·,u)|W_n(·,0)=w]

and by Lemma <ref> we readily obtain that:

    (T_n,uf)(w)=E[f(we^-u+σ(u)A_n(·)].

We can define the generator by: 𝒜_n=lim_u↘ 0T_n,u-I/u. We also have that for f∈ M:

    |(T_n,uf)(w)-f(w)-EDf(w)[σ(u)A_n-w(1-e^-u)].
       -.1/2ED^2f(w)[{σ(u)A_n-w(1-e^-u)}^(2)]|
    (<ref>)≤   K_fEσ(u)A_n-w(1-e^-u)^3
    ≤   4K_f[σ^3(u)EA_n^3+(1-e^-u)^3w^3]
    ≤ K_3(1+w^3)u^3/2

for a constant K_3 depending only on f, where the last inequality follows from the fact that for u≥ 0, σ^3(u)≤ 3u^3/2 and (1-e^-u)^3≤ u^3/2.
So:


    |(T_n,uf-f)(w)+uDf(w)[w]-uED^2f(w)[A_n^(2)]|
    ≤   |(T_n,uf)(w)-f(w)-EDf(w)[σ(u)A_n-w(1-e^-u)].
       -.1/2ED^2f(w)[{σ(u)A_n-w(1-e^-u)}^(2)]|+|σ(u)EDf(w)[A_n]|
       +|(u-1+e^-u)Df(w)[w]|+|(σ^2(u)/2-u)ED^2f(w)[A_n^(2)]|
       +|(1-e^-u)^2/2D^2f(w)[w^(2)]|+|σ(u)(1-e^-u)ED^2f(w)[A_n,w]|
    ≤   K_1(1+w^3)u^3/2+|σ(u)EDf(w)[A_n]|+|u-1+e^-u|Df(w)w
       +|σ^2(u)/2-u|D^2f(w)EA_n^2
       +(1-e^-u)^2/2D^2f(w)w^2+σ(u)(1-e^-u)D^2f(w)wEA_n
    (<ref>)≤   K_1(1+w^3)u^3/2+|σ(u)EDf(w)[A_n]|+|u-1+e^-u|K_f(1+w^2)w
       +|σ^2(u)/2-u|K_f(1+w)EA_n^2+(1-e^-u)^2/2K_f(1+w)w^2
       +σ(u)(1-e^-u)(1+w)wEA_n
    ≤   3u^3/2(K_1(1+w^3)+K_f(1+w^2)w+K_f(1+w)EA_n^2.
       .+K_f(1+w)w^2+(1+w)wEA_n)+|σ(u)EDf(w)[A_n]|
    ≤    K_4(1+w^3)u^3/2,

for some constant K_4 depending only on f. The last inequality follows from the fact that:

    EDf(w)[A_n]=   EDf(w)[n^-1/2∑_k=1^⌊ ns(1)⌋Z_k
    1_[s^-1(k/n),1]]
    
    =   n^-1/2∑_k=1^⌊ ns(1)⌋Df(w)[1_[s^-1(k/n),1]]E[Z_k]
    
    =   0.

It follows that for any f∈ M:

    𝒜_nf(w)=lim_u↘0T_n,uf(w)-f(w)/u=-Df(w)[w]+ED^2f(w)[A_n^(2)].




 §.§ Solving Stein's equation


Suppose that g∈ M satisfies Eg(A_n)=0. Then the equation: 𝒜_nf=g
is solved by:

    f=ϕ_n(g)=-∫_0^∞ T_n,ugdu

for T_n,u defined by (<ref>).
Furthermore, ϕ_n(g)∈ M and the following inequalities hold:

    A)  Dϕ_n(g)(w)≤(1+4/3EA_n^2)g_M(1+w^2),
       B)  D^2ϕ_n(g)(w)≤(1/2+EA_n/3)g_M(1+w),
       C)  D^2ϕ_n(g)(w+h)-D^2ϕ(g)(w)≤g_M/3h.



The proof will follow the procedure used to prove <cit.>.

Step 1: First, we show that ϕ_n(g)∈ M and that (<ref>) holds. Assume g∈ M and E[g(A_n)]=0. 
Note that if, for instance:

    |g(w)-g(x)|≤ C_g(1+w^2+x^2)w-x

uniformly in w,x∈ D[0,1] then:

    lim_t→∞∫_0^t|T_n,ug(w)|du=lim_t→∞∫_0^t|Eg(we^-u+σ(u)A_n)|du
    ≤   lim_t→∞[∫_0^t|E[g(we^-u+σ(u)A_n)-g(σ(u)A_n)]|du+∫_0^t|E[g(σ(u)A_n)-g(A_n)]|du]
    ≤   C_glim_t→∞[∫_0^tE[(1+e^-uw+σ(u)A_n^2+σ^2(u)A_n^2)e^-uw]du.
       .+∫_0^tE|(1+(σ^2(u)+1)A_n^2)|(σ(u)-1)A_ndu] 
    ≤    C_glim_t→∞[∫_0^t[ e^-uw+2e^-3uw^3+3σ^2(u)e^-uwEA_n^2]du.
       .+∫_0^t(σ(u)-1)E|(1+(σ^2(u)+1)Z^2)|A_ndu]
    ≤   C(1+w^3),

for some constant C. For such g,
f=ϕ_n(g)=-∫_0^∞ T_n,ugdu
exists.
Note that every g∈ M satisfies condition (<ref>) because for such g:

    |g(w)-g(x)|(<ref>)≤ K_gw-x^3+|Dg(x)[w-x]+1/2D^2g(x)[w-x,w-x]| 
    ≤    K_gw-x^3+Dg(x)w-x+1/2D^2g(x)w-x^2 
    (<ref>)≤    K_gw-x(w-x^2+1+x^2+1/2w-x(1+x))
    ≤    K_gw-x(2w^2+2x^2+1+x^2+1/2(w+x+wx+x^2))
    ≤    C_g(1+w^2+x^2)w-x

uniformly in w,x because w≤ 1+w^2, x≤ 1+x^2 and wx≤w^2+x^2. 

Now take g∈ M, such that Eg(A_n)=0 and note that for ϕ_n defined in (<ref>) we get:

    ϕ_n(g)(w+h)-ϕ_n(g)(w)(<ref>)=-E∫_0^∞[g((w+h)e^-u+σ(u)A_n)-g(we^-u+σ(u)A_n)]du

and so dominated convergence (which can be applied because of (<ref>)) gives:

    D^kϕ_n(g)(w)=-E∫_0^∞e^-kuD^kg(we^-u+σ(u)A_n)du,   k=1,2.


Also, observe that:

    A)  Dϕ_n(g)(w)
       ≤∫_0^∞ e^-uEDg(we^-u+σ(u)A_n)du
       (<ref>)≤g_M∫_0^∞ e^-u(1+Ewe^-u+σ(u)A_n^2)du 
       ≤g_M∫_0^∞(e^-u+2w^2e^-3u+2EA_n^2(e^-u-e^-3u))du
       ≤(1+4/3EA_n^2)g_M(1+w^2),
       B)  D^2ϕ_n(g)(w)
       ≤∫_0^∞ e^-2uED^2g(we^-u+σ(u)A_n)du
       (<ref>)≤g_M∫_0^∞ e^-2u(1+Ewe^-u+σ(u)A_n)du
       ≤g_M∫_0^∞ e^-2u(1+e^-uw+σ(u)EA_n)du
       ≤g_M(1/2+w/3+EA_n/3)
       ≤(1/2+EA_n/3)g_M(1+w),
       C)  D^2ϕ_n(g)(w+h)-D^2ϕ_n(g)(w)
       ≤E∫_0^∞ e^-2uD^2g((w+h)e^-u+σ(u)A_n)-e^-2uD^2g(we^-u+σ(u)A_n)du
       g∈ M,(<ref>)≤g_M∫_0^∞ e^-2ue^-uhdu=g_M/3h,

uniformly in g∈ M, which proves (<ref>). It follows by (<ref>) and (<ref>) that ϕ_n(g)∈ M. Thus, ϕ_n(g) is in the domain of the semigroup and of 𝒜_n. Also, similarly, for any t>0, ∫_0^tT_n,ugdu∈ M.

Step 2: We now show that for all t>0 and for all g∈ M:

    T_n,tg-g=𝒜_n(∫_0^t T_n,ugdu).

We will prove it by following the steps of the proof of Proposition 1.5 on p. 9 of <cit.>. Observe that for all w∈ D[0,1] and h>0:

    1/h[T_n,h-I]∫_0^tT_n,ug(w)du=1/h∫_0^t[T_n,u+hg(w)-T_n,ug(w)]du
    
    =   1/h∫_t^t+hT_n,ug(w)du-1/h∫_0^hT_n,ug(w)du
    (<ref>)=   1/h∫_t^t+hE[g(w e^-u+σ(u)A_n)]du-1/h∫_0^hE[g(w e^-u+σ(u)A_n)]du.

Taking h→ 0 on the left-hand side gives 𝒜_n(∫_0^t T_n,ug(w)du) because ∫_0^t T_n,ug(w)du is in the domain of 𝒜_n, as shown in Step 1. In order to analyse the right-hand side note that:


    |1/h∫_0^hE[g(w e^-u+σ(u)A_n)]-g(w)du|
    MVT≤   1/h∫_0^hE[w(e^-u-1)+σ(u)A_nsup_c∈ [0,1]Dg(cw+(1-c)(we^-u+σ(u)A_n))]du
    ≤   g_M/h∫_0^hE[(w(1-e^-u)+σ(u)A_n)(1+3w^2+3w^2e^-2u+3σ^2(u)A_n^2)]du
    
    =   g_M/hE{(1+3w^2+3A_n^2)(w(-1+h+cosh(h)-sinh(h))..
       .+A_ne^-h(-√(e^2h-1)+e^h(h+log(1+e^-h√(-1+e^2h))))
       +3w(w^2-A_n^2)(e^-3h/6(e^h-1)^2(e^h+2))
       +.3(w^2A_n-A_n^3)1/3(√(1-e^-2h)-√(e^-6h(e^2h-1)))}0.

Similarly:

    |1/h∫_t^t+hE[g(we^-u+σ(u)A_n)]du-E[g(we^-t+σ(t)A_n)]|0

Therefore, as h→ 0, the right-hand side of (<ref>) converges to T_n,tg-g, which proves (<ref>).

Step 3: We note that for any h>0 and for any f∈ M:

    1/h[T_n,s+hf-T_n,sf]=T_n,s[T_n,h-I/hf]

and therefore for any w∈ D[0,1], by dominated convergence (which can be applied because of (<ref>)):

    (d/ds)^+T_n,sf(w)   =lim_h↘ 0T_n,s[T_n,h-I/hf(w)]=lim_h↘ 0E[T_n,h-I/hf(we^-s+σ(s)A_n)]
       =E[lim_h↘ 0T_n,h-I/hf(we^-s+σ(s)A_n)]=T_n,s𝒜_nf(w)

and, similarly, for s>0, (d/ds)^-T_n,sf=T_n,s𝒜_nf because:

    lim_h↘01/-h[T_n,s-hf-T_n,sf](w)-T_n,s𝒜_nf(w)
    
    =   lim_h↘0T_n,s-h[(T_n,h-I/h-𝒜_n)f](w)+lim_h↘0(T_n,s-h-T_n,s)𝒜_nf(w)
    
    =   lim_h↘ 0E[(T_n,h-I/h-𝒜_n)f(we^-s+h+σ(s-h)A_n)]
       +lim_h↘ 0E[𝒜_nf(we^-s+h+σ(s-h)A_n)-𝒜_nf(we^-s+σ(s)A_n)]
    (<ref>)=   0

again, by dominated convergence. It can be applied because of (<ref>) and the observation that:

    |𝒜_nf(we^-s+h+σ(s-h)A_n)-𝒜_nf(we^-s+σ(s)A_n)|
    
    =   |-Df(we^-s+h+σ(s-h)A_n)[we^-s+h+σ(s-h)A_n].
       +∑_i=1^nD^2f(we^-s+h+σ(s-h)A_n)[1_[i/n,1]^(2)]
       .-Df(we^-s+σ(s)A_n)[we^-s+σ(s)A_n]+∑_i=1^nD^2f(we^-s+σ(s)A_n)[1_[i/n,1]^(2)]|
    ≤   f_M(1+we^-s+h+σ(s-h)A_n^2)we^-s+h+σ(s-h)A_n
       +nf_M(1+we^-s+h+σ(s-h)A_n)
       +f_M(1+we^-s+σ(s)A_n)we^-s+σ(s)A_n+f_Mn(1+we^-s+σ(s)A_n)
    ≤   f_M(1+2w^2e^-2s+2+2σ^2(s-1)A_n^2)(we^-s+1+σ(s-1)A_n)
       +nf_M(1+we^-s+1+σ(s-1)A_n)
       +f_M(1+we^-s+σ(s)A_n)we^-s+σ(s)A_n+nf_M(1+we^-s+σ(s)A_n)

for all h∈[0,1].

Thus, for all w∈ D[0,1] and s>0: d/dsT_n,sf(w)=T_n,s𝒜_nf(w) and so, by the Fundamental Theorem of Calculus:

    T_n,rf(w)-f(w)=∫_0^rT_n,s𝒜_nf(w)ds.

Applying (<ref>) to f=∫_0^tT_n,ugdu we obtain:

    T_n,r∫_0^tT_n,ug(w)du-∫_0^tT_n,ug(w)du=∫_0^rT_n,s𝒜_n(∫_0^tT_n,ug(w)du)ds

Now, we take t→∞. We apply dominated convergence, which is allowed because of (<ref>) and the following bound for h_t,u(w)=∫_0^tT_n,ug(w)du:

    |𝒜_nh_t,u(w)|
    ≤   ∫_0^tE|e^-uDg(we^-u+σ(u)A_n)[w]|du
       +∑_i=1^n∫_0^tE|e^-2uD^2g(we^-u+σ(u)A_n)[1_[i/n,1]^(2)]|du
    ≤   ∫_0^∞E|e^-uDg(we^-u+σ(u)A_n)[w]|du
       +∑_i=1^n∫_0^∞E|e^-2uD^2g(we^-u+σ(u)A_n)[1_[i/n,1]^(2)]|du
    (<ref>)A,B≤   (1+4/3EA_n^2)g_M(1+w^2)w+n(1/2+EA_n/3)g_M(1+w),

where the first inequality follows again by dominated convergence applied because of (<ref>) in order to exchange integration and differentiation in a way similar to (<ref>). As a result, we obtain:

    T_n,r∫_0^∞ T_n,ug(w)du-∫_0^∞ T_n,ug(w)du   =∫_0^r T_n,slim_t→∞𝒜_n(∫_0^tT_n,ug(w)du)ds
       (<ref>)=-∫_0^r T_n,sg(w)ds.

Now, dividing both sides by r and taking r→ 0, we obtain:

    𝒜_n(∫_0^∞ T_n,ug(w))   =-lim_r→ 01/r∫_0^r T_n,sg(w)ds
       =-lim_r→ 0[1/r∫_0^rEg(we^-s+σ(s)A_n)ds]
       (<ref>)=-g(w),

which finishes the proof.


It is an easy consequence of Propositions <ref> and <ref> that for g∈ M: 
    𝒜_nϕ_n(g)(w)=-Dϕ_n(g)(w)[w]+ED^2ϕ_n(g)(w)[A_n^(2)].




§ PROOFS OF THEOREMS <REF>, <REF> AND <REF>



 §.§ Proof of Theorem <ref>



  §.§.§ Discretisation of Brownian Motion

Let A_n be as in (<ref>).
Now, note that we can first realise B and then set A_n(t)=B(⌊ ns(t)⌋/n) for t∈[0,1] so that:

    sup_t∈[0,1]|A_n(t)-Z(t)|   =sup_t∈[0,1]|B(⌊ ns(t)⌋/n)-B(s(t))|=sup_t∈[0,s(1)]|B(t)-B(⌊ nt⌋/n)|.

By Lemma 3 of <cit.> we get:

    A)  EA_n-Z   ≤E[sup_t,s∈[0,s(1)],|t-s|≤1/n|B(t)-B(s)|]
       ≤5/√(π)·6/√(log 2)n^-1/2√(log (2s(1)n))
    B)  EA_n-Z^2   ≤E[(sup_t,s∈[0,s(1)],|t-s|≤1/n|B(t)-B(s)|)^2]
       ≤5/2·(6/√(log 2))^2n^-1log (2s(1)n)
    C)  EA_n-Z^3   ≤E[(sup_t,s∈[0,s(1)],|t-s|≤1/n|B(t)-B(s)|)^3]
       ≤5/√(π)·(6/√(log 2))^3n^-3/2(log (2s(1)n))^3/2

and therefore we obtain, for any g∈ M:

    |Eg(A_n)-Eg(Z)|
    MVT≤   E[sup_c∈[0,1]Dg((1-c)Z+cA_n)Z-A_n]
    ≤   g_ME[sup_c∈[0,1](1+Z+c(A_n-Z)^2)Z-A_n] 
    ≤   g_ME[(1+2Z^2+2Z-A_n^2)Z-A_n]
    Hölder≤   g_M{EZ-A_n+2EZ-A_n^3+2(EZ^3)^2/3(EA_n-Z^3)^1/3}
    (<ref>)≤   g_M{5/√(π)·6/√(log 2)n^-1/2√(log (2s(1)n)).
       +10/√(π)·(6/√(log 2))^3n^-3/2(log (2s(1)n))^3/2
       .+2(EZ^3)^2/3(5/√(π))^1/3·6/√(log 2)n^-1/2√(log (2s(1)n))}
    Doob's L^3≤   g_M(30/√(πlog 2)+2· 5^1/3· 6/π^1/6√(log 2)((3/2)^3· 2√(2/π))^2/3s(1))n^-1/2√(log (2s(1)n))
       +g_M10/√(π)·(6/√(log 2))^3n^-3/2(log (2s(1)n))^3/2
    
    =   g_M30+54· 5^1/3s(1)/√(πlog 2)n^-1/2√(log (2s(1)n))
       +g_M2160/√(π)(log 2)^3/2n^-3/2(log (2s(1)n))^3/2.




  §.§.§ Applying Stein's method

Let g∈ M and g_n=g-E[g(A_n)]. Let f_n=ϕ_n(g_n), as in (<ref>). First, note that:

    EDf_n(Y_n)[Y_n]   =n^-1/2∑_j=1^⌊ ns(1)⌋EDf_n(Y_n)[X_j1_[s^-1(j/n),1]].

We now let Y_n^j=n^-1/2∑_k≠ j X_k1_[s^-1(k/n),1]=Y_n-n^-1/2X_j1_[s^-1(j/n),1] and observe that, by Taylor's theorem:

    | n^-1/2EX_jDf_n(Y_n)[1_[s^-1(j/n),1]].-E{ n^-1/2X_jDf_n(Y_n^j)[1_[s^-1(j/n),1]].
       ..+n^-1(X_j)^2D^2f_n(Y_n^j)[(1_[s^-1(j/n),1]e_i)^(2)]}|
    
    =   |E[ n^-1/2X_jDf_n(Y_n^j+n^-1/2X_j1_[j/n,1])[1_[s^-1(j/n),1]]..
       - n^-1/2X_jDf_n(Y_n^j)[1_[s^-1(j/n),1]]
       .. -n^-1(X_j)^2D^2f_n(Y_n^j)[(1_[s^-1(j/n),1])^(2)]]|
    (<ref>)C)≤   n^-3/2/6g_n_ME|X_j|^3

because, clearly, 1_[s^-1(j/n),1]=1. Also, in the last inequality we have used the fact that X_j is independent of Y_n^j. We can now sum (<ref>) over j=1,2,...,⌊ ns(1)⌋ and use the fact that X_j's are independent of Y_n^j's and that X_j's have mean 0 and variance 1 to obtain:

    |EDf_n(Y_n)[Y_n]-n^-1∑_j=1^⌊ ns(1)⌋D^2f_n(Y_n^j)[(1_[s^-1(j/n),1])^(2)]|≤n^-1/2/6s(1)g_n_ME|X_1|^3.

We notice that for 𝒜_n defined in Proposition <ref>, using Remark <ref>, we obtain:


    |E𝒜_nf_n(Y_n)|=|EDf_n(Y_n)[Y_n]-ED^2f_n(Y_n)[A_n^(2)]|
    ≤   |EDf_n(Y_n)[Y_n]-n^-1∑_j=1^⌊ ns(1)⌋ED^2f_n(Y^j_n)[(1_[s^-1(j/n),1])^(2)]|
       +n^-1|∑_j=1^⌊ ns(1)⌋E{ D^2f_n(Y_n)[(1_[s^-1(j/n),1])^(2)]-D^2f_n(Y_n^j)[(1_[s^-1(j/n),1])^(2)]}|
    ≤   n^-1/2/6g_n_ME|X_1|^3
       +n^-1|∑_j=1^⌊ ns(1)⌋E{ D^2f_n(Y_n^j+n^-1/2X_j1_[s^-1(j/n),1])[(1_[s^-1(j/n),1])^(2)]..
       -..D^2f_n(Y_n^j)[(1_[s^-1(j/n),1])^(2)]}|
    (<ref>)C)≤   n^-1/2/6g_n_ME|X_1|^3+n^-1g_n_M/3∑_j=1^⌊ ns(1)⌋n^-1/2EX_j1_[s^-1(j/n),1]
    ≤   n^-1/2/6s(1)g_n_ME|X_1|^3+n^-1/2/3s(1)g_n_MEX_11_[s^-1(j/n),1]
    ≤   n^-1/2s(1)g_n_M/2E|X_1|^3.

The last inequality follows by Jensen's inequality:

    E|X_1|≤√(E|X_1|^2)=1=(E|X_1|^2)^3/2≤E|X_1|^3.


Now, note that this gives:

    |Eg(Y_n)-Eg(A_n)|=|Eg_n(Y_n)|=|E𝒜_nf_n(Y_n)|
    ≤   n^-1/2/2s(1)g_n_ME|X_1|^3
    ≤   n^-1/2/2s(1)(g_M+Eg(A_n))E|X_1|^3
    ≤   n^-1/2/2s(1)(2+EA_n^3)g_ME|X_1|^3.

Also, recall that, by Doob's L^p inequality (see, e.g. Theorem 1.7 of Chapter II in <cit.>), if M is a right-continuous martingale then for every t>0 and p>1:

    E[(sup_s∈[0,t]|M_s|)^p]≤(p/p-1)^pE|M_t|^p.

Note that A_n is an integrable process, adapted to its natural filtration and for any t∈ [s^-1(m/n),s^-1((m+1)/n)) and r∈ [s^-1(l/n),s^-1((l+1)/n)) and r<t:

    E[A_n(t)|{ A_n(u):u≤ r}.]   =n^-1/2E[.∑_k=1^mZ_k|Z_1,...,Z_l]
       =n^-1/2∑_k=1^lZ_k + n^-1/2E[∑_k=l+1^m Z_k]
        = A_n(r)

and so A_n is a right-continuous martingale. Applying Doob's L^3 inequality to it yields:

    EA_n^3≤(3/2)^3E|A_n(1)|^3=(3/2)^3· 2√(2/π)s(1)^3/2,

because A_n(1)∼𝒩(0,⌊ ns(1)⌋/n).
Therefore (<ref>) gives:

    |Eg(Y_n)-Eg(A_n)|≤ n^-1/2(1+(3/2)^3·√(2/π)s(1)^3/2)g_Ms(1)E|X_1|^3.


Combining this with (<ref>):

    |Eg(Y_n)-Eg(Z)|≤   g_M30+54· 5^1/3s(1)/√(πlog 2)n^-1/2√(log (2s(1)n))
       +g_Ms(1)(1+(3/2)^3√(2/π)s(1)^3/2)E|X_1|^3n^-1/2
       +g_M2160/√(π)(log 2)^3/2n^-3/2(log (2s(1)n))^3/2,

which proves Theorem <ref>.

 §.§ Proof of Theorem <ref>

Note that (P(⌊ nS^(n)(t)⌋),t∈[0,1]) can be expressed in the following way:

    P(⌊ nS^(n)(t)⌋)-⌊ nS^(n)(t)⌋ =∑_i=1^⌊ nS^(n)(t)⌋ X_i,

where (X_i+1)'s are i.i.d. Poisson(1). 
Therefore, we can express (Ỹ_n(t),t∈[0,1]) in the following way:

    Ỹ_n(t)=   n^-1/2{∑_i=1^⌊ nS^(n)(t)⌋ X_i+P(nS^(n)(t))-P(⌊ nS^(n)(t)⌋)-(nS^(n)(t)-⌊ nS^(n)(t)⌋)}.

We also define:

    Y_n(t)=n^-1/2∑_i=1^⌊ nS^(n)(t)⌋ X_i.

Note that |nS^(n)(t)-⌊ nS^(n)(t)⌋|≤ 1 for all t≥ 0. Also, observe that for all t≥ 0: 

    |P(nS^(n)(t))-P(⌊ nS^(n)(t)⌋)|≤ P(⌊ nS^(n)(t)⌋+1)-P(⌊ nS^(n)(t)⌋).

By the independence of increments of a Poisson process:

    A)  EỸ_n-Y_n≤ n^-1/2[1+E[max_1≤ i≤ nP̅_i]]
       B)  EỸ_n-Y_n^3≤ n^-3/2[4+4E[max_1≤ i≤ nP̅_i^3]],

where P̅_1,⋯,P̅_ni.i.d∼Poisson(1). Using the trick from <cit.>, we note that, by Jensen's inequality applied to function exp(xloglog (n+2)):

    exp(loglog (n+2)·E[max_1≤ i≤ nP̅_i])   ≤E[exp(loglog (n+2)·max_1≤ i≤ nP̅_i)]
       =E[max_1≤ i≤ nexp(loglog (n+2)·P̅_i)]
       ≤ n E[exp(loglog n·P̅_1)]
       =nexp(log (n+2)-1)
       ≤(1+2e^-1)n^2

and by Jensen's inequality applied to function exp(x^1/3loglog (n+3)), which is convex for x≥8/(loglog (n+3))^3:


    exp(loglog (n+3)·{E[.max_1≤ i≤ nP̅_i^3|(max_1≤ i≤ nP̅_i^3)≥8/(loglog (n+3))^3]}^1/3)
    ≤   E[.exp(loglog (n+3)·max_1≤ i≤ nP̅_i)|(max_1≤ i≤ nP̅_i^3)≥8/(loglog (n+3))^3]
    ≤   nE[exp(loglog (n+3)·P̅_1)|(max_1≤ i≤ nP̅_i^3)≥8/(loglog (n+3))^3.]
    ≤   nexp(log (n+3)-1)/P[(max_1≤ i≤ nP̅_i)≥2/(loglog (n+3))]
    ≤   nexp(log (n+3)-1)/P[P̅_1≥2/(loglog 4))]
    ≤   (1+3e^-1)n^2/1-1957/720e.

Now, combining (<ref>), (<ref>) and (<ref>), we obtain:

    A)     EỸ_n-Y_n≤ n^-1/2[1+log(1+2e^-1)+2log n/loglog(n+2)]
    B)     EỸ_n-Y_n^3≤ n^-3/2[4+4(log(1+3e^-1/1-1957/720e)+2log n/loglog(n+3))^3+32/(loglog(n+3))^3]
       ≤ n^-3/2[4+16701+128(log n)^3/(loglog(n+3))^3].

We also note that:

    [EỸ_n^3]^2/3   Doob≤ n^-1(3/2)^2[E|P(nS^(n)(1))-nS^(n)(1)|^3]^2/3
       ≤(3/2)^2n^-1[E|P(nS^(n)(1))-nS^(n)(1)|^4]^1/2
       =(3/2)^2n^-1/2√(S^(n)(1))(1+3nS^(n)(1))^1/2.


Then, for every g∈ M:

    |Eg(Y_n)-Eg(Ỹ_n)|MVT≤E[sup_c∈[0,1]Dg((1-c)Ỹ_n+cY_n)Y_n-Ỹ_n]
    ≤   g_ME[sup_c∈[0,1](1+Ỹ_n+c(Y_n-Ỹ_n)^2)Y_n-Ỹ_n]
    ≤   g_ME[(1+2Ỹ_n^2+2Y_n-Ỹ_n^2)Y_n-Ỹ_n]
    Hölder≤   g_M{EY_n-Ỹ_n+2EY_n-Ỹ_n^3+2(EỸ_n^3)^2/3(EY_n-Ỹ_n^3)^1/3}
    (<ref>),(<ref>)≤   g_M{ n^-1/2[1+log(1+2e^-1)+2log n/loglog(n+2)]+n^-3/2[8+33402+256(log n)^3/(loglog(n+3))^3].
       .+9/2n^-1√(S^(n)(1))(1+3nS^(n)(1))^1/2[4+16701+128(log n)^3/(loglog(n+3))^3]^1/3}.

Let A_n(t)=n^-1/2∑_i=1^⌊ nS^(n)(t)⌋ Z_i,t∈[0,1] for Z_ii.i.d∼𝒩(0,1). By (<ref>):

    |Eg(Y_n)-Eg(A_n)|   ≤ n^-1/2(1+(3/2)^3√(2/π)S^(n)(1)^3/2)S^(n)(1)g_ME|X_1|^3
       ≤ n^-1/2(1+(3/2)^3√(2/π)S^(n)(1)^3/2)S^(n)(1)g_M(1+2e^-1)

because X_1𝒟=P(1)-1.
Now let Ã_n(t)=n^-1/2∑_i=1^⌊ nS(t)⌋ Z_i,t∈[0,1]. Then:

    A)  EA_n-Ã_n=n^-1/2E[sup_t∈[0,1]|∑_i=⌊ nS(t)∧ S^(n)(t)⌋+1^⌊ nS(t)∨ S^(n)(t)⌋ Z_i|]
       =n^-1/2E[sup_t∈[0,1]|∑_i=1^⌊ nS(t)∨ S^(n)(t)⌋-(⌊ nS(t)∧ S^(n)(t)⌋+1) Z_i|]
       Doob,Jensen≤ 2n^-1/2√(E|∑_i=1^sup_t∈[0,1](⌊ nS(t)∨ S^(n)(t)⌋-(⌊ nS(t)∧ S^(n)(t)⌋+1)) Z_i|^2)≤ 2√(S-S^(n))
       B)  EA_n-Ã_n^3=n^-3/2E[sup_t∈[0,1]|∑_i=⌊ nS(t)∧ S^(n)(t)⌋+1^⌊ nS(t)∨ S^(n)(t)⌋ Z_i|^3]
       =n^-3/2E[sup_t∈[0,1]|∑_i=1^⌊ nS(t)∨ S^(n)(t)⌋-(⌊ nS(t)∧ S^(n)(t)⌋+1) Z_i|^3]
       Doob≤(3/2)^3n^-3/2E[|∑_i=1^sup_t∈[0,1](⌊ nS(t)∨ S^(n)(t)⌋-(⌊ nS(t)∧ S^(n)(t)⌋+1)) Z_i|^3]
       ≤ 2√(2/π)(3/2)^3S-S^(n)^3/2
       C)  (EÃ_n^3)^2/3Doob≤ 2n^-1(3/2)^2π^-1/3(S(1)^3/2)^2/3=9/2π^1/3n^-1S(1).

Therefore:

    |Eg(A_n)-Eg(Ã_̃ñ)|MVT≤E[sup_c∈[0,1]Dg((1-c)Ã_n+cA_n)A_n-Ã_n]
    ≤   g_ME[sup_c∈[0,1](1+Ã_n+c(A_n-Ã_n)^2)A_n-Ã_n]
    ≤   g_ME[(1+2Ã_n^2+2A_n-Ã_n^2)A_n-Ã_̃ñ]
    Hölder≤   g_M{EA_n-Ã_n+2EA_n-Ã_n^3+2(EÃ_n^3)^2/3(EA_n-Ã_n)^1/3}
    (<ref>)≤   g_M{ 2√(S-S^(n))+27√(2)/2√(π)S-S^(n)^3/2+27√(2)/2√(π)S(1)√(S-S^(n))}.


By (<ref>) we get for Z=B∘ S:

    |Eg(Ã_n)-Eg(Z)|≤g_M30+54· 5^1/3S(1)/√(πlog 2)n^-1/2√(log (2S(1)n))
       +g_M2160/√(π)(log 2)^3/2n^-3/2(log (2S(1)n))^3/2.

Theorem <ref> now follows from (<ref>), (<ref>), (<ref>), (<ref>).

 §.§ Proof of Theorem <ref>

Note that X_n jumps up by 1/n with intensity 1/2n^2X_n(t)(1-X_n(t))+nν_2(1-X_n(t)) and down by 1/n with intensity 1/2n^2X_n(t)(1-X_n(t))+nν_1X_n(t). To see this observe that a jump occurs with intensity n 2 and it is an up-jump if the first gene chosen was of type a, the second of type A and the one with type A died (which happens with probability 1/2X_n(t)n(1-X_n(t))/n-1) or if the first one chosen was of type A, the second of type a and the type A gene died (which happens with probability 1/2(1-X_n(t))nX_n(t)/n-1). In addition, there are n(1-X_n(t)) genes of type A and each of them mutates into type a at rate ν_2. Hence:

    P[X_n(t+Δ t)-X_n(t)=1/n]=1/2n^2X_n(t)(1-X_n(t))Δ t+nν_2(1-X_n(t))Δ t
    P[X_n(t+Δ t)-X_n(t)=-1/n]=1/2n^2X_n(t)(1-X_n(t))Δ t+nν_1X_n(t)Δ t.

Therefore:

    M_n(t)=   P_1(n^2R^(n)_1(t))-n^2R^(n)_1(t)/n-P_-1(n^2R^(n)_-1(t))-n^2R^(n)_-1(t)/n
       +∫_0^t(ν_2-(ν_1+ν_2)M_n(s))ds,

where P_1,P_-1 are i.i.d. Poisson processes with rate 1, independent of X_n, and  

    R_1^(n)(t):=∫_0^t (1/2X_n(s)+ν_2/n)(1-X_n(s))ds
    
    R_-1^(n)(t):=∫_0^t (1/2(1-X_n(s))+ν_1/n)X_n(s)ds
    for  t∈[0,1].

Also let:

    R_1(t)=R_-1(t):=∫_0^t1/2X(s)(1-X(s))ds
    
    I_n(t):=∫_0^t(ν_2-(ν_1+ν_2)X_n(s))ds
    
    I(t):=∫_0^t(ν_2-(ν_1+ν_2)X(s))ds
    for t∈[0,1].


Let us denote Z_1=B_1∘ R_1, Z_-1=B_-1∘ R_-1, where B_1 and B_-1 are i.i.d. standard Brownian Motions, independent of X and:

    Y_n^1(·):=P_1(n^2R_1^(n)(·))-n^2R_1^(n)(·)/n,     Y_n^-1(·):=P_-1(n^2R_-1^(n)(·))-n^2R_-1^(n)(·)/n.
 

Now, for any g∈ M:

    |Eg(M_n)-Eg(M)|
    ≤   |E[E[g(Y_n^1-Y_n^-1+I_n)-g(Z_1-Y_n^-1+I_n)|Y_n^-1,I_n,R_1^(n),R_-1^(n),R_1]].
       +E[E[g(Z_1-Y_n^-1+I_n)-g(Z_1-Z_-1+I_n)|Z_1,I_n,R_1^(n),R_-1^(n),R_1]]
       .+E[E[g(Z_1-Z_-1+I_n)-g(Z_1-Z_-1+I)|Z_1,Z_-1,R_1]]|
    
    =   |E[E[g^(1)(Y_n^1)-g^(1)(Z_1)|Y_n^-1,I_n,R_1^(n),R_1]].
       +E[E[g^(-1)(Y_n^-1)-g^(-1)(Z_-1)|Z_1,I_n,R_-1^(n),R_-1]]
       .+E[E[g^(0)(I_n)-g^(0)(I)|Z_1,Z_-1,R_1,R_-1]]|,

where g^(1)(x)=g(x-Y_n^-1+I_n), g^(-1)(x)=g(Z_1-x+I_n), g^(0)(x)=g(Z_1-Z_-1+x). Note that, given R_1 and R_1^(n), Y_n^1 and Z_1 are independent of Y_n^-1 and I_n. Similarly, given R_-1 and R_-1^(n), Y_n^-1 and Z_-1 are independent of Z_1 and I_n. Also, given R_1 and R_-1, I_n and I are independent of Z_1 and Z_-1.

We now apply Theorem <ref> to obtain:

    A)  E[E[g^(1)(Y_n^1)-g^(1)(Z_1)|Y_n^-1,I_n,R_1^(n),R_1]]
    ≤   E[E[g^(1)_M|R_1,R_1^(n)].
       ·{(2+27√(2)/2√(π)R_1(1))√(R_1-R_1^(n))+27√(2)/2√(π)R_1-R_1^(n)^3/2.
       + n^-1[30+54· 5^1/3R_1(1)/√(πlog 2)√(log (2R_1(1)n^2)).+(1+(3/2)^3√(2/π)R_1^(n)(1)^3/2)
       · R_1^(n)(1)(1+2e^-1)
    .+1+log(1+2e^-1)+4log n/loglog(n^2+2)]
       +n^-29√(R_1^(n)(1))/2(1+3n^2R_1^(n)(1))^1/2[4+16701+1024(log n)^3/(loglog(n^2+3))^3]^1/3
       +.n^-3[2160/√(π)(log 2)^3/2(log (2R_1(1)n^2))^3/2+8+33402+2048(log n)^3/(loglog(n^2+3))^3]}
    ≤   E[E[g^(1)_M|R_1,R_1^(n)]{(2+27√(2)/16√(π))√(R_1-R_1^(n))+27√(2)/2√(π)R_1-R_1^(n)^3/2..
       + n^-1[30+27/4· 5^1/3/√(πlog 2)√(log (n^2/4)).+(1+(3/2)^3√(2/π)(1/8+ν_2/n)^3/2)
       ·.(1/8+ν_2/n)(1+2e^-1)+1+log(1+2e^-1)+4log n/loglog(n^2+2)]
       +n^-29√(1/8+ν_2/n)/2(1+3n^2/8+3nν_2)^1/2[4+16701+1024(log n)^3/(loglog(n^2+3))^3]^1/3
       +..n^-3[2160/√(π)(log 2)^3/2(log (n^2/4+ν_2n))^3/2+8+33402+2048(log n)^3/(loglog(n^2+3))^3]}]
       B)  |E(E{.E[.g^(-1)(Y_n^-1)-g^(-1)(Z_-1)|Z_1,I_n]|R_-1,R_-1^(n)})|
    ≤   E[E[.g^(-1)_M.|R_-1,R_-1^(n)].
       ·{(2+27√(2)/2√(π)R_-1(1))√(R_-1-R_-1^(n))+27√(2)/2√(π)R_-1-R_-1^(n)^3/2.
       + n^-1[30+54· 5^1/3R_-1(1)/√(πlog 2)√(log (2R_-1(1)n^2))+(1+(3/2)^3√(2/π)R_-1^(n)(1)^3/2).
       ·. R_-1^(n)(1)(1+2e^-1)+1+log(1+2e^-1)+4log n/loglog(n^2+2)]
       +n^-29√(R_-1^(n)(1))/2(1+3n^2R_-1^(n)(1))^1/2[4+16701+1024(log n)^3/(loglog(n^2+3))^3]^1/3
       +.n^-3[2160/√(π)(log 2)^3/2(log (2R_-1(1)n^2))^3/2+8+33402+2048(log n)^3/(loglog(n^2+3))^3]}
    ≤   E[E[g^(1)_M|R_-1,R_-1^(n)]{(2+27√(2)/16√(π))√(R_-1-R_-1^(n))+27√(2)/2√(π)R_-1-R_-1^(n)^3/2..
       + n^-1[30+27/4· 5^1/3/√(πlog 2)√(log (n^2/4)).+(1+(3/2)^3√(2/π)(1/8+ν_1/n)^3/2)
       .·(1/8+ν_1/n)(1+2e^-1)+1+log(1+2e^-1)+4log n/loglog(n^2+2)]
       +n^-29√(1/8+ν_1/n)/2(1+3n^2/8+3nν_1)^1/2[4+16701+1024(log n)^3/(loglog(n^2+3))^3]^1/3
       +..n^-3[2160/√(π)(log 2)^3/2(log (n^2/4+ν_1n))^3/2+8+33402+2048(log n)^3/(loglog(n^2+3))^3]}],

where we have used the fact that R_1(1),R_-1(1)≤1/8 and R_1^(n)≤1/8+ν_2/n, R_-1^(n)≤1/8+ν_1/n. We also note that:

    |E[E[g^(0)(I_n)-g^(0)(I)|Z_1,Z_-1,R_1,R_-1]]|
    ≤   E[E[g^(0)_M|R_1,R_-1][I_n-I+2I_n-I^3+2II_n-I]]
    ≤   E[E[g^(0)_M|R_1,R_-1][I_n-I+2I_n-I^3+2ν_2I_n-I]].

Now note that:

    A)  E[.sup_w,h∈ DD^2g(w+h-Y_n^-1+I_n)-D^2g(w-Y_n^-1+I_n)/h|R_1,R_1^(n),R_-1^(n)]
    ≤   sup_x,y∈ DD^2g(x+y)-D^2g(x)/y
       B)  E[.sup_w∈ DD^2g(w-Y_n^-1+I_n)/1+w|R_1,R_1^(n),R_-1^(n)]
    ≤   E[.sup_w∈ DD^2g(w-Y_n^-1+I_n)/1+w-Y_n^-1+I_n·1+w-Y_n^-1+I_n/1+w|R_1,R_1^(n),R_-1^(n)]
    Doob≤   (sup_w∈ DD^2g(w)/1+w)[1+2√(E[.(Y_n^-1(1))^2| R_1,R_1^(n),R_-1^(n)])+ν_2]
    
    =   (sup_w∈ DD^2g(w)/1+w)[1+2√(R_-1^(n)(1))+ν_2]
    ≤   (1+2√(1/8+ν_1/n)+ν_2)(sup_w∈ DD^2g(w)/1+w)
       C)  E[.sup_w∈ DDg(w-Y_n^-1+I_n)/1+w^2|R_1,R_1^(n),R_-1^(n)]
    ≤   E[.sup_w∈ DDg(w-Y_n^-1+I_n)/1+w-Y_n^-1+I_n^2·1+w-Y_n^-1+I_n^2/1+w^2|R_1,R_1^(n),R_-1^(n)]
    Doob≤    3(sup_w∈ DDg(w)/1+w^2)[1+4E[.(Y_n^-1(1))^2|R_1,R_1^(n),R_-1^(n)]+ν_2^2]
    
    =   3(sup_w∈ DDg(w)/1+w^2)[1+4R_-1^n(1)+ν_2^2]
    ≤   (9/2+12ν_1/n+3ν_2^2)(sup_w∈ DDg(w)/1+w^2)
       D)  E[.sup_w∈ D|g(w-Y_n^-1)|/1+w^3|R_1,R_1^(n), R_-1^(n)]
    ≤   E[.sup_w∈ D|g(w-Y_n^-1+I_n)|/1+w-Y_n^-1+I_n^3·1+w-Y_n^-1+I_n^3/1+w^3|R_1,R_1^(n),R_-1^(n)]
    Doob≤    9(sup_w∈ D|g(w)|/1+w^3)[1+27/8E[.|Y_n^-1(1)|^3|R_1,R_1^(n),R_-1^(n)]+ν_2^3]
    ≤   9(sup_w∈ D|g(w)|/1+w^3)[1+27/8(3(R_-1^(n)(1))^2+n^-1R_-1^(n)(1))^3/4+ν_2^3]
    ≤   (9+243/8·(3(1/8+ν_1/n)^2+1/8n+ν_1/n^2)^3/4+9ν_2^3)(sup_w∈ D|g(w)|/1+w^3).

Therefore:

    E[g^(1)_M|R_1,R_1^(n),R_-1^(n)]
    ≤   [9+2√(1/8+ν_1/n)+12ν_1/n+243/8·(3/64+6ν_1+1/8n+ν_1+ν_1^2/n^2)^3/4+ν_2+3ν_2^2+9ν_2^3]g_M

Similarly:

    A)  E[.sup_w,h∈ DD^2g(Z_1-(w+h)+I_n)-D^2g(Z_1-w+I_n)/h|R_1,R_1^(n),R_-1^(n)]
    ≤   sup_x,y∈ DD^2g(x+y)-D^2g(x)/y
       B)  E[.sup_w∈ DD^2g(Z_1-w+I_n)/1+w|R_1,R_1^(n),R_-1^(n)]
    ≤   E[.sup_w∈ DD^2g(Z_1-w+I_n)/1+Z_1-w+I_n·1+Z_1-w+I_n/1+w|R_1,R_1^(n),R_-1^(n)]
    Doob≤   (sup_w∈ DD^2g(w)/1+w)[1+2√(E[.(Z_1(1))^2| R_1,R_1^(n),R_-1^(n)])+ν_2]
    
    =   (sup_w∈ DD^2g(w)/1+w)[1+2√(R_1(1))+ν_2]
    ≤   (1+√(2)/2+ν_2)(sup_w∈ DD^2g(w)/1+w)
       C)  E[.sup_w∈ DDg(Z_1-w+I_n)/1+w^2|R_1,R_1^(n),R_-1^(n)]
    ≤   E[.sup_w∈ DDg(Z_1-w+I_n)/1+Z_1-w+I_n^2·1+Z_1-w+I_n^2/1+w^2|R_1,R_1^(n),R_-1^(n)]
    Doob≤   3(sup_w∈ DDg(w)/1+w^2)[1+4E[.(Z_1(1))^2|R_1,R_1^(n),R_-1^(n)]+ν_2^2]
    
    =   3(sup_w∈ DDg(w)/1+w^2)[1+4R_1(1)+ν_2^2]≤(9/2+3ν_2^2)(sup_w∈ DDg(w)/1+w^2)
       D)  E[.sup_w∈ D|g(Z_1-w+I_n)|/1+w^3|R_1,R_1^(n), R_-1^(n)]
    ≤   E[.sup_w∈ D|g(Z_1-w+I_n)|/1+Z_1-w+I_n^3·1+Z_1-w+I_n^3/1+w^3|R_1,R_1^(n),R_-1^(n)]
    Doob≤   9(sup_w∈ D|g(w)|/1+w^3)[1+27/8E[.|Z_1(1)|^3|R_1,R_1^(n),R_-1^(n)]+ν_2^3]
    ≤    9(sup_w∈ D|g(w)|/1+w^3)[1+27√(2)/4√(π)R_1(1)^3/2+ν_2^3]
    ≤   (9+243/64√(π)+9ν_2^3)(sup_w∈ D|g(w)|/1+w^3).

Therefore:

    E[g^(-1)_M|R_1,R_1^(n),R_-1^(n)]≤(9+243/64√(π)+ν_2+3ν_2^2+9ν_2^3)g_M.

Also:

    A)  E[.sup_w,h∈ DD^2g(Z_1-Z_-1+(w+h))-D^2g(Z_1-Z_-1+w)/h|R_1,R_1^(n),R_-1^(n)]
    ≤   sup_x,y∈ DD^2g(x+y)-D^2g(x)/y
       B)  E[.sup_w∈ DD^2g(Z_1-Z_-1+w)/1+w|R_1,R_1^(n),R_-1^(n)]
    ≤   E[.sup_w∈ DD^2g(Z_1-Z_-1+w)/1+Z_1-Z_-1+w·1+Z_1-Z_-1+w/1+w|R_1,R_1^(n),R_-1^(n)]
    Doob≤   (sup_w∈ DD^2g(w)/1+w)[1+2√(E[.(Z_1(1)-Z_-1(1))^2| R_1,R_1^(n),R_-1^(n)])]
    
    =   (sup_w∈ DD^2g(w)/1+w)[1+2√(2R_1(1))]
    ≤   2(sup_w∈ DD^2g(w)/1+w)
       C)  E[.sup_w∈ DDg(Z_1-Z_-1+w)/1+w^2|R_1,R_1^(n),R_-1^(n)]
    ≤   E[.sup_w∈ DDg(Z_1-Z_-1+w)/1+Z_1-Z_-1+w^2·1+Z_1-Z_-1+w^2/1+w^2|R_1,R_1^(n),R_-1^(n)]
    Doob≤   2(sup_w∈ DDg(w)/1+w^2)[1+4E[.(Z_1(1)-Z_-1(1))^2|R_1,R_1^(n),R_-1^(n)]]
    
    =   2(sup_w∈ DDg(w)/1+w^2)[1+8R_1(1)]
    ≤   4(sup_w∈ DDg(w)/1+w^2)
       D)  E[.sup_w∈ D|g(Z_1-Z_-1+w)|/1+w^3|R_1,R_1^(n), R_-1^(n)]
    ≤   E[.sup_w∈ D|g(Z_1-Z_-1+w)|/1+Z_1-Z_-1+w^3·1+Z_1-Z_-1+w^3/1+w^3|R_1,R_1^(n),R_-1^(n)]
    Doob≤   4 (sup_w∈ D|g(w)|/1+w^3)[1+27/8E[.|Z_1(1)-Z_-1(1)|^3|R_1,R_1^(n),R_-1^(n)]]
    ≤   4(sup_w∈ D|g(w)|/1+w^3)[1+27√(2)/4√(π)(2R_1(1))^3/2]
    ≤   (4+27√(2)/8√(π))(sup_w∈ D|g(w)|/1+w^3).

So:

    E[g^(0)_M|R_1,R_1^(n),R_-1^(n)]≤(4+27√(2)/8√(π))g_M.



Now, the Moran model and the Wright-Fisher diffusion can be coupled using the Donnelly-Kurtz look-down construction (see the discussion below Theorem <ref>). In this construction first the Wright-Fisher diffusion M is realised and then the Moran model M_n is constructed by describing nM_n(s) as a Binomial(n,M(s)) random variable. Note that:

    A)  E√(R_1-R_1^(n))
    ≤   E√(∫_0^1 |1/2X(s)(1-X(s))-(1/2X_n(s)+ν_2/n)(1-X_n(s))|ds)
    Jensen≤   √(∫_0^1 E|1/2X(s)(1-X(s))-(1/2X_n(s)+ν_2/n)(1-X_n(s))|ds)
    ≤   {∫_0^1 (1/2E[E[|X_n(s)-X(s)||X(s)]]+1/2E[E[|X_n^2(s)-X^2(s)||X(s)]]..
       ..+ν_2/nE[E[|1-X_n(s)||X(s)]])ds}^1/2
    ≤   {∫_0^1 (1/2E[√(Var[.X_n(s)|X(s)])]+1/2E[√(Var[.X^2_n(s)|X(s)])]..
       ..+1/2E[X(s)(1-X(s))/n]+ν_2/nE[1-X(s)])ds}^1/2
    
    =   {∫_0^1 (1/2E[√(X(s)(1-X(s))/n)]+1/2E[(X(s)(1-7X(s)+6nX(s))/n^3....
       +X(s)(12X^2(s)-20nX^2(s)+4n^2X^2(s)-6X^3(s)+10nX^3(s)/n^3
       ....-8X^4(s)/n)^1/2]+E[(X(s)+2ν_2)(1-X(s))/2n])ds}^1/2
    ≤   √(1/4n^-1/2+1/2√(13n^-3+16n^-2+4n^-1)+1+2ν_2/2n^-1)
       B)  ER_1-R_1^(n)^3/2
    ≤   E[(∫_0^1 |1/2X(s)(1-X(s))-(1/2X_n(s)+ν_2/n)(1-X_n(s))|ds)^3/2]
    Hölder≤   E[∫_0^1 |1/2X(s)(1-X(s))-(1/2X_n(s)+ν_2/n)(1-X_n(s))|^3/2ds]
    Jensen≤   √(3)∫_0^1{√(2)/4E[.E|X_n(s)-X(s)|^3/2|X(s)].
       +√(2)/4E[.E|X_n^2(s)-X^2(s)|^3/2|X(s)]
       .+ν_2^3/2/n^3/2E[E[|1-X_n(s)|^3/2|X(s)]]} ds
    Jensen≤   √(3)∫_0^1{√(2)/4E[(Var[.X_n(s)|X(s)])^3/4].
       +1/2E[.E|X_n^2(s)-(X(s)(1-X(s))/n+X^2(s))|^3/2|X(s)]
       +1/2E[(X(s)(1-X(s))/n)^3/2]
       .+ν_2^3/2/n^3/2E[((1-X(s))X(s)/n+(1-X(s))^2)^3/4]} ds
    ≤   √(3)∫_0^1{√(2)/4E[(Var[.X_n(s)|X(s)])^3/4]+(1/16+ν_2^3/2)n^-3/2.
       .+1/2E[.E|X_n^2(s)-(X(s)(1-X(s))/n+X^2(s))|^3/2|X(s)]} ds
    Jensen≤   √(3)∫_0^1{√(2)/4E[(Var[.X_n(s)|X(s)])^3/4].
       .+1/2E[(Var[.X_n^2(s)|X(s)])^3/4]+(1/16+ν_2^3/2)n^-3/2} ds 
    
    =   √(3)∫_0^1{√(2)/4E[(X(s)(1-X(s))/n)^3/4]+1/2E[(X(s)(1-7X(s))/n^3...
       +X(s)(6nX(s)+12X^2(s)-20nX^2(s)+4n^2X^2(s)-6X^3(s))/n^3
       ...+X(s)(10nX^3(s)-8n^2X^3(s))/n^3)^3/4]+(1/16+ν_2^3/2)n^-3/2} ds
    ≤   √(3)(√(2)/32n^-3/4+1/2(13n^-3+16n^-2+4n^-1)^3/4+(1/16+ν_2^3/2)n^-3/2)
       C)  E√(R_-1-R_-1^(n))
    ≤   E√(∫_0^1 |1/2X(s)(1-X(s))-(1/2(1-X_n(s))+ν_1/n)X_n(s)|ds)
    Jensen≤   √(∫_0^1 E|1/2X(s)(1-X(s))-(1/2(1-X_n(s))+ν_1/n)X_n(s)|ds)
    ≤   {∫_0^1 (1/2E[E[|X_n(s)-X(s)||X(s)]]+1/2E[E[|X_n^2(s)-X^2(s)||X(s)]]..
       ..+ν_1/nE[E[X_n(s)|X(s)]])ds}^1/2
    ≤   {∫_0^1 (1/2E[√(Var[.X_n(s)|X(s)])]+1/2E[√(Var[.X^2_n(s)|X(s)])]..
       ..+1/2E[X(s)(1-X(s))/n]+ν_1/nE[X(s)])ds}^1/2
    
    =   {∫_0^1 (1/2E[√(X(s)(1-X(s))/n)]+1/2E[(X(s)(1-7X(s)+6nX(s))/n^3....
       +X(s)(12X^2(s)-20nX^2(s)+4n^2X^2(s)-6X^3(s)+10nX^3(s))/n^3
       ....-8X^4(s)/n)^1/2]+E[X(s)(1-X(s)+2ν_1)/2n])ds}^1/2
    ≤   √(1/4n^-1/2+1/2√(13n^-3+16n^-2+4n^-1)+1+2ν_1/2n^-1)
       D)  ER_-1-R_-1^(n)^3/2
    ≤   E[(∫_0^1 |1/2X(s)(1-X(s))-(1/2(1-X_n(s))+ν_1/n)X_n(s)|ds)^3/2]
    Hölder≤   E[∫_0^1 |1/2X(s)(1-X(s))-(1/2(1-X_n(s))+ν_1/n)X_n(s)|^3/2ds]
    Jensen≤   √(3)∫_0^1{√(2)/4E[.E|X_n(s)-X(s)|^3/2|X(s)].
       +√(2)/4E[.E|X_n^2(s)-X^2(s)|^3/2|X(s)]
       .+ν_1^3/2/n^3/2E[E[|X_n(s)|^3/2|X(s)]]} ds
    Jensen≤   √(3)∫_0^1{√(2)/4E[(Var[.X_n(s)|X(s)])^3/4].
       .+1/2E[.E|X_n^2(s)-(X(s)(1-X(s))/n+X^2(s))|^3/2|X(s)].
       +1/2E[(X(s)(1-X(s))/n)^3/2]
       .+ν_1^3/2/n^3/2E[((1-X(s))X(s)/n+X^2(s))^3/4]} ds
    ≤   √(3)∫_0^1{√(2)/4E[(Var[.X_n(s)|X(s)])^3/4]+(1/16+ν_1^3/2)n^-3/2.
       .+1/2E[.E|X_n^2(s)-(X(s)(1-X(s))/n+X^2(s))|^3/2|X(s)]} ds
    Jensen≤   √(3)∫_0^1{√(2)/4E[(Var[.X_n(s)|X(s)])^3/4]+1/2E[(Var[.X_n^2(s)|X(s)])^3/4].
       +.(1/16+ν_1^3/2)n^-3/2} ds 
    
    =   √(3)∫_0^1{√(2)/4E[(X(s)(1-X(s))/n)^3/4]+1/2E[(X(s)(1-7X(s))/n^3...
       +X(s)(6nX(s)+12X^2(s)-20nX^2(s)+4n^2X^2(s)-6X^3(s))/n^3
       ...+X(s)(10nX^3(s)-8n^2X^3(s))/n^3)^3/4]+(1/16+ν_1^3/2)n^-3/2} ds
    ≤   √(3)(√(2)/32n^-3/4+1/2(13n^-3+16n^-2+4n^-1)^3/4+(1/16+ν_1^3/2)n^-3/2).

Also:

    A)  EI-I_n≤∫_0^1(ν_1+ν_2)E[E[|X_n(s)-X(s)||X(s)]]ds
    ≤   ∫_0^1(ν_1+ν_2)E[√(Var[X_n(s)|X(s)])]ds
    ≤   ∫_0^1(ν_1+ν_2)E[√(X(s)(1-X(s))/n)]ds≤1/2(ν_1+ν_2)n^-1/2
       B)  EI-I_n^3Hölder≤∫_0^1E|(ν_1+ν_2)(X_n(s)-X(s))|^3ds
    ≤   (ν_1+ν_2)^3∫_0^1[E[(X_n(s)-X(s))^4]]^3/4ds
    ≤    (ν_1+ν_2)^3∫_0^1 E[(X(s)(1-7X(s)+7nX(s)+12X^2(s)-18nX^2(s)/n^4..
       ..+X(s)(6n^2X^2(s)-6X^3(s)+11nX^3(s)-6n^2X^3(s)+n^3X^3(s))/n^4)^3/4]ds 
    ≤    (ν_1+ν_2)^3 (13n^-4+18n^-3+6n^-2+n^-1)^3/4.

We now combine (<ref>),(<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) to obtain:

    |Eg(M)-Eg(M_n)|
    ≤   g_M[9+2√(1/8+ν_1/n)+12ν_1/n+243/8·(3/64+6ν_1+1/8n+ν_1+ν_1^2/n^2)^3/4+ν_2+3ν_2^2+9ν_2^3]
       ·{(2+27√(2)/16√(π))√(1/4n^-1/2+1/2√(13n^-3+16n^-2+4n^-1)+1+2ν_2/2n^-1).
       +27√(2)/2√(π)√(3)(√(2)/32n^-3/4+1/2(13n^-3+16n^-2+4n^-1)^3/4+(1/16+ν_2^3/2)n^-3/2)
       + n^-1[30+27/4· 5^1/3/√(πlog 2)√(log (n^2/4))+(1+(3/2)^3√(2/π)(1/8+ν_2/n)^3/2).
       ·(1/8+ν_2/n)(1+2e^-1).+1+log(1+2e^-1)+4log n/loglog(n^2+2)]
       +n^-29√(1/8+ν_2/n)/2(1+3/8n^2+3ν_2n)^1/2[4+16701+1024(log n)^3/(loglog (n^2+3))^3]^1/3
       +.n^-3[2160/√(π)(log 2)^3/2(log (n^2/4+ν_2n))^3/2+8+33402+2048(log n)^3/(loglog (n^2+3))^3]}
       +g_M(9+243/64√(π)+ν_2+3ν_2^2+9ν_2^3)
       ·{(2+27√(2)/16√(π))√(1/4n^-1/2+1/2√(13n^-3+16n^-2+4n^-1)+1+2ν_1/2n^-1).
       +27√(2)/2√(π)√(3)(√(2)/32n^-3/4+1/2(13n^-3+16n^-2+4n^-1)^3/4+(1/16+ν_1^3/2)n^-3/2)
       + n^-1[30+27/4· 5^1/3/√(πlog 2)√(log (n^2/4))+(1+(3/2)^3√(2/π)(1/8+ν_1/n)^3/2).
       ·(1/8+ν_1/n)(1+2e^-1).+1+log(1+2e^-1)+4log n/loglog(n^2+2)]
       +n^-29√(1/8+ν_1/n)/2(1+3/8n^2+3ν_1n))^1/2[4+16701+1024(log n)^3/(loglog (n^2+3))^3]^1/3
       .+n^-3[2160/√(π)(log 2)^3/2(log (n^2/4+ν_1n))^3/2+8+33402+2048(log n)^3/(loglog(n^2+3))^3]}
       +g_M(4+27√(2)/8√(π))
       ·[1/2(1+2ν_2)(ν_1+ν_2)n^-1/2+2(ν_1+ν_2)^3 (13n^-4+18n^-3+6n^-2+n^-1)^3/4]
    ≤   g_M{(18+ν_1^1/2+47ν_1^3/4+31ν_1^3/2+ν_2+3ν_2^2+9ν_2^3).
       ·(1.02· 10^6+425ν_2^1/2+623ν_2+39ν_2^3/2+7ν_2^5/2)
       + (12+3ν_2+3ν_2^2+9ν_2^3)(1.02· 10^6+425ν_1^1/2+623ν_1+39ν_1^3/2+7ν_1^5/2)
       .+7(1/2(1+2ν_2)(ν_1+ν_2)+31(ν_1+ν_2)^3)} n^-1/4
       +g_M2112[(18+ν_1^1/2+47ν_1^3/4+31ν_1^3/2+ν_2+3ν_2^2+9ν_2^3)(log(n^2/4+ν_2n))^3/2.
       +.(12+3ν_2+3ν_2^2+9ν_2^3)(log(n^2/4+ν_1n))^3/2]n^-3.


The term of order n^-1/4 appearing in the bound obtained in Theorem <ref> is unexpected. It comes from the comparison of the two time changes R_1^(n) and R_1 applied at certain points in the proof to the Poisson process and to Brownian Motion respectively. 

One would expect n^-1√(log n^2) to be closer to the true speed of convergence. This is because our process M_n can be expressed as a difference of two scaled Poisson processes with parameters of order n^2, which resemble scaled random walks stopped at the n^2-th step. Order n^-1√(log n^2) would therefore be in line with the results presented in <cit.> in the context of scaled random walks and indeed with Theorem <ref>.

We can therefore guess that our bound is not sharp, yet a method avoiding the comparison of the aforementioned time changes is needed to improve it.


A strategy similar to the one used in the proof of Theorem <ref> may be used to obtain bounds on the distance between other continuous-time Markov chains and diffusions. A key ingredient in the proof is, however, a way of coupling the two for any fixed time.




§ APPENDIX: PROOF OF PROPOSITION <REF>

Note that the proof of Proposition 3.1 of <cit.> readily applies in this case up to and excluding (3.4) and it suffices to prove that lim inf_n→∞P[Y_n∈ B]≥P[Z∈ B] for all sets B of the form B=⋂_1≤ l≤ LB_l, where B_l={ w∈ D:w-s_l<γ_l}, s_l∈ C([0,1],R) and γ_l is such that P[Z∈∂ B_l]=0.

We will condition on the fact that the minimum holding time (interval of constancy of Y_n) is of length greater that r_n=λ_n^-3. It follows from Theorems 2.1 and 2.2 of Chapter 5 in <cit.> that if we condition on the number of holding times being equal to i, their lengths are distributed uniformly over the simplex A_i={ (x_1,...,x_i):x_j≥ 0, ∑_j=1^i x_j≤ 1}. Note that the probability of the minimum of them being greater or equal to r_n is (1-ir_n)^i if i≤ r_n and 0 otherwise. This is because Vol(A_i)=1/i! and Vol({ (x_1,...,x_i):x_j≥ r_n, ∑_j=1^i x_j≤ 1})=(1-ir_n)^i/i!. Therefore:

    P[minimal waiting time≥ r_n]
    
    =   ∑_i=1^∞P[minimal waiting time≥ r_n|#waiting times=i]P[#waiting times=i]
    
    =   ∑_i=1^⌊λ_n^3⌋(1-iλ_n^-3)^ie^-λ_n(λ_n)^i-1/(i-1)! 1.

To see this note the following:

    A)∑_i=⌈λ_n^5/4⌉^⌊λ_n^3⌋(1-iλ_n^-3)^ie^-λ_n(λ_n)^i-1/(i-1)!   ≤ e^-λ_n(λ_n^3-λ_n^5/4)(1-λ_n^-7/4)^λ_n^5/4λ_n^λ_n^5/4-1/(⌈λ_n^5/4⌉-1)!
       ≤λ_n^2⌈λ_n^5/4⌉/e^λ_n·λ_n^-1/8⌈λ_n^5/4⌉+9/8⌈λ_n^9/8⌉0


    B)∑_i=1^⌈λ_n^5/4⌉-1(1-iλ_n^-3)^ie^-λ_n(λ_n)^i-1/(i-1)!≥ (1-λ_n^-7/4)^⌈λ_n^5/4⌉e^-λ_n∑_i=1^⌈λ_n^5/4⌉-1(λ_n)^i-1/(i-1)!1,

where the convergence in B) holds since (1-λ_n^-7/4)^⌈λ_n^5/4⌉→ 1, e^-λ_n∑_i=1^∞(λ_n)^i-1/(i-1)!=1 and:


    e^-λ_n∑_i=⌈λ_n^5/4⌉^∞(λ_n)^i-1/(i-1)!≤ e^-λ_nλ_n^⌈λ_n^5/4⌉/⌈λ_n^5/4⌉!·⌈λ_n^5/4⌉+1/⌈λ_n^5/4⌉+1-λ_n0

for instance, by Proposition A.2.3(ii) of <cit.>.
Furthermore, note that for g_l,n^* defined by (3.6) in <cit.>:


    lim inf_n→∞E[∏_l=1^L g^*_l,n(Y_n)]
    
    =   lim inf_n→∞E[.∏_l=1^L g^*_l,n(Y_n)|minimal waiting time≥ r_n]P[minimal waiting time≥ r_n]
       +lim inf_n→∞E[.∏_l=1^L g^*_l,n(Y_n)|minimal waiting time< r_n]P[minimal waiting time< r_n]
    
      =   lim inf_n→∞E[.∏_l=1^L g^*_l,n(Y_n)|minimal waiting time≥ r_n]P[minimal waiting time≥ r_n]
    because:
    
    0≤   lim inf_n→∞E[.∏_l=1^L g^*_l,n(Y_n)|minimal waiting time< r_n]P[minimal waiting time< r_n]
    ≤   lim inf_n→∞P[minimal waiting time< r_n](<ref>)=0.

Following the same steps as in <cit.>, we obtain:

    lim inf_n→∞P[Y_n∈ B]≥lim inf_n→∞P[Y_n∈ B and minimal waiting time≥ r_n]
       ≥lim inf_n→∞E[.∏_l=1^L g^*_l,n(Y_n)|minimal waiting time≥ r_n]P[minimal waiting time≥ r_n]
       (<ref>),(<ref>)≥lim inf_n→∞{E[∏_l=1^L g^*_l,n(Z_n)]-Cτ_n∏_l=1^L g^*_l,n_M^0}
       ≥lim inf_n→∞{E[∏_l=1^L g^*_l,n(Z_n)]-C”τ_np_n^2(ϵγ)^-2η_n^-3}
       Fatou≥E[lim inf_n→∞∏_l=1^L g^*_l,n(Z_n)]≥P[⋂_1≤ l≤ L(Z-s_l<γ_l(1-θ))]
    .



§ ACKNOWLEDGEMENTS

The author would like to thank Professor Gesine Reinert, Professor Alison Etheridge and Professor Andrew Barbour for many helpful discussions and Professor Gesine Reinert for constructive comments on the early versions of the paper. The author is also grateful to Dr Sebastian Vollmer for pointing out a mistake in an early version of the proof of Proposition <ref>.


alpha


