On negative association of some finite point processes on general state spaces, extended version
Last Günter Szekli Ryszard













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On negative association of some finite point processes on general state spaces
    Günter Last
Department of Mathematics, Karlsruhe Institute of Technology, Englerstr. 2,  D-76131, Karlsruhe, Germany.


Karlsruhe Institute of Technology
Ryszard Szekli
Work supported by National Science Centre, Poland grant 2015/19/B/ST1/01152, address: University of Wrocław, Mathematical Institute, pl. Grunwaldzki 2/4,  50-384 Wrocław, Poland.
 

  University of Wrocław

    
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Abstract: 
We study negative association (NA) for mixed sampled point processes and show that NA holds for such processes if the random number of points of them fulfills ULC  property. We connect NA property of point processes with dcx  dependence ordering and show some consequences of it for mixed sampled and determinantal point processes. 
Some applications illustrate general theory.






Keywords: Finite point processes; association; negative association; strong Rayleigh measure; ULC;




§ INTRODUCTION


The questions studied in this paper are motivated by several negative dependence properties which are present in combinatorial probability, stochastic processes,  statistical mechanics, reliability and statistics. We focus our study on point processes theory which is a natural tool in many of these fields.  For each of these fields, it seems desirable to get a  better understanding of what it means for a collection of random variables to be repelling or mutually negatively dependent.  It is known that it is not  possible to copy the theory of positively dependent random variables.


Early history of various concepts of multivariate negative dependence  are based on topics considered in   Block and Savits <cit.>, Block, Savits and Shaked <cit.>, Ebrahimi and Ghosh <cit.> and Karlin and Rinott <cit.>. One of the fundamental results discussed in <cit.> is that if a distribution satisfies an intuitive structural condition called Condition N (recalled in the next section), then it satisfies all of the other conditions introduced there.  Condition N is satisfied by the multinomial, hypergeometric and Dirichlet distribution as well as several others. It also implies negative association introduced by Joag-Dev and Proschan <cit.>. A slightly stronger version of Condition N implies a condition based on stochastic ordering (NDS) due to Block, Savits and Shaked <cit.>. Negative association has one distinct advantage over the other types of negative dependence. Non-decreasing functions of disjoint sets of negatively associated random variables are also negatively associated. This closure property does not hold for the other types of negative dependence studied in the above mentioned papers. Condition N appears in a natural way in the context of queueing networks implying negative association of the population vector in Gordon-Newell networks and negative association of the sojourn times  vector in cyclic networks, see <cit.>, and <cit.>.



Pemantle in <cit.> in his negative dependence study confined himself  to binary-valued random variables, in the hope that eliminating the metric and order properties of the real numbers in favour of the two point set, will better reveal what is essential to the questions about negative dependence.
The list of examples that  motivated him to develop techniques for proving that measures have negative dependence properties such as negative association include the uniform random spanning tree, where the vector of indicator functions of the events that the edges of a graph belong to randomly chosen spanning tree is a random vector which is negatively associated, which was proved by   Feder and Mihail <cit.>. Similar properties hold for weighted spanning trees.  Further items on this list are simple exclusion processes, random cluster models and the occupation status of competing urns. 
Dubhashi and Ranjan <cit.> consider the competing urns example in detail and show negative association of the numbers of balls in each bin and some consequences such as Chernoff bounds for various models (for extensions see <cit.>).
From this fact it follows negative association of the indicators of exceeding any prescribed thresholds in bins. Occupation numbers of urns under various probability schemes have appeared in many places. Instead of multinomial probabilities, one can postulate indistinguishability  of urns or balls and arrive at  Bose-Einstein or other statistics. Negative association  arise in the multinomial models, where Mallows <cit.> was one of the first ones to observe negative dependence.



In Borcea et al. <cit.>  several conjectures related to negative dependence made by Liggett <cit.>, Pemantle <cit.>, and Wagner <cit.>, respectively, were solved  and also  Lyons' main results <cit.> on negative association  for determinantal probability measures induced by positive contractions were extended. The authors used several new classes of negatively dependent measures for zero-one valued vectors related to the theory of polynomials and to determinantal measures (for example  strongly Rayleigh measures related to  the notion of proper position for multivariate stable polynomials).
The problem of describing natural negative dependence properties that are preserved by symmetric exclusion evolutions has attracted some attention in the theory of interacting particle systems and Markov processes. In <cit.> the authors  provide an answer to the aforementioned problem and show that if the initial distribution of a symmetric exclusion process is strongly Rayleigh, then so is the distribution at any time; therefore,  the latter distribution is strongly negatively associated. In particular, this solved an open problem of Pemantle  and Liggett  stating that the distribution of a symmetric exclusion process at any time, with non-random/deterministic initial configuration is negatively associated, and shows that the same is actually true whenever the initial distribution is strongly Rayleigh. In a later paper <cit.>, Liggett has applied these results to prove convergence to the normal and Poisson laws for various functionals of the symmetric exclusion process.
Another  possible scenario when utilising negative association is for example : first, show that a model is negatively associated; second, use that negatively associated measures have sub - Gaussian tails; finally use that negative association is known to imply the Chernoff - Hoeffding tail bounds. That kind of approach via strong Rayleigh measures has been realised by Pemantle and Peres <cit.>, and a way of finding negative association via strong Rayleigh property by Peres et al <cit.>.



For point processes a negative association result is known in a quite general setting for so called determinantal point processes on locally compact Polish spaces generated by locally trace class positive contractions on natural L^2 space, see e.g.<cit.>, Theorem 3.7. A broad list of interesting examples of determinantal point processes can be found in <cit.>.
Negative dependence for finite point processes via determinantal and/or strongly Rayleigh measures  have interesting applications in various applied fields such as machine learning, computer vision, computational biology, 
natural language processing, combinatorial bandit learning, neural network compression  and matrix approximations, see for example <cit.>, <cit.>, <cit.>, <cit.>, and references therein.


Another approach to study dependence has been used in finance models. Positive and negative dependence may be seen as some stochastic ordering relation to independence. Such stochastic orderings are called dependence orderings (see <cit.> or <cit.>). Typical orderings used are supermodular ordering and directionally convex ordering. Relations of these orderings to association and negative association with some applications to concentration inequalities and to the theory of copula functions are given by Christofides and Vaggelatou <cit.>, and Ruechendorf <cit.>, respectively. Related results in the theory of point processes and stochastic geometry, where the directionally convex ordering is used to express more clustering in point patterns, are obtained by Blaszczyszyn and Yogeshwaran <cit.>.


Positive and negative association may be used to obtain information on the distribution of
functionals such as the sum of coordinates. Newman  <cit.> shows that under either a positive or a negative dependence assumption the joint characteristic function of the variables   is well approximated by the product of individual characteristic functions. This allows him to obtain central limit theorems  for stationary
sequences of associated variables. In the positive association case one needs to assume summable covariances, whereas in the negative case one gets this for free. The list of references on central limit theorems for positively/negatively associated variables is very long, a recent reference dealing with point processes is for example the paper by Poinas et al <cit.>. 

The central concept of negative dependence in the present paper is negative association (NA) which we show for some point processes. The theory and application of NA  are not
simply the duals of the theory and application of positive association, but
differ in important respects. Negative association has one distinct advantage
over the other known types of negative dependence. Non-decreasing functions of
disjoint sets of NA  random variables are also NA. 
Apart from NA property of determinantal point processes not much is known about NA property of other point processes. 
Therefore  it might be interesting to characterise NA property of a different, elementary but very useful class of finite point processes with iid locations of points, so called  mixed sampled point processes.
In order to obtain NA property in this general class of point processes we use some results from the theory of strongly Rayleigh measures on the unit cube (see theorems 3.3 and 3.6). Consequences of NA property of point processes to ordering of dependence in point processes are described in Proposition 4.7 and Corollary 4.8.








§ NEGATIVE ASSOCIATION AND RELATED DEFINITIONS

For the distribution of a real random variable X we say that its density (or probability function) is PF_2 if it is log-concave (discrete log-concave) on its support, see eg. Block et al <cit.> for a detailed description.

The random vector =(X_1,…, X_n) satisfies Condition N if there exist n +1 independent real random variables S_0,S_1,…,S_n,  each having a PF_2 density  (or probability function) and a real number s such that

    d= [(S_1,…, S_n)|S_0 +S_1 +⋯ +S_n = s]

where  d= denotes equality in distribution, and [(S_1,…, S_n)|S_0 +S_1 +⋯ +S_n = s] denotes a random variable having the distribution of (S_1,…, S_n) conditioned on the event S_0 +S_1 +⋯ +S_n = s.


The multinomial distribution is conditional distribution of sums of independent Poisson random variables given that their sum is fixed, and  the Dirichlet is the conditional distribution of independent gammas given that its sum is fixed. 


Condition N is related to many other definitions of negative dependence, and for example, as summarised in <cit.>, it is stronger than being completely RR_2 in pairs, than having RR_2 in pairs measure, than being S-MRR_2, and finally stronger than NA. We recall the definition of NA.



A random vector X=(X_1,…,X_n)  is negatively associated  Negatively associated
(NA)  if, for every subset A⊆{1,… 
,n}

    (f(X_i,i∈ A),g(X_j,j∈ A^c) ) ≤  0,

whenever f,g  are real non-decreasing functions.

NA  may also refer to the set of random variables {X_1,… 
,X_n}, or to the
underlying distribution of  X.

Negative association possesses the following properties (see  Joag-Dev and
Proschan <cit.>)



  (i) A pair (X,Y)  of random variables is NA  if and only if

    (X≤ x,Y≤ y) ≤(X≤ x) (Y≤ y),

i.e. (X,Y)  is negatively quadrant dependent  (NQD).

  (ii) For disjoint subsets A_1,… 
,A_m  of {1,… 
,n}, and non-decreasing
positive functions f_1,… 
,f_m,  X  is NA  implies

    ∏  ^m_i=1f_i( X_A_i) ≤∏  ^m_i=1 f_i( X_A_i),

where X_A_i = (X_j,j∈ A_i).


  (iii) Any (at least two element) subset of NA  random variables is NA.

  (iv)  If X  has independent components then it is NA.

  (v)  Increasing (non-decreasing) real functions defined on disjoint subsets of a set of NA
random variables are NA.

  (vi)  If X  is NA and Y  is NA, and X is independent of Y  then ( X, Y)
is NA.



In some applications negative association appears, when the random
variables are subjected to conditioning.

Let X_1,… 
,X_n  be independent, and suppose that

    (f( X_A) |∑^_i∈ A X_i=s)

is increasing in s, for every nondecreasing f, and every A⊆{1,… 
,n}.
Then the  distribution of ( X |∑^n_i=1X_i=s)
is NA, for almost all s.


The above theorem takes on added interest, when considered in conjunction with
the following theorem from <cit.>. For a queueing theoretical proof of this theorem see <cit.>.
 
Let X_1,… 
,X_n  be mutually independent with PF_2  densities and S_n=∑^n_i=1X_i. Then

    (ϕ ( X) | S_n=s )

is increasing in  ( almost every ) s, provided ϕ  is non-decreasing.


If X_1,… 
,X_n   are independent with PF_2  densities then the conditional
distribution of ( X | S_n=s )  is NA, for almost all  s.

Conditioning with respect to sums is not the only way to obtain NA property by conditioning, as it is shown in <cit.>, where conditioning on order statistics were used.

The property of negative association is reasonably useful but hard to verify.
Negatively correlated probability measures appear naturally in many different contexts. Here are some examples related to NA property.

Let x=(x_1,… 
,x_n)  be a set of real numbers. A permutation
distribution is the joint distribution of the vector X, which takes as
values all permutations of x  with equal probabilities 1/n!. Such a
distribution is NA. Negatively correlated normal random variables are NA. See Joag-Dev and Proschan <cit.>) for these and many other examples.

There are some examples directly related to Condition N and conditioning.

NA property of multinomial distributions can be seen from Condition N, since it is the conditional distribution of independent
Poisson random variables given their sum. 


One can see that multivariate hyper-geometric distribution is NA because it is the conditional distribution of independent binomial random variables given their sum, see <cit.>. 

The population vector in Gordon-Newell closed queueing networks
is NA, see   <cit.>,  Section 3.8, Theorem E.


The sojourn times vector in cyclic queues has got property NA, see
<cit.>.


A plethora of negative dependence properties and conjectures for zero-one valued vectors  ( CNA, CNA+, JNRD, JNRD+, h-NLC+, S-MRR_2, ULC, strongly Rayleigh, Rayleigh, PHR) were introduced and studied in<cit.> and <cit.> with an application to symmetric exclusion processes. These classes are related to 
determinantal probability measures described for example in  <cit.> and <cit.>. We shall not recall all of these definitions, and only recall NA property of determinantal point processes.


We shall utilize a slightly broader class than NA in our formulations on dependence orderings. We define this new class of distributions as an analog of weak association in sequence class (WAS) introduced by Rüchendorf in <cit.>.  We say that a random vector 𝐗 (or its distribution) is sNA (negatively associated in sequence) if  

    (_(X_i>t,f(X_i+1,… ,X_n))≤ 0,

for all f real non-decreasing functions and t∈, i=1,…,n-1. 

This condition  is equivallent to [(X_i+1,… ,X_n)| X_i>t)] <_st (X_i+1,… ,X_n) for all t∈ and i=1,…, n-1 Here <_st denotes the usual strong stochastic ordering on ^n. For the definitions of strong stochastic orderings we refer to <cit.>.



 §.§ Rayleigh measures and related classes

In order to study NA property of point processes we  need some other (stronger) properties of negative dependence, which we shall take from <cit.>. In this section, we will reduce the context of general point processes to point processes which can be represented in distribution by distributions of zero-one valued variables. Therefore we will be first interested  in probability measures on the unit cube. More precisely, let us consider probability function μ: 2^[n]→ [0,1], for [n]={1,…,n}, such that ∑_S∈ 2^[n]μ (S)=1. 
For i∈ [n]  the i-th coordinate function on 2^[n] is a binary random variable given by X_i(S) = 1 if i ∈ S and 0 otherwise, where S ⊆  [n].
There is a 1-1 correspondence between probability functions  on  2^[n] and the corresponding generating functions. For μ: 2^[n]→ [0,1] its generating function (multi-affine polynomial), is given by

    P_μ() = ∑_S⊆ [n]μ (S)^S,

where = (z_1,…,z_n),  S⊆[n], and ^S=∏_i∈ Sz_i.
One can define then that μ is  NA if 

    ∑_S∈ 2^[n] f(S)μ (S)∑_S∈ 2^[n] g(S)μ (S)≥∑_S∈ 2^[n] f(S)g(S) μ(S)

for any increasing functions f, g on 2^[n] (with respect to the inclusion) that depend on disjoint sets of coordinates, which is equivalent to the fact that the vector =(X_1,…,X_n)  is NA on the probability space (2^[n], _μ), where _μ is the probability measure corresponding to probability function (density) μ.

 A polynomial with all real coefficients P ∈ [z_1,…,z_n] is called (real) stable if P(z_1,…, z_ n)= 0 whenever Im (z_j) > 0,  for 1 ≤ j ≤  n. 

The following property implies NA.

 A measure _μ with density μ on 2^[n] (equivalently the corresponding 0-1 vector ) is called strongly Rayleigh if its generating polynomial P_μ is (real) stable. 

A product measure on 2^[n] i.e. a measure with density μ having its  generating function of the form

    P_μ()=∏_i=1^n (p_iz_i+1-p_i),   0≤ p_i ≤ 1, i∈ [n],

is strongly Rayleigh. □

A useful characterisation of strongly Rayleigh measures comes from the fact that multi-affine polynomial P∈[z_1, …, z_n] is (real) stable if and only if

    ∂ P/∂ z_i () ∂ P/∂ z_j () ≥ P()∂ ^2/∂ z_iz_j P (),

for all ∈^n, and 1≤ i,j≤ n, see <cit.>, Theorem 5.6.

A measure _μ with density μ on 2^[n] (equivalently the corresponding 0-1 vector ) is called  Rayleigh if its generating polynomial P_μ fulfills (<ref>) for all ∈^n_+.

A complex measure μ  on 2^[n] is called symmetric  if its generating polynomial P_μ is symmetric in all n variables. The measure μ  is almost symmetric or almost symmetric if P_μ is symmetric in all but possibly one variable.

One says that μ is conditionally negatively associated  (CNA) if each measure obtained from μ by conditioning on some (or none) of the values of the variables is NA. Finally, μ is called strongly conditionally negatively associated 
(CNA+) if each measure obtained from μ by imposing external fields and projections is CNA, for details see <cit.>, where one of the implications in Conjecture 2.6. there, states that  Rayleigh property implies CNA+ property of μ.

Pemantle's result (<cit.>, Theorem 3.7 in 3.5), says that  for symmetric measures, Rayleigh, CNA, and CNA+ properties are equivalent, and in addition they are equivalent to the fact that the sequence ( μ (∑_i=0^nX_i=k)/nk) _k=0^n is a log-concave sequence. 

A partial answer to the implication from the mentioned Conjecture 2.6 is given in <cit.> in
Corollary 6.6. where this implication is established for μ  almost symmetric. This implies that (almost) symmetric  Rayleigh measures are NA. Strongly Rayleigh measures are NA without assumptions on symmetry. 

Properties for more general polynomials than multi-affine (and distributions of vectors  of random variables with values in finite sets of natural numbers) can be reduced to properties of multi-affine polynomials by so called polarization.

Let P∈[z_1,…, z_n] be a polynomial of degree d_i in the variable z_i for 1≤ i ≤ n. The polarization P̃, is the unique polynomial in the variables z_ij,  1≤ i ≤ n, 1≤  j ≤ d_i, satisfying


  * P̃ is multi-affine, 

  * P̃ is symmetric in the variables z_i1,… ,z_id_i,
for 1≤ i ≤ n,

  * if we let z_ij =z_i  for all i, j  in P̃ ,  we recover P .

It is known (see e.g. <cit.>, Corrolary 4.7) that P∈[z_1,…, z_n] is stable iff P̃ is stable.
More precisely, if ( X_i,j : i ≤ n, j≤  d_i) is a finite family of non-negative integer variables with real stable generating polynomial, the aggregate variables X_i=∑_j=1^d_i X_i,j
 will also have a real stable generating polynomial. This is because it follows from the definition that stability is preserved by substituting z_i,j = z_i for all i,j. Conversely, if (X_i : 1 ≤  i ≤ n) are random variables whose joint law has a real stable generating polynomial P, one can define the polarization of P by the substitutions z_i^j=d_ij^-1 e_j(z_i,1, …, z_i,d_i ) where d_i are upper bounds for the values of X_i  and e_j  is the elementary
symmetric function of degree j  on d_i variables. Stability of P implies stability of the polarization P̃  of P, hence the strong Rayleigh property of a collection of binary variables (X_i,j) with probability generating function P̃.






§ NA FOR MIXED SAMPLED POINT PROCESSES


We shall adopt our  notation from the book by Last and Penrose <cit.>. Let (Ω,  F, ℙ) be a probability space and  (𝕏 , 𝒳) a measurable state space.
Denote by 𝐍_<∞(𝕏)=𝐍_<∞  the space of all measures μ on (𝕏 , 𝒳) such that μ(B) ∈_+ for all B ∈𝒳. Let 𝐍(𝕏)=𝐍 be the space of all measures that can be written as a countable sum of measures
from 𝐍_<∞. An example is the Dirac measure δ _x for a point x ∈𝕏 , given by δ_x(B):=_B (x). 

We define a point process η as a measurable mapping from (Ω,  F, ℙ) to (𝐍, 𝒩) (𝒩 is the smallest σ-field on 𝐍 such that μ↦μ(B) is measurable for all B ∈𝒳).

We restrict our attention in this paper to  finite point process with points located in a complete separable metric space  with 𝒳 being the Borel σ field.

We define NA property of point processes as follows.

A point process η is negatively associated (NA) or negatively associated in sequence (sNA)  if for each collection of disjoint sets B_1,…, B_n∈𝒳 the vector (η (B_1),…,η (B_n)) is NA or sNA, respectively, as defined for random vectors.

For a more general definition of negative association for point processes (random measures) we refer to <cit.> or <cit.>. 

For a Borel set A ⊆𝕏, let 𝒩_A denote the σ-field on 𝐍 generated by the functions
μ↦μ (B)  for Borel B ⊆ A.  The natural (inclusion) partial order on 𝐍 allows us to define f: 𝐍→ which is increasing.  We say that a point process η  has negative associations if (f (η)g(η))≤ (f(η))(g(η)) for every pair f, g of real bounded increasing functions that are measurable with respect to complementary subsets A, A^c
of 𝕏, meaning that a function  is measurable with respect to A if it is measurable with respect to 𝒩_A. It is clear that with such a general definition we have that if η has negative associations then η is NA. Let us recall Theorem 3.7 from <cit.>. Let λ be a Radon measure on a locally compact Polish space 𝕏. Let K be a locally trace-class positive contraction on L_2(𝕏, λ). By η_K we denote the determinantal point process generated by K, for details see <cit.>, section 3.2.

 The determinantal point process η_K defined above has negative associations.



Apart from determinantal point processes not much is known about NA property of point processes. 

Therefore  we concentrate our efforts  on characterising NA property for an elementary but very useful class of finite point processes with iid locations of points, more precisely
our main focus in this paper is on the  class of so  called mixed sampled point processes on 𝕏,  defined by 

    η =∑_i=1^τδ_X_i,

where (X)_i≥ 1 is iid with distribution F, and τ∈∪{0} is independent of (X_i)_i≥ 1.

For this process, given any finite partition A_1,…,A_k of 𝕏, conditionally on τ, the joint distribution of the number of points is given by

    (η (A_1)=n_1,…,η (A_k)=n_k|τ =N)=Nn_1⋯ n_kF(A_1)^n_1⋯ F(A_k)^n_k,

and unconditionally

    (η (A_1)=n_1,…,η (A_k)=n_k)=∑_N=0^∞ (τ=N)Nn_1⋯ n_kF(A_1)^n_1⋯ F(A_k)^n_k.

The joint probability generating function is therefore given by

    (z_1^η(A_1)⋯ z_k^η(A_k))=P _τ(F(A_1)z_1+⋯+F(A_k)z_k),

where P _τ(z)= (z^τ), z∈ [0,1].

First, we shall consider mixed point processes defined by (<ref>) for which random variables τ are of the form 
    τ=∑_i=1^nU_i

where n∈ and 
U_1,…, U_i are independent Bernoulli variables with possibly different success probabilities. The class of random variables which are the sums of n Bernoulli variables we denote by 
𝒬_n.
Moreover we denote by

    𝒬:=  (∪_n=1^∞_n)

the class of all distributions with supports contained in {0,1,…} appearing as  weak limits of distributions from _n, n≥ 1, i.e. the weak closure of ∪_n=1^∞_n 
The main results of this paper are  contained in Theorems <ref> and <ref>.

Suppose that η is a mixed sampled point process on 𝕏, defined by (<ref>), for which τ∈𝒬. Then η is NA.

Let B_1,…,B_n ∈𝒳 be a partition of , and q_i:=F(B_i), i=1,…,m. Define by

    _i:=(_{X_i∈ B_1},… , _{X_i∈ B_m},

the vector generated by the i-th sample X_i∈, i≥ 1. Note that each _i has multinomial distribution with success parameters q_1,…,q_m and the number of trials equal 1, and as such is NA. Moreover _i, i≥ 1 are independent.  
Let =(U_1,…, U_n) be a vector of zero-one valued, independent random variables which is independent of _i, i≥ 1. The vector composed as (, _1,…, _n) is NA because of properties (iv) and (vi) of NA.

 Now using property (v) we get that the vector (U_1_1,…,U_n_n) is NA as a monotone transformation (multiplication)  of disjoint coordinates of (, _1,…, _n). Again using property (v), this time to (U_1_1,…,U_n_n) and using appropriately addition we get that the vector ∑_i=1^n U_i_i is NA. It is clear that ∑_i=1^n U_i_i has got the same distribution as ∑_i=1^τ_i if τ∈𝒬_n, which in turn is the same as for (η (B_1),…, η (B_m)). This finishes the proof for τ∈𝒬_n for arbitrary n∈. 

For τ∈𝒬 there exist a sequence τ_kd⟶τ,  k→∞, for τ_k∈∪_n=1^∞𝒬_n, and

    [ f(∑_i=1^τ_k_i) g(∑_i=1^τ_k_i)]≤[f(∑_i=1^τ_k_i)][g(∑_i=1^τ_k_i)],

for f,g supported by disjoint coordinates, which are non-decreasing and bounded. Letting k→∞ gives 

    [f(∑_i=1^τ_i)g(∑_i=1^τ_i)]≤[f(∑_i=1^τ_i)][g(∑_i=1^τ_i)].

Since each non-decreasing function can be monotonically approximated by non-decreasing and bounded functions, we get NA property of η. □

The class 𝒬 can be completely characterized, see e.g. <cit.>.


    τ∈𝒬   iff  τ=^d τ_1 +τ_2,

where τ_1, τ_2 are independent and τ_1 has Poisson distribution and τ_2=^d ∑ _i=1^∞ U_i for independent zero-one valued variables U_i with p_i=(U_i=1)≥ 0, i≥ 1, such that ∑_i=1^∞ p_i<∞.

It is interesting to note that hypergeometric random variables belong to the class 𝒬, see e.g. <cit.>

We say that a  real sequence (a_i)_i=0^n has no internal zeros if the indices of its non-zero terms form a discrete interval. Following Pemantle <cit.> we shall use the following class of sequences and distributions.

W say that a finite real sequence (a_i)_i=0^n of non-negative real numbers with no internal zeros is ultra log-concave (ULC(n)) if 
    (a_i/ni)^2≥a_i-1/ni-1a_i+1/n i+1,   i=1,…,n-1.


The class of random variables for which their probability functions have the above property we denote by 𝒮_n, i.e.

    𝒮_n:={τ : ((τ=i))_i=0^n   is   ULC(n)}.

It is known that if a non-nengative sequence (a_i)_i=0^n is ULC(n), and a nonnegative sequence (b_i)_i=0^m is ULC(m) then the convolution of these sequences is ULC(m+n), see <cit.>, Theorem 2. Let

    𝒮:= (∪_n=1^∞𝒮_n).

Sums of independent variables from the class 𝒮 are in 𝒮. We shall see below that 𝒬⊆𝒮. Utilizing the class 𝒮, the Theorem <ref> can be generalised with a use of elementary symmetric functions.

Suppose that η is a mixed sampled point process on 𝕏, defined by (<ref>), for which τ∈𝒮. Then η  is NA.

Assume first that τ∈𝒮_n.
Let B_1,…,B_n ∈𝒳 be a partition of , and q_i:=F(B_i), i=1,…,m. Define by

    _i:=(_{X_i∈ B_1},… , _{X_i∈ B_m},

the zero-one valued vector generated by the i-th sample X_i∈, i≥ 1. Note that each _i has multinomial distribution with success parameters q_1,…,q_m, and the number of trials equal 1, and as such is NA. Moreover _1,… are independent.  

For fixed n∈,  let =(U_1,…, U_n) be, independent of _i, i≥ 1,  vector,  of 0-1 valued random variables obtained in the following way.
For the generating function of τ, P_τ(z)=(z^τ), we define  by providing its multidimensional generating function. It is obtained by substituting in P_τ(z), for each k=0,…, n, 

    z^k:=nk^-1 e_k(z_1, …, z_n ),

 where e_k(z_1, …, z_n ) is  the k-th elementary symmetric polynomial. Note that immediately from the definition of the elementary symmetric polynomials, for each k, the function nk^-1 e_k(z_1, …, z_n ) of variables z_1, …, z_n is the multivariate generating function of a vector of n 0-1 valued variables which takes on exactly k values 1 with the same probability nk^-1 defined for all possible selections of k  coordinates on which the values 1 are obtained. The distribution of  defined in such a way is the mixture with the coefficients a_k:=(τ=k) of the distributions corresponding to nk^-1 e_k(z_1, …, z_n ), k=0,…, n. Since each function e_k is symmetric in variables z_1, …, z_n, the same is true for the generating function of =(U_1,…, U_n), therefore  (U_1,…, U_n) are exchangeable. Moreover  ∑_i=1^nU_id=τ, since by setting z_1=⋯=z_n:=z we obtain P_τ(z). In other words  the sequence ((τ=i))_i=0^n is the rank sequence for the vector =(U_1,…, U_n).

From our assumption  the rank sequence for =(U_1,…, U_n) is ULC(n) and from Theorem 2.7 in  <cit.> we obtain that  is NA. Now the vector composed as (, _1,…, _n) is NA because of property (vi) of NA. Using property (v) we get that the vector (U_1_1,…,U_n_n) is NA as a monotone transformation (multiplication)  of disjoint coordinates of (, _1,…, _n). Again using property (v), this time to (U_1_1,…,U_n_n) and using appropriately addition we get that the vector ∑_i=1^n U_i_i is NA. It is clear that ∑_i=1^n U_i_i has got the same distribution as ∑_i=1^τ_i, which in turn has the same  distribution as (η (B_1),…, η (B_m)). This finishes the proof for τ∈𝒮_n, for arbitrary n∈. For τ∈𝒮, we apply an analogous limiting argument as in Theorem <ref>. □


The following lemma may be regarded as known since it is an immediate consequence of the classical Newton inequalities.  We put its formulation in the setting of the introduced in this paper classes of random variables. 

For all n>1

    𝒬_n ⊆𝒮_n.


Suppose τ∈𝒬_n. Then, for its generating function, 

    P_τ(z)=(1-p_1+p_1z)⋯ (1-p_n+p_nz)=p_1⋯ p_n(1-p_1/p_1+z)⋯ (1-p_n/p_n+z).

For a_k:=1-p_k/p_k we have

    P_τ(z)=p_1⋯ p_n(a_1+z)⋯ (a_n+z)=p_1⋯ p_n[x^n+c_1x^n-1+⋯+c_n],

where c_n=a_1+⋯+a_n, c_2=a_1a_2+⋯+ a_n-1a_n, ⋯, c_n=a_1⋯ a_n, i.e. c_k, k=0,…, n, coefficients are given by the corresponding elementary symmetric polynomials in variables a_i, i=1,…, n. It is known from the classical Newton's inequalities that for k=1,…,n-1

    ( c_k/nk) ^2≥c_k-1/nk-1c_k+1/nk+1,

and since (τ=k)=p_1⋯ p_n c_n-k, we get that the sequence ((τ=k))_k=0^n   isULC(n), and therefore τ∈𝒮_n. □ 

Lemma <ref> implies the announced before inclusion  𝒬⊂𝒮.

It is interesting to note that arguments utilised in  Theorem <ref> can be used for random vectors with arbitrary positive values.


Assume that _i=(Z_i^1,…,Z_i^m), i≥ 1 is a sequence of independent, identically distributed  random vectors
with components in _+ such  that for each i≥ 1
∑^m_j=1{Z_i^j>0}≤ 1, that is, at most one of the
components can be positive. Then for τ∈𝒮 which is independent of _i,  i≥ 1, the vector :=∑^τ_i=1_i is NA.

We use basically the same argument as in Theorem <ref>. Let =(U_1,…, U_n) be, independent of _i, i≥ 1, vector of 0-1 valued random variables obtained by its generating function obtained as follows. In  P_τ(z), we substitute z^k:=nk^-1 e_k(z_1, …, z_n ), k=1,…,n, where e_k(z_1, …, z_n ) are the elementary symmetric polynomials. This substitution defines a generating function of variables z_1,…,z_n. It is then immediate  that ∑_i=1^nU_i=^d τ, that is the sequence ((τ=i))_i=0^n is the rank sequence for (symmetric) =(U_1,…, U_n). From our assumption  we have that the rank sequence for =(U_1,…, U_n) is ULC(n) and from Theorem 2.7 in  <cit.> we obtain that  is NA. Now the vector composed as (, _1,…, _n) is NA because of the following lemma and property (vi) of NA.

Assume that =(Z^1,…,Z^m) is a random vector
with components in _+. Assume that
∑^m_j=1{Z^i>0}≤ 1, that is, at most one of the
components can be positive. Then  is NA.

Proof of lemma <ref>. In order to show that
(f(Z^1,…,Z^k),g(Z^k+1,…,Z^m))≥ 0
for non-decreasing f and g, it suffices to assume that f(0)=g(0)=0. Otherwise one can consider f-f(0) and g-g(0). Because only of the coordinates can be non-zero, we get 
E[f(Z^1,…,Z^k)g(Z^k+1,…,Z^m))]=0,
while the  product of the expectations is non-negative
since, f≥ 0 and g≥ 0. □

Now, using property (v),  we get that the vector (U_1_1,…,U_n_n) is NA as a monotone transformation (multiplication)  of disjoint coordinates of (, _1,…, _n). Again using property (v), this time to (U_1_1,…,U_n_n) and using appropriately addition we get that the vector ∑_i=1^n U_i_i is NA. It is clear that ∑_i=1^n U_i_i has got the same distribution as ∑_i=1^τ_i. □

The above proposition can be used to study random measures other than point processes. We shall pursue this topic elsewhere. 




§ DEPENDENCE ORDERINGS  FOR POINT PROCESSES


An extensive study of dependence orderings for multivariate point processes on  is contained in <cit.>. Related results in the theory of point processes and stochastic geometry, where the directionally convex ordering is used to express more clustering in point patterns, are obtained by Blaszczyszyn and Yogeshwaran <cit.>, see also references therein.
We shall use NA property of point processes we study to obtain comparisons related to dependency properties. First we recall some  basic facts on dependence orderings of vectors and their relation to NA which can be directly utilized for point processes.



 §.§ Dependence orderings and negative correlations for vectors


For a function f:^n→ define the difference operator Δ^ϵ_i, ϵ >0, 1≤ i≤ n
by

    Δ^ϵ_i f() = f(+ϵ_i)-f()

where _i is the i-th unit vector. Then f is called
super-modular if for all 1≤  i < j ≤ n and ϵ, δ > 0 
    Δ^δ_jΔ^ϵ_i f()≥ 0.
 for all ∈ , and directionally convex if this inequality holds for all 1≤ i ≤ j ≤ n.
Let ^sm, ^dcx denote the classes of super-modular, and directionally convex functions.
 Then of course ^dcx⊆^sm. 
Typical examples from  ^dcx class of functions are f()=ψ (∑_i=1^nx_i), for ψ convex, or f()=max_1≤ i≤ nx_i, but there are many other useful functions in this class, see for example <cit.>.

The corresponding stochastic orderings one defines by
<_sm if f()≤ f() for all f∈^sm, and analogously for <_dcx. For differentiable functions f one obtains f ∈^sm iff ∂ ^2/∂ x_ix_jf≥ 0, for i < j,  and f ∈^dcx iff this inequality holds for i ≤ j. While comparison of  and   with respect to <_sm  implies (and is restricted to the case of)  identical marginals X_i=^dY_i, the comparison with respect to the smaller class ^dcx implies  convexly increasing marginals X_i <_cx Y_i (which means by definition that ψ(X_i)≤ψ (Y_i) for all ψ:→ convex). Both of these orderings belong to the class of so called dependency orderings, see e.g. <cit.>, which is defined by a list of suitable properties, among them the property that (X_i,X_j)≤ (Y_i,Y_j). 

In <cit.> another stochastic ordering related to dependence comparisons was introduced by the condition <_wcs (weakly conditional increasing in sequence order) iff 

    (_(X_i>t,f(X_i+1,… ,X_n))≤ (_(Y_i>t, f(Y_i+1,… ,Y_n)),

for all f monotonically non-decreasing, and all t∈, 1≤ i≤ n-1.
The following theorem from <cit.> connects the above defined orderings.

Let , be n-dimensional random vectors.



  * If X_id=Y_i,  1≤ i≤ n then <_wcs implies that <_sm.

  * If X_i<_cxY_i, 1≤ i≤ n then <_wcs implies that <_dcx.
 

Dependence orderings can be used to define some classes of distributions with negative or positive covariances when applied to vectors with independent components. More precisely, denote by ^*  a vector with independent components, and X_i^*d= X_i.
Then
 is called weakly associated in sequence  if ^* <_wcs.
Note that  is sNA, as defined by (<ref>), iff <_wcs^*.
Note also  that  sNA is further equivalent to

    [(X_i+1,… ,X_n)| X_i>t)] <_st (X_i+1,… ,X_n) ,
 
for all i=1,…, n-1, t>0, where [(X_i+1,… ,X_n)| X_i>t)] denotes a random vector which has got the distribution of (X_i+1,… ,X_n) conditioned on the event {X_i>t}, and <_st is the usual (strong) stochastic order.
It is clear directly from the definition that NA property implies sNA property. For  being sNA, Theorem  <ref> implies  that, for example, (see also <cit.> for the case NA)
∑_i=1^n X_i <_cx∑_i=1^n X^*_i,
and
max_1≤ k≤ n∑_i=1^k X_i<_icxmax_1≤ k≤ n∑_i=1^k X^*_i,
 where <_icx is defined similarly as <_cx but with the use of non-decreasing convex functions.
Taking other super-modular functions it is possible to get maximal inequalities for sNA vectors as in <cit.> . For <_dcx ordering we get the following corollary from Theorem <ref> which will be used for point processes with NA property.

Suppose  is sNA and ^* has independent coordinates with X_i<_cxY_i^* then <_dcx^*.







 §.§ NA and dependence orderings for point processes



We shall connect the approach by dependence orderings with negative association and by using Corollary <ref> we shall be able to compare dependence for  point processes which posses NA property with some other processes. This will   imply in particular comparison results for void probabilities and/or moment measures.





Two point processes η_1,η_2 on a complete, separable metric space 𝕏 are ordered in directionaly convex order (weakly conditional increasing sequence order) 
    η_1<_dcxη_2   (η_1<_wcsη_2)   iff (η_1(B_1),…, η_1  (B_n))<_dcx(<_wcs)  (η_2(B_1),…, η_2 (B_n)),
 
as defined for random vectors, for all disjoint, bounded  Borel sets B_1,…, B_n, n≥ 1.


The following consequence of <_dcx ordering for point processes is known from <cit.>, Proposition 6.

Let η_1,η_2 be two point process on ^d. If η_1<_dcxη_2 then



  * (moment measures) (η_1(B_1)⋯η_1 (B_n)))≤ (η_2(B_1)⋯η_2 (B_n))) 
 for all disjoint, bounded Borel sets B_1,…, B_n,
  * (void probabilities) (η_1(B)=0)≤(η_2 (B)=0) 
 for all bounded Borel sets B.


Using <_wcs criterion for <_dcx from Theorem <ref>, we get

Let η_1,η_2 be two point process on ^d. If η_1(B)<_cxη_2(B) and η_1<_wcsη_2 then the moment measures and void probabilities comparisons from the above lemma hold.

An interesting case for such comparisons is when η_2 is a Poisson point process. 

Suppose η is a simple point process on ^d which is sNA  then 



  * (moment measures) (η (B_1)⋯η  (B_n)))≤ (η (B_1))⋯(η (B_n)) 
 for all disjoint, bounded Borel sets B_1,…, B_n, which implies 
    (exp(∫_^dh()η(d)≤exp(∫_^d(e^h()-1)η(d)),

for all h≥ 0.

  * (void probabilities) (η (B)=0)≤exp(-η(B))
  for all bounded Borel sets B, which implies 

    (exp(-∫_^dh()η(d)≤exp(∫_^d(e^-h()-1)η(d)),

for all h≥ 0.


From our assumption (η (B_1)⋯η  (B_n)) is sNA. Since  is sNA, as defined by (<ref>) iff <_wcs^* then  from  Theorem <ref>, (η (B_1)⋯η  (B_n))<_sm^*, where ^* denotes the corresponding independent version. From the definition of <_sm we obtain directly that (η (B_1)⋯η  (B_n)))≤ (η (B_1))⋯(η (B_n)). Now from Proposition 1 in <cit.> we obtain that this implies (exp(∫_^dh()η(d)≤exp(∫_^d(e^h()-1)η(d)), for all h≥ 0.
 
Regarding void probabilities, since (η (B_1)⋯η  (B_n)) is sNA for all  bounded Borel sets B_1,…, B_n, it follows that (η(B),η(B')) is also sNA for all bounded Borel B,B'. Directly from the definition of sNA we conclude that (η (B)=0),η (B')=0)≤(η (B)=0)(η (B')=0). Now, from Proposition 3.1 in <cit.> we get that (η (B)=0)≤exp(-η(B)), for all bounded Borel sets B. Moreover, from Proposition 2 in <cit.> this inequality  holds if and only if (exp(-∫_^dh()η(d)≤exp(∫_^d(e^-h()-1)η(d)),
for all h≥ 0.
□

From Corollary <ref>, the above result can be  modified (using the same argument)  to obtain  <_dcx ordering for processes on general state spaces.  For point processes on ^d, this result can also be used to obtain  comparison of moment measures and void probabilities for sNA processes with Poisson processes.

Suppose η_1 is a  point process on  a complete, separable metric space 𝕏  which is sNA,  and η_1 (B)<_cx Y, for all bounded Borel sets B, where Y is a random variable with Poisson distribution Po(η_1(B)). Then 
    η_1<_dcxη_2,
 where η_2 is a Poisson point process with intensity measure η_1. 







 §.§ <_dcx comparisons for mixed sampled and determinantal point processes


Before we formulate  more general results, which use our results on NA properties of mixed sampled point processes,  we start with an example which gives a direct approach to a comparison of void probabilities.

[Comparison of binomial mixed sampled p.p. with Poisson p.p. on ^d]Let η be mixed sampled point process on ^d with τ being binomially distributed random variable. We shall compare void probabilities for this process with void probabilities of Poisson point process with the same intensity measure.  In general,
for a simple point processes η on ^d in order to get (η (B)=0)≤exp(-η(B)), it is enough to check (see Proposition 3.1, <cit.>) whether

    (η(B)=0, η(B')=0)≤(η(B)=0))(η(B')=0),

for disjoint B, B'.
We have
for arbitrary, measurable disjoint sets B, B'

    (η (B)=η (B')=0)
       =(τ =0)+∑_n=1^∞(τ =n) (X_1∉ B∪ B', … . X_n∉ B∪ B')
       = (τ =0)+∑_n=1^∞(τ =n)(1-F(B∪ B'))^n
       =P_τ (1-F(B∪ B')) =P_τ (1-(F(B)+F( B')) .







 Since τ has a binomial distribution with, say,  parameters n (number of trials), and p∈ (0,1) (success probability) then
P_τ (1-s)=(p(1-s)+(1-p))^n. It is easy to see by differentiaition that ϕ (s):=-log P_τ (1-s) is then an increasing and convex function such that ϕ(0)=0.
It is known that such a function is superadditive (see e.g. Bruckner and Ostrow (1963)), therefore ϕ(s+t)≥ϕ(s)+ϕ (t), and then
P_τ(1-(F(B)+F(B'))≤ P_τ (1-F(B))P_τ (1-F(B')). In this case, for disjoint B, B' we obtain

    (η (B)=η (B')=0)≤(η (B)=0)P(η (B')=0).

Therefore for this process we obtain (η_1(B)=0)≤exp(-η(B)).
  □

For mixed sampled point processes on general spaces we get the following comparison result.

Suppose η_1 is a  mixed sampled point process on  a complete, separable metric space 𝕏  defined by (<ref>) for which τ∈𝒮. Then

    η_1<_dcxη_2
 where  η_2 denotes a Poisson point process on 𝕏, with the intensity measure η_1.

From Theorem <ref> we know that η_1 is NA, and from Proposition <ref> we shall get the conclusion of the present proposition if we  show that for such processes η_1 (B)<_cxY, where Y denotes a random variable with Poisson distribution Po(η_1(B)). From the definition of mixed sampled point processes we know that η_1(B) is distributed as a random sum ∑_i=1^τ U_i, where (U_i, i≥ 1) is an iid sequence of Bernoulli, i.e. 0-1 valued  variables with  success probability F(B). Since  τ∈𝒮_n, from the definition of the log-concave ordering <_lc in <cit.> it follows that  τ <_lc Po((τ)), which  implies that τ<_cx Po((τ), see Theorem 1 in <cit.>. It follows that ∑_i=1^τ U_i<_cx∑_i=1^Po((τ))U_i, where Po((τ)) denotes a Poisson random variable which is independent of (U_i, i≥ 1) (see e.g. <cit.>, Corollary 4.5). From this we get η _1(B)<_cx Y, for Y described above since η_1(B)=(τ)(U_i) and ∑_i=1^Po((τ))U_i has Poisson distribution.
For arbitrary τ∈𝒮 we apply weak approximation by τ's in 𝒮_n, n≥ 1.
 □

From Proposition <ref> we get the following corollary.

Suppose that η is a simple mixed sampled point process on ^d, defined by (<ref>), for which τ∈𝒮. Then



  * (moment measures) (η (B_1)⋯η  (B_n)))≤ (η (B_1))⋯(η (B_n)),
 for all disjoint bounded Borel sets B_1,…, B_n, n≥ 1,

  * (void probabilities) (η (B)=0)≤exp(-η(B)), for all bounded Borel sets B, n≥ 1. 


Let λ be a Radon measure on a locally compact Polish space 𝕏. Let K be a locally trace-class positive contraction on L_2(𝕏, λ), and  η_K be  the determinantal point process generated by K. From  Proposition <ref> we obtain the following corollary.

 Suppose that η_K is the determinantal point process described above. Then 
    η_K<_dcxη_2
 where η_2 denotes a Poisson point process with intensity measure η_K.

Fix a bounded Borel set B. From  Theorem <ref> we know that η_K is NA so in order to get the conclusion of this corollary it is enough (see Proposition <ref>) to show that η _K(B)<_cx Y, where Y is a random variable with Po(η_K(B)) distribution.
From Hough et al. <cit.>, Proposition 9, we know that η_K (B) is distributed as a sum of independent Bernoulli random variables.  From the definition of the log-concave ordering <_lc in <cit.> it follows that  η_K(B) <_lc Y, where Y is described above, which in turn  implies that η_K (B)<_cxY, see Theorem 1 in <cit.>.
□

The above corollary  for the case of jointly observable sets and 𝕏 =^d was observed by Blaszczyszyn and Yogeshwaran in <cit.>, Proposition 5.3, using a different argument.


  





§ SOME APPLICATIONS
 

Let η be a point process on  a complete, separable metric space 𝕏.  Using Chebyshev's inequality we have

    (|η (B)-(η(B))| ≥ϵ) ≤ (η (B)/(ϵ ^2),
 for all bounded Borel sets  B and ϵ >0. 

Similarly, using Chernoffs bound,

    (η (B)- (η (B))≥ϵ)≤ e^-t((η(B))+ϵ)(e^tη (B)),

for any t,a >0, and the upper bounds can be replaced by the values taken from the dominating in <_dcx process. If η is determinantal or NA mixed sampled point processes, the corresponding Poisson processes can be used to obtain  upper bounds and  concentration inequalities using Proposition <ref>.

Similarly, we have for all bounded Borel sets  B and ϵ , t>0. 

    ((η(B))-η(B)≥ϵ)≤ e^t((η(B))-ϵ) (e^-tη(B)).


Using Corollary 2 from <cit.> we can get, from negative association of η, Kolmogorov type inequalities.

Suppose that η is a mixed sampled point process  on  a complete, separable metric space 𝕏, for which τ∈𝒮.  Then for any increasing sequence b_k, k≥ 1 of positive numbers, any collection of disjoint bounded Borel sets B_1,…, B_n∈𝒳, and ϵ >0



  * 
    (max_k≤ n|1/b_k∑_i=1^k(η (B_i)-(η (B_i))|≥ϵ)≤ 
    8ϵ ^-2∑_i=1^n(η (B_i))/b_i^2,


  * for any integer m<n

    (max_m≤ k≤ n|1/b_k∑_i=1^k(η (B_i)-(η (B_i))|≥ϵ)≤ 
    32ϵ ^-2(∑_i=m+1^n(η (B_i))/b_i^2+∑_i=1^m(η (B_i))/b_m^2).








plain
