ELECTRONIC COMMUNICATIONS in PROBABILITY

In this note we show how the properties of association and negative association can be combined with Stein's method for compound Poisson approximation.  Applications include $k$--runs in iid Bernoulli trials, an urn model with urns of limited capacity and extremes of random variables.


Introduction and main results
In recent years Stein's method has proved to be an effective technique for probability approximation, often yielding explicit error bounds and working well in the presence of dependence.Stein's method may be applied in a wide variety of settings: in this note we consider compound Poisson approximation.See [1] and references therein for an introduction to Stein's method for compound Poisson approximation, and Stein's technique more generally.
Our purpose in this note is to show how assumptions of association or negative association may be combined with Stein's method in a compound Poisson approximation setting.This provides an analogue of the idea of a 'monotone coupling' in Stein's method for Poisson approximation.See, for example, [3,Section 2.1].In a Poisson approximation setting, the existence of a monotone coupling means that error bounds obtained via Stein's method are often simpler to state and easier to evaluate in practice than they would otherwise be.The same is true if we make assumptions of association or negative association in a compound Poisson approximation setting, as will be demonstrated in the applications of Section 2.
This work is organised as follows.The remainder of Section 1 is devoted to introducing the notation and ideas we will need, and stating our main results.Applications of these results are discussed in Section 2, with the proof of our main theorems being given in Section 3.
Throughout this work, we assume that X 1 , . . ., X n are (possibly dependent) nonnegative integer valued random variables.We consider compound Poisson approximation for their sum W = X 1 + • • • + X n .Recall that X 1 , . . ., X n are said to be associated if for all non-decreasing functions f and g.On the other hand, X 1 , . . ., X n are said to be negatively associated if for all non-decreasing functions f and g, and all Γ 1 , Γ 2 ⊆ {1, . . ., n} with Γ 1 ∩ Γ 2 = ∅.
See [7] and references therein for further discussion of these properties.
We will say that U ∼ CP(λ, µ) has a compound Poisson distribution if where the Y i are positive integer valued random variables with P (Y i = j) = µ j for each i, N ∼ Po(λ) has a Poisson distribution and each of these random variables are independent.We write µ = (µ 1 , µ 2 , . ..) and λ j = λµ j .Thus, λ = j≥1 λ j and µ j = λ −1 λ j .We follow, for example, [13] and for each i we consider a 'neighbourhood of dependence' consisting of those indices j ∈ {1, . . ., i − 1, i + 1, . . ., n} for which X i and X j are strongly dependent, in some sense.These neighbourhoods of dependence are chosen to suit the problem at hand.We will denote by J (i) the neighbourhood of dependence of X i .We then define As in the work of Barbour et al. [1] or Roos [13], we then define our approximating compound Poisson random variable U by setting for each j ≥ 1.
In this note we consider approximation in total variation distance, although our results may also be applied to approximation in other probability metrics.The total variation distance between random variables W and U supported on Z + is defined by

Stein's method for compound Poisson approximation
Before stating our main results, we give a brief outline of Stein's method for compound Poisson approximation.For further details, see [1].Letting U ∼ CP(λ, µ), we begin by finding for each A ⊆ Z + a function f A such that f A (0) = 0 and for each x ∈ Z + .Note that the functions f A will depend on the choices of λ and µ.Given such functions f A , by replacing x by W and taking expectation in the above we may then write In proving our main results in Section 3, we proceed by bounding the right-hand side of (1.2).In doing so, it will be essential to have bounds on where we use ∆ to denote the forward difference operator, so that ∆f (x) = f (x + 1) − f (x) for any function f .Good bounds on H(λ, µ) can be hard to find.Barbour et al. [1,Theorem 4] show that and, furthermore, that we cannot do better than this in general.This bound is useful only for very small λ.However, under particular conditions we can find much better bounds.For example, if we define and assume that then Barbour et al. [1,Theorem 5] show that log + denoting the positive part of the natural logarithm.Alternatively, under the as- Barbour and Xia [5,Theorem 2.5] show that (1.6)

Main results
We are now in a position to state our main results.The proofs of Theorems 1.1 and 1.2 are deferred until Section 3.
Theorem 1.1.Suppose that X 1 , . . ., X n are negatively associated, and define λ j as in (1.1) for each j ≥ 1.Then Theorem 1.2.Suppose that X 1 , . . ., X n are associated, and define λ j as in (1.1) for each j ≥ 1.Then We have stated our results for approximation in total variation distance, although they may easily be adapted for use with other probability metrics.For example, we may wish to use the Kolmogorov distance, defined for non-negative integer valued random variables by

Compound Poisson approximation with association or negative association
In this case, the bounds of Theorems 1.1 and 1.2 continue to hold, with H(λ, µ) replaced by Section 3] and [6, Section 1] for further bounds on K(λ, µ).
Remark 1.4.Boutsikas and Koutras [7] also discuss compound Poisson approximation for a sum of associated or negatively associated random variables.The bounds we establish in this note offer a greater flexibility in the choice of approximating compound Poisson distribution than their results: we are able to choose the sets J (i) to suit the problem at hand.The approximating distribution chosen by [7] is the same as that obtained by us when setting J (i) = ∅ for each i.Furthermore, our bounds have the advantage of including the so-called 'Stein factor' H(λ, µ), giving good bounds when this Stein factor is small.

Independent summands
Suppose that X 1 , . . ., X n are independent, non-negative integer valued random variables.In line with the definitions of Section 1, we choose Since X 1 , . . ., X n are independent, they are also negatively associated.We apply Theorem 1.1 to immediately obtain the bound This bound has also been obtained in the independent case by various other authors, for example [13,Corollary 1].See also Lemma 8 of [7].We do not concern ourselves with evaluating bounds on the Stein factor H(λ, µ) for this example, since better bounds in compound Poisson approximation for a sum of independent random variables are available by means other than Stein's method.See also Section 4.1 of [1].

k-runs
We now turn our attention to compound Poisson approximation for the number of runs in iid Bernoulli trials.This problem is discussed by Barbour et al. [2, Section 2.1].We show that our Theorem 1.2 can be used to improve the bounds of their work.
We let ξ 1 , . . ., ξ n be iid Bernoulli random variables with P (ξ , where all indices are written modulo n to avoid edge effects.Thus, W = X 1 + • • • + X n counts the number of k-runs in our Bernoulli trials.Our random variables X 1 , . . ., X n are associated, so Theorem 1.2 may be applied in this case.
Following [2], we choose We have that for each i, EX i = p k , EZ i = 2(k − 1)p k and that EW = np k .We also have, ECP 18 (2013), paper 30.
Combining these expressions, Theorem 1.2 easily yields the bound This improves upon the bound of [2], who show that the above total variation distance is bounded by H(λ, µ)(6k − 5)np 2k .

An urn model with overflow
We consider the following model of [8].Suppose that n balls are distributed into m urns, with each ball equally likely to be assigned to any urn.We fix some k ≥ 2, and assume that each of our m urns can hold at most k − 1 balls.If a ball is assigned to an urn which is already full, that ball is placed in an additional 'overflow urn' of unlimited capacity.We consider a compound Poisson approximation for W , the number of balls allocated to the overflow urn.Boutsikas and Koutras [8, pg 278] give a bound for such an approximation in Kolmogorov distance.Here, we consider approximation in the stronger total variation distance.
We write W = X 1 + • • • + X m , where, for each 1 ≤ j ≤ m, X j = (S j − k + 1)I(S j ≥ k) and S j is the number of balls allocated to urn j.Our random variables X 1 , . . ., X m are negatively associated, so we may apply Theorem 1.1.See [8, pg 278].

Extremes
Suppose that ξ 1 , . . ., ξ n is a stationary sequence of negatively associated random variables.For simplicity, we will assume that each of the ξ i have the same distribution function, with F (x) = P (ξ i ≤ x) for each i = 1, . . ., n.Note that throughout this section we will treat all indices modulo n.
We fix some a 1 , . . ., a n ∈ R and let X i = I(ξ i > a i ).Since ξ 1 , . . ., ξ n are negatively associated, X 1 , . . ., X n also have this property.See [11, page 288].We define W = n i=1 X i , so that W counts the number of the ξ i exceeding the threshold a i .In line with our earlier work, we consider the approximation of W by a compound Poisson distribution CP(λ, µ), with λ j given by (1.1) for j ≥ 1.For further discussion on the choice and calculation of the λ j see [10] and references therein. 2 , we use the definition of Z i and immediately obtain from Theorem 1.1 that ECP 18 (2013), paper 30.
where λ, µ and Λ are as above.Similar bounds apply if we wish to consider the probability that exactly k of our random variables ξ 1 , . . ., ξ n exceed a.
We consider now an example in which we may compare (2.4) with a result of [10].
Example 2.2.Assume, in the setting of this section, that the random variables ξ 1 , . . ., ξ n are also m-dependent.That is, we may choose J (i) = {i − m, . . ., i − 1, i + 1, . . ., i + m} for i = 1, . . ., n and we have that for j = i, ξ i is independent of ξ j for all j ∈ J (i).We will assume, for simplicity, that a i = a ∈ R for each i = 1, . . ., n.In this case, we have and where this final inequality uses the negative association property.We thus obtain the bound (2.5) We compare this to the bound of Proposition 2.1 of [10], which states that if ξ 1 , . . ., ξ n is a stationary m-dependent (but not necessarily negatively associated) sequence of random variables then where the same approximating compound Poisson distribution is used as in (2.5).When ξ 1 , . . ., ξ n are negatively associated in addition to being m-dependent, our bound (2.5) slightly improves upon this result.Note also that we do not need a condition of mdependence in order to apply the more general bound (2.4).
For further details in this example, including comments on the Stein factor H(λ, µ), we refer the reader to [10].
Finally, we note that it is straightforward to construct examples with m-dependence where (2.4) does better than (2.5).For example, if m = 1 take pairs of random variables (ξ i , η i ) for i = 1, . . ., n/2 such that for each i, ξ i and η i are independent of all random variables except each other.Now take W = n/2 i=1 {I(ξ i > a) + I(η i > a)}.If our negative association condition holds, an argument analogous to that used to derive (2.5) gives a bound smaller than that result.
In the setting of this section, if we suppose that our random variables ξ 1 , . . ., ξ n are associated rather than negatively associated, we may employ Theorem 1.2 rather than than Theorem 1.1 to obtain a bound analogous to (2.4).One application of such a bound would be to extremes of moving average processes.We illustrate this by considering a special case in the following example.
With our choice of J (i) we have that EZ i = 2λ/n for each i.Further straightforward calculations show that Combining the above expressions, Theorem 1.2 gives (2.6) Similar calculations to the above also give us the parameters of our approximating compound Poisson distribution.Using (1.1) we have that

Proofs of Theorems 1.1 and 1.2
Our proofs are based on techniques developed by [12] and [9].We begin by defining the size-biased distribution.For any non-negative integer valued random variable X with EX > 0, we let X have the X-size-biased distribution.That is, X satisfies E[Xg(X)] = (EX)Eg(X ) , (3.1) for all functions g for which the above expectations exist.We have that P (X = j) = (EX) −1 jP (X = j) , for all j ≥ 1.Throughout this section we let λ j be defined by (1.1) for each j ≥ 1 and Y be a random variable, independent of all else, with P (Y = j) = µ j for j ≥ 1, where µ j = λ −1 λ j and λ = j≥1 λ j .We note that with these choices of parameters, we have that i≥1 iλ i = EW , Recalling that f A (0) = 0, we write f A (j) = ∆f A (0) + • • • + ∆f A (j − 1).Substituting this in the above and interchanging the order of summation, we have that Then, using (1.2), we obtain To proceed further with our proofs, we need the following lemmas.Lemma 3.1 treats the case of negative association, while Lemma 3.2 considers the case of association.The proofs of these lemmas are given in Sections 3.1 and 3.2, respectively, before which we show how they are used to prove Theorems 1.1 and 1.2.Note that, since we are assuming Y is independent of all other random variables, the size-biased version Y used in Lemmas 3.1 and 3.2 is, in particular, independent of W . Lemma 3.1.Suppose that X 1 , . . ., X n are negatively associated.Then, for all nondecreasing functions g, Eg(W ) ≤ Eg(Y + W ) .
Lemma 3.2.Suppose that X 1 , . . ., X n are associated and let V be a random index, independent of all else, chosen according to the distribution P (V = i) = (EW ) −1 EX i for i = 1, . . ., n.Then, for all non-decreasing functions g, To complete the proof of Theorem 1.1, we combine (3.3) with Lemma 3.1.Since I(j > k) is non-decreasing in j we get that, in the negatively associated case, Using (3.1) and (3.2), we have that and Combining these expressions with (3.4), we easily obtain that Employing the definition of V then gives us the bound of Theorem 1.2.To establish our Theorems 1.1 and 1.2, it therefore remains only to prove Lemmas 3.1 and 3.2.

Proof of Lemma 3.1
Our proof is based upon that of Lemma 3.1 of [12].We begin by recalling the definition of W i from Section 1 and observing that, under the conditions of Theorem 1.1, E[X i I(X i + Z i = j)g(j for any j ∈ Z + , i = 1, . . ., n and g non-decreasing.This can be seen by conditioning on X i + Z i and using the negative association property.
To prove Lemma 3.1, we note that it is enough to show that under our negative association assumption for all non-decreasing functions g.Using (3.1) and (3.2), this is equivalent to the statement of our lemma.We have that where we use (3.6) for this inequality.Now, since g is non-decreasing and W i ≤ W This completes the proof of Lemma 3.1.
Suppose that a i = a ∈ R for each i.We note that W = 0 if and only if ξ 1 , . . ., ξ n are all at most a.Hence, an immediate corollary of (2.4) is the bound Page 6/12 ecp.ejpecp.orgCompound Poisson approximation with association or negative association Remark 2.1.