COMPOUND POISSON APPROXIMATION FOR DISSOCIATED RANDOM VARIABLES VIA STEIN'S METHOD PETER EICHELSBACHER AND MALGORZATA ROOS Abstract. In the present paper we consider compound Poisson approximation by Stein's method for dissociated random variables. We present some applications to problems in system reliability such as k-runs, counting isolated vertices in the rectangle lattice on the torus as well as consecutive 2-systems, coloured graphs, birthday problems, connected s-systems and two dimensional consecutive-k-out-of-n systems. In particular our examples have the structure of an incomplete U -statistics. For nonnegative integer valued complete U -statistics improvements of the Poisson approximation results can not be expected in general. We mainly apply techniques from Barbour and Utev, who gave new bounds for the solutions of the Stein equation in compound Poisson approximation in two recent papers.
1. Introduction Let ? denote an arbitrary nite collection of indices, usually denoted by , and soPon. Let X be 0-1-valued, possibly dependent, random variables and let W = 2? X . If the X are weakly dependent and the value 1 occurs with small probability, Poisson approximation in total variation distance between the law of W , denoted by L(W ), and the Poisson distribution with parameter E W , denoted by Po(E W ), can be successfully established via the Stein-Chen method, which at the same time gives estimates of the approximation error (see the book of Barbour, Holst and Janson [7]). If the random variables X can take other positive integer values or if the dependence is stronger (clumps of 1's tend to occur), the compound Poisson approximation should provide better approximation. The importance of developing a compound Poisson approach has been discussed in Aldous [1]. In [3] Barbour, Chen and Loh introduced a Stein equation for compound Poisson approximation. Roos developed the local version of the basic method in [16] and in [15] the coupling approach. The theoretical results were successfully applied to many examples in reliability theory (see [4] and [5]) and to the problem of uniform m-spacings on the unit circle (see [13] and [14]). The solutions of the corresponding Stein equation may in general grow exponentially with the mean of clumps. For a special class of distributions in [3, Theorem 5] a better bound was given which 1991 Mathematics Subject Classi cation. 05C80, 60C05, 60F05, 62E17, 90B25. Key words and phrases. Stein's method, compound Poisson distribution, dissociated random variables, locally dependent random variables. 1
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P. EICHELSBACHER AND M. ROOS
is comparable in sharpness to the corresponding bound in the Poisson Stein-Chen method (apart from a logarithmic term in the numerator of the bound). Recently, Barbour and Utev [9] proved bounds for the solutions of the Stein equation with respect to the Kolmogorov distance which enabled them to carry out compound Poisson approximation in this distance with the same eciency as in the Poisson case. They assumed that the compound Poisson limit distribution has a fourth moment and that it satis es an aperiodicity condition. In [8] the same authors proved the counterparts in total variation distance under the assumption that the compound Poisson limit distribution is aperiodic of nite exponential moment, which is hardly restrictive. The contribution of [9] and [8] is fundamental. Barbour and Utev applied the local approach in the way developed by Roos [16] in a more general setting. Consequently, the results derived for indicator random variables in [15] and [16] became a special case of the more general results in [8]. The aim of the present paper is to improve some results stated in [4, 5, 16] by using the new bounds from [9] and [8]. We observe that the random variables in the examples considered up to now are actually dissociated ones. Moreover, we present Stein's method for compound Poisson approximation both in Kolmogorov distance and in total variation distance for this special class of random variables. In view of the theoretical work by Barbour and Utev we provide the following examples and applications: k-runs, isolated vertices in the rectangular lattice on the torus, consecutive 2-systems, coloured graphs, connected-s systems, and the two dimensional consecutive-k-out-of-n system. In [9, Section 5] the k-run example was already discussed brie y. In particular all the examples have the structure of an incomplete U -statistics. After having calculated the bound for compound Poisson approximation for U -statistics, we see that improvements of the Poisson approximation results on U -statistics (see [6]) cannot be expected in general. In the examples we are able to obtain estimates with the right behaviour when the mean of the clump tends to in nity. As a consequence of the good bounds in [9] and [8] no unwanted logarithmic factor and no unwanted e , where is the mean of the clumps, come into play. Most of our examples deal with models from reliability theory. In this theory evaluation is an important and integral feature of planing, designing and operating engineering systems. In these models the independent items constitute the system. In the de nition of the system there are special critical combinations consisting of underlying independent items. A critical combination is said to fail if all items in it are failed. A failure of a critical combination is dangerous for the system. Let W be the random variable which counts the number of failed critical combinations in the system. The system is said to fail if there are at least m occurrences of failed critical combinations, where m 1 is a xed integer. The reliability of the system is P(W m ? 1). Thus estimates of P(W m ? 1) are estimates of the reliability
COMPOUND POISSON APPROXIMATION FOR DISSOCIATED RANDOM VARIABLES
3
of the system. Since it is not always possible to compute the reliability of a system exactly, it is reasonable to look instead for good approximations. Because the critical combinations may overlap, the indicators of their failure can be dependent, in spite of the independence of individual underlying items. Frequently, the indicators of the failure of critical combinations are dissociated or locally dependent random variables. It is a setting where a compound Poisson approximation is promising. In our paper we denote 0-1-valued random variables X by I . In Section 2, we state known theoretical results about Stein's method and give the theorem for dissociated random variables as well as locally dependent ones. In Section 3 examples are presented. 2. Elaboration
2.1. Stein's method, compound Poisson distribution and approximation. P We denote by CP (; ) the compound Poisson distribution of Nj=1 Xj , where (Xj ; j 1) are nonnegative valued, independent and identically distributed variables with distribution (a probability measure on (0; 1)) and are independent of N , distributed according to the usual Poisson distribution with mean . We brie y summarize Stein's method for this class of distributions. If W is a nonnegative integer valued random variable satisfying X E ii g(W
i1
+ i) ?
Wg(W )
"0 M0(g) + "1 M1 (g)
(2.1)
! R and for some small "0 and "1 , where M0(g) = sup jg(j )j; M1 (g) = sup jg(j + 1) ? g(j )j;
for all bounded g :
N
j 1
j 1
then it follows that
dTV (L(W ); CP (; )) "0 H0 (; ) + "1 H1 (; );
(2.2)
P
where = i1 i , i = i = (that means that the measure has the form = P i1 i i , where i denotes the Dirac measure concentrated on i 2 N ), dTV (P; Q) denotes the total variation distance between two probability measures P and Q on Z+ (dTV (P; Q) = supAZ+fjP (A) ? Q(A)jg) and Hl (; ) = supAZ+ Ml (gA), l = 0; 1, with gA being the solution of the Stein equation X
i1
ii g(w + i) ? wg(w) = 1A (w) ? CP (; )fAg; w 0; A Z+:
(2.3)
The method works as follows: if we can bound the left hand side of (2.1) with suitably small "0 and "1 and the constants Hl (; ) are not too large, the distribution of W is close to an appropriate compound Poisson distribution. The bounds on Hl (; ) will depend on and and not on the given random variable W itself.
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P. EICHELSBACHER AND M. ROOS
There may not be any good bounds for the Hl (; ) (see discussion in the introduction of [8]), but for the terms Hl(a) (; ) = sup Ml (gA( + a)); l = 0; 1; AZ+
and by inequality (4.4) in [9] (2.2) can be replaced by
2H0(a) (; ) m1 (1 ? c1 ) (2.4) (a) (; ) ? 1 2 m H 2 0 + P W 2 (1 + c1 )m1 1 + m (1 ? c ) ; 1 1 P where c1 2 (0; 1), a = c1 m1 and ml denotes i1 il i for l 1. If (z) = P i i1 i z has radius of convergence R() > 1 (meaning that we assume that has a nite exponential moment), 2 and
dTV (L(W ); CP (; )) "0 H0(a) (; ) + "1
H1(a) (; ) +
Condition 2.5 (Aperiodicity). for any 0 < : %( ) > 0, where % ( ) =P min(%1 ( ); 21 %2 ( ); 1) with %k ( ) =P inf %k (), k = 1; 2, and %1() = 1 ? i1 i cos i and %2() = 1 ? m11 i1 ii cos i, then there exist constants Cl (), l = 0; 1; 2, given explicitly in terms of (see [8, (1.20)-(1.28)]) such that for any a C2 ()m1 + 1 H0(a) (; ) p1 C0 (); H1(a) (; ) 1 C1 (): (2.6) Note that C2 () < 1, so that there are feasible choices for c1 in (2.4). Note that p C2 () = 1 ? % (0)=4 with 0 = m2 =m4 . In [9], where the Kolmogorov distance is considered, the Hl (; ) are replaced by Jl (; ) = sup Ml (g[0;m]); l = 0; 1; m0
which play the same role as the Hl (; ) if one considers the smaller family of sets A = [0; m], m 1, on the right hand side of the Stein equation (2.3). Then the estimate for approximation in terms of Kolmogorov distance is the analogue of (2.2): ? dK L(W ); CP (; ) "0 J0 (; ) + "1 J1 (; ); where dK (L(W ); CP (; )) = supt2R jP(W t) ? CP (; )f(?1; t)gj. The quantities Jl(a) (; ), l = 0; 1, are obtained in the same way as the Hl(a) (; ) but with g[0;m] instead of gA, A Z+. Under weaker assumptions, namely Condition 2.5 and the niteness of the fourth moment, Theorem 4.3 in [9] gives a bound in the Kolmogorov distance: (a) (; ) 4 J ( a ) ( a ) 0 dK (L(W ); CP (; )) "0 J0 (; ) + "1 J1 (; ) + m % ( ) 1 0 (a) (; ) (2.7) ? 4 m J + P W m1 (1 ? %(0)=4 1 + m2 %0 ( ) ; 1
0
COMPOUND POISSON APPROXIMATION FOR DISSOCIATED RANDOM VARIABLES
5
p
where 0 = m2 =m4 , % is as de ned in Condition 2.5 and J0(a) (; ) p1 D0 (); J1(a) (; ) 1 D1 (); where the constants Dl (), l = 0; 1, are given in (4.8) and (4.9) in [9]. 2.2. Dependence structure. We recall the de nition of dissociation introduced by McGinley and Sibson in [12] which can be seen as the most natural and useful generalisation of the assumption of independence. Suppose that ? is a collection of k-subsets = f 1; : : : ; k g of f1; 2; : : : ; ng. Then a family (X ; 2 ?) of nonnegative integer valued random variables is said to be dissociated if the subsets of random variables (X ; 2 A) and (X ; 2 B ) are independent whenever ([ 2A ) \ ([ 2B ) = ;. If k = 1, the notions of dissociated and independent coincide. The most important setting is the one in which X = (Y 1 ; : : : ; Y k ) for some nonnegative integer valued symmetric functions ,Pwhere Y1 ; : : : ; Yn are independent random elements of some space X . Here W = 2? X is called an incomplete U -statistics with kernel class ( ) 2? . If ? is the set of all k-subsets of f1; : : : ; ng, W is called a U -statistics. Many statistics commonly used are in fact members of this class (see for example [10] and [11]). Now we state a result on bounds for "0 and "1 in the case of nonnegative valued dissociated random variables (X ; 2 ?). We require some further notation. For ? xed, we de ne the dissociated partition for each 2 ?: 0 w ? = f g [ ?vs [ ? [ ? and set X X X Z = X ; U = X ; W = X : 2?vs
2?0
2?w
Here ?vs contains indices of those X which strongly in uence X . Depending on the choice of this region we de ne for each vs ?0 = f 2 ? : 2= ?vs [ f g; \ 6= ; for some 2 ? [ f gg and ?w = f 2 ? : \ = ; for all 2 ?vs [ f gg: Note that, by de nition of dissociation, the random variables with indices in the region ?w do not in uence X and Z . Moreover, the de nition of ?vs provide us with enough freedom of choice in de ning the clumps. In particular the random variables with indices in ?0 may be dependent on X . Frequently, a \maximal" ?vs is chosen, as for example, in the local dependence below.
Theorem 2.8. If (X ; 2 ?) are nonnegative valued dissociated random variables, 0 w ?vs , ? and ? are as de ned above, then we can take "0 = 0
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P. EICHELSBACHER AND M. ROOS
and
"1 =
X
2?
E (X ) E (X
+ Z + U ) + E (X U ) :
(2.9)
Proof. We only have to choose the special dissociated partition ?b = ?0 in Lemma 1.8 in [8]. Since the X are dissociated, W is independent of Z and X and therefore, 1 = 2 = 3 = 0 and 4 = "1 . The theorem follows immediately. Remark 2.10. We are able to prove the same result in case of indicator random variables by applying the dissociated partition to Theorem 2 in [16]. We have chosen, however, to use the more general formulation for nonnegative random variables of Lemma 1.8 in [8]. Remark 2.11. With the de nition of ?s and ?i in [7, Chapter 2.3], taking ?vs = ;, 0 s w i ? = ? and ? = ? we obtain the same result as Theorem 2.N in [7] (where the (X ) are indicator random variables). For Poisson approximation to be good, the -clump must be negligible, and ?vs = ; is an adequate choice. Remark 2.12. There are cases when one is able to nd some probabilistic structure in ?0 in the case of dissociated random variables. In such situation it might be advantageous to use the coupling approach instead of the local one, see Remark 2.1.5 in [7].
Let ? be an arbitrary index set (for applications it is normally sucient to consider P nite sets; in the in nite case one has to ensure that sums 2? are convergent). A family of nonnegative integer valued random variables (X ; 2 ?) is said to be locally dependent if for each 2 ?, there exist A B ? with 2 A such that X is independent of (X ; 2 Ac ) and (X ; 2 A ) is independent of (X ; 2 B c ). It is well known (see [3, Section 4]) that m-dependence is a special case of local dependence. Moreover, nite dependence (see De nition in [3]) is a special case of local dependence. If ? is a collection of k-subsets than a locally dependent family (X ; 2 ?) is also a dissociated family by de nition: for a xed we choose A = f 2 ? : \ 6= ;g and B = A [ f 2 ? : 2= A and \ 6= ; for some 2 A g. Note that, by de nition, 2 A . Actually a dissociated family is also a locally dependent family. For a locally dependent family we obtain
Corollary 2.13. If (X ; 2 ?) are nonnegative integer valued locally dependent random variables, then we can take
"0 = 0 and
"1 =
X X
2? 2B
E X E X
+
X
X
2? 2B nA
E X E X :
(2.14)
COMPOUND POISSON APPROXIMATION FOR DISSOCIATED RANDOM VARIABLES
7
b Proof. For any 2 ? we choose the partition f g [ ?vs = A , ? = B n A and ?w = B c in Theorem 2.8. By the de nition of local dependence X and X with 2 B n A are independent.
Remark 2.15. Note that "1 in (2.8) and in (2.14) has the form X
2?
X
2f g[?vs [?0
E X E X
+
X
2?0
E (X X )
:
It is easily seen that (2.8) takes the form (2.14) whenever the random variables with indices in ?0 are independent of X (X is independent of U ). Remark 2.16. Corollary 2.13 is an improvement of Theorem 8 in [3]: we are dealing with the case of nonnegative integer valued random variables. In [3] a slightly P P weaker bound "1 = 2 2? 2B E X E X is proven. Remark 2.17. Note that the approximating distribution CP (; ) can be calculated P explicitly: i = i1 2? E (X I [X + Z = i]), i 1 and
=
X E
2?
X : X + Z
Remark 2.18. One can consider an approximation by a CP (; 0) distribution instead of the naturally emerging distribution CP (; ). Theorem 1.10 in [8] states that for any choice of 0 : (a) (; 0 ) 2 H ( a ) ( a ) 0 0 0 0 0 0 dTV (L(W ); CP (; )) "0 H0 (; ) + "1 H1 (; ) + m0 (1 ? c ) 1 1 (2.19) ( a ) 0 0 ? ) + P W 21 (1 + c1 )m01 1 + 2mm20H(10 ?(; c1 ) 1 with "00 = jm1 ? m01 j + "0 and "01 = m1 dW ( ; 0 ) + "1 ; here is used to denote a size-biased distribution i = ii =m1 and dW denotes the Wasserstein L1 metric on probability measures over R . One can choose the distribution 0 to be of the following form:
01 = 1 +
1 X
i=l+1
i i ;
(2.20) 0j = j for j = 2; : : : ; l 0j = 0 for j l + 1: InPsuch a case m01 = m 1 . Consequently jm1 ? m01 j = 0 and m1 dW ( ; 0 ) = P 1 0 1 i=l+1 i(i ? 1)i = i=l+1 i(i ? 1)i . This choice of was implicitly suggested in Theorem 3 of [16]. In her notation i = 0i for i 1. ?
Remark 2.21. There are di erent approaches to bound the truncation term P W (1 ? ")E W ; e.g., one might use the Chebyshev-inequality. For a further re nement,
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P. EICHELSBACHER AND M. ROOS
Bernstein-type inequalities may be applied. In our examples we mostly apply Janson's inequality which works only for indicator random variables built up from independent ones. For the sake of completeness we state the inequality (see [7, Theorem 2.S]). Consider a collection of independent random indicator variables (Ji ; i 2 Q) and a nite family of subsets (Q( ); 2 ? ) of the index set Q and Q P de ne I = i2Q( ) Ji and W = 2? I . Partition ? P into ?+ [ ?i , where P ?+ = f 6= : Q( ) \ Q( ) 6= ;g and de ne = EW1 2? 2?+ E I I . Then for any 0 " 1 Janson obtained ? 1 2 (2.22) P W (1 ? ")E W exp ? 2(1 + ) " E W : 3. Applications 3.1. k-runs. Consider the problem of k-runs of 1's in a series of independent identically distributed Bernoulli random variables. When approximating the distribution of the number of overlapping k-runs of heads, in [2, Section 4.1.2] Arratia, Goldstein and Gordon give a bound on the total variation distance from a compound Poisson approximation of order nkp2k (1 ? p) and Roos improves this in [14] to order kpk log(npk ) when p < 1=2. Theorem 4.3 in [9] leads to a bound on the Kolmogorov distance from the same compound Poisson distribution of order kpk + exp(?cnpk ) for some c > 0, as was already mentioned in [9]. We want to show the calculations for the bound on the Kolmogorov distance as well as on the total variation distance. Suppose n satis es n > 4k ? 3. Consider a collection of independent Bernoulli random variables (Jj ; j 2 f1; : : : ; ng) with P(Jj = 1) = p. To avoid edge e ects assume that indices of the form i + nl are identi ed with i whenever 1 i n and l 2 Z. Using this circular convention de ne the family (Q( ); 2 ?) of subsets of index set f1; : : : ; ng consisting of k consecutive indices. These are the critical combinations in the reliability theory. We identify each index set consisting of k consecutiveQindices with the rst P index on its left + k ? 1 hand side. Thus j?j = n. We de ne I = j= Jj and W = 2? I . Then E I = pk and E W = npk . The random variables (I ; 2 ?) are dissociated. 0 w Choose ?vs = f ? (k ? 1); : : : ; ? 1; +1; : : : ; + k ? 1g and ? and ? as in the dissociated partition. Then applying Corollary 2.13 we obtain "1 = (6k ? 5)n(pk )2. Furthermore we have to calculate the limiting measure CP (; ): Lemma 3.3.4 in [14] states 8 > npk pi?1 (1 ? p)2 for i = 1; : : : ; k ? 1, > < k i = > npi (2pi?1 (1 ? p) + (2k ? i ? 2)pi?1(1 ? p)2) for i = k; : : : ; 2k ? 2, > : npk 2k?2 for i = 2k ? 1. 2k?1 p (3.1) P2k?1 Thus we can calculate = i=1 i , the i 's, and the moments of ; for example, m1 = npk =. It is obvious that m4 is nite and Condition 2.5 is ful lled, since
COMPOUND POISSON APPROXIMATION FOR DISSOCIATED RANDOM VARIABLES
9
1 = 1 = = npk (1 ? p)2= > 0 (see [9, Remark before Lemma 4.1]). So we (a) (; ), l = 0; 1, can apply (2.7): therefore, we have to determine the order of J l ? and to calculate the remaining contribution P W m1 (1 ? %(0)=4) using an appropriate inequality. Since = O(n pk ), the moments m1 and m2 are of order O(1) and the same holds for 0 and %(0). Therefore, the order of J0(a) (; ) is O(1=(n pk )1=2) and J1(a) (; ) is of order O(1=(n pk )). Since "1 = O(k n(pk )2 ), the second summand in (2.7) has order O(k pk ). To calculate the order of the remaining contribution (the third summand in (2.7)), we apply Janson's inequality (2.22): With ?+ = ?vs we obtain k in the k-run example = (4k ? 4)p and get ? 1 2 P W m1 (1 ? % (0 )=4) exp ?m1 (% (0 )=4) 2(1 + (4k ? 4)pk ) : Summarizing we nd that in Kolmogorov distance the right hand side of (2.7) has the order ? O kpk + exp(?const. npk ) (3.2) when the mean number of runs n pk satis es the condition n pk 1. The bound in total variation is only available in the case 2, which roughly means that n pk 2. Of course, the assumption R() > 1 is ful lled. We have to check the in uence on the order of the constants Cl () for l = 0; 1. We see that this order is dependent only on the order of the moments and therefore in our example H0(a) (; ) is of order O(1=(n pk )1=2) and H1(a) (; ) of order O(1=(n pk )). The main signi cance is that the constants Cl () exist. They are not optimal. We use the fact that C2 () < 1 to choose c1 suitably and therefore, we can apply ? (2.4) and bound P W 21 (1 + c1 )m1 again by Janson's inequality and obtain the same order (3.2) of the bound in total variation distance, possibly with less favorable constants, and under the condition that n pk 2. Note that apart from this one, there is no other restriction on the value p. We see that this bound is quite applicable when E W = npk ! 1 which is not the case for the bounds in [2] and [14]. The importance of k-runs for the reliability theory called there consecutive-k-out-of-n systems has been discussed in [4]. 3.2. Isolated vertices in the rectangular lattice on the torus. Consider a rectangular lattice on the torus with n vertices and N = 2n edges (see [16, Section 3]). Note that it is a 4-regular graph without triangles. Assume that the edges can be deleted independently of each other with a constant probability 1 ? p = q. De ne a family (Q( ); 2 ?) of subsets of the vertex set f1; 2; : : : ; ng consisting of a vertex and its four neighbours. These are the critical combinations of the system. We identify each index set consisting of 5 vertices with the center. Thus j?j = n P and we de ne I = I [v is isolated ] and W = 2? I . With the above de nitions P(v is isolated ) = q 4 and E W = nq 4 . The random variable W counts the number
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P. EICHELSBACHER AND M. ROOS
of isolated vertices in the graph. The idea is to give a bound for reliability in the system. By construction and the choice of ?vs the random variables (I ; 2 ?) vs are dissociated. Choose ? = f 6= : v and v are neighboursg and ?0 and ?w as in the dissociated partition. Applying Corollary 2.13 we obtain "1 = 21nq8 and by [16, Section 3] n 4 i = i i ? 1 q3i+1(1 ? q3)5?i ; for i = 1; : : : ; 5: Since 1 = nq4 (1 ? q3 )4= > 0 and m4 is nite, = O(nq4), and the moments mi are of order O(1), the second summand in (2.7) has order O(q4 ). We can apply Janson's inequality (2.22) with = 12q4 and obtain in Kolmogorov distance that the upper bound is of order ?
O q4 + exp(?const. nq4 )
(3.3)
provided nq4 1. Since R() > 1, the same order holds in total variation distance when nq4 2. It is easily seen that our approximating distribution has the form CP (; ) = P5 according to i=1 i Zi , where the Zi are independent random variables distributed ? Po(i ). It follows from [16, Section 3] that in such a case an order O (log+ 2nq4)q4 of the upper bound for q3 1=5 could be expected when the local approach of [16] would be applied directly. The restrictive condition q3 1=5 is needed there to assure that ii & 0. For any q another compound Poisson approximation was suggested in [16]. Her CP ( ) can be expressed in our notation as CP (; 0) for a particular choice of 0 suggested in (2.20) in Remark 2.18 with l = 2. In such a case 5 X
i=3
?
i(i ? 1) i = 12 n q10 1 + O(q3 ) : ?
By (2.19) in Remark 2.18 we obtain for dTV L(W ); CP (; 0) the same order as in (3.3). Note that our result improves the? one of [16] as shewas only able to prove ? 0 that the order of the upper bound on dTV L(W ); CP (; ) is O (log+ 2 n q4) q4 . Our condition n q4 2 is hardly restrictive. Moreover, our bound (3.3) becomes rapidly smaller than the earlier estimate when n q4 increases. Note that for Poisson approximation, when n q4 ! const., the approximation is only accurate to order O(n?3=4). Similarly to [16, Section 3] our result allows us to approximate the reliability of the system by the distributions CP (; ) or CP (; 0 ) if q4 is small. 3.3. Consecutive 2-systems, coloured graphs, birthday problems. Given a graph G(V; E ) with n vertices and N edges, where the vertices are subject to fail. Take ? = E and associate with each vertex vj a Bernoulli random variable Yj where ? P(Yj = 0) = P the item in vj fails (is down) = qj
COMPOUND POISSON APPROXIMATION FOR DISSOCIATED RANDOM VARIABLES 11
for j = 1; : : : ; n. We consider the case when Y1 ; : : : ; Yn are independent. For 2 ? set I = I (Yi = Yj = 0) = I (the two items ofPthe edge fail) with = fi; j g. Edges are critical combinations here. Let W = 2? I be the number of all edges in the graph, in which both vertices are failed. Of course the family (I ; 2 ?) is dissociated. De ne the strong neighbourhood of dependence by ?vs = f 2 ? nf g : j \ j =Q1g consisting of edges with one vertex of and one other vertex. We write = i2 qi for the probability that both vertices of the edge fail. Then we obtain from Corollary 2.13
"1 =
X
2?
+
X Y
2?vs j 2
qj + 2
X Y
2?0 j 2
qj :
If D(l) denotes the degree of the vertex vl in the graph and D = max1ln D(l), we 0 2 obtain j?vs j 2(D ? 1) and j? jP 2(D ? 1) . With qmax = max1in qi we get the 2 bound "1 (4D2 ? 6D +3)qmax 2? (see [5]). If all the failure probabilities are equal to q, it follows that "1 (4D2 ? 6D +3)q4 N . In the notation of [5] we denote for each = fi; j g by S2 ( ) the set of vertices s such that fs; ig and fs; j g are in E and by S1 ( ) the set of vertices s such that there is only one edge connecting s P P with . So Z can be written as 2?vs I + 2?vs : j \S2( )j=1 I =: : j \S1 ( )j=1 1 2 Z + Z and we have X ? i = 1i P Z 1 + 2Z 2 = i ? 1jI = 1 : (3.4) 2? Since Z 1 is distributed according to a Binomial distribution with parameters jS1 ( )j and q and Z 2 to a Binomial distribution with parameters jS2 ( )j and q and Z 1 and Z 2 are independent, has order O(Nq2 ) and therefore the second summand in (2.7) has order O(poly(D) q2), where poly(D) denotes a positive polynomial of degree 2 in D. We can apply Janson's inequality (2.22) with = 2(D ? 1)2q2 and thus the ? order of the bound in total variation is O poly(D)q2 + exp(? const. N poly(D)q2 ) which holds for Nq2 2. This again is an improvement of existing results (see [5, Section 2] and [14, Theorem 3.F]). In [5] the underlying graph to be a complete d-regular tree was??considered. If d denotes the degree of any non-leaf vertex, the + 2 2 2 upper bound is O 1 + log (Nq ) q (5d ? 5d ? 1) , so for constant E W = Nq2 the order of approximation is O(q2). Again we get this order even in the case when E W ! 1. Provided Nq2 1 we obtain the same order of the bound in Kolmogorov's distance. The consecutive 2-system has another interpretation: colour the vertices of the xed graph G(V; E ) at random, independently of each other and with the same probability distribution over the colours. Let Xi denote the colour of vertex vi and P set pr = P(Xi = r) and take I = I (Xi = Xj ) with = fi; j g. Then W = 2? I is the number of edges connecting two vertices of the same colour. Else, the vertices of the graph denote people, the colours their birthdays, and the edges pairs of people who are acquainted. Now W counts the number of pairs of people who are
12
P. EICHELSBACHER AND M. ROOS
2 acquainted and have the same birthday. Here = = P(I = 1) = m r=1 pr . So we are in the situation of the consecutive 2-system with equal failure probability . The bound in total variation distance is of order P
O poly(D)
m X r=1
p2 + exp(?const. r
poly(D)
m X r=1
p2 ) r
;
which is an improvement of [7, Theorem 5.G and 5.H]. The parameters i can be P calculated explicitly by (3.4). The Kolmogorov distance has the same order if N mr=1 p2r 1. 3.4. Connected s-systems. Observe that we can get improved bounds also for the so called connected s-systems which are studied in [5] and [14]. Here one assumes that critical combinations are a subset of all the connected subgraphs with s vertices in the underlying graph. Moreover, one assumes that the system fails if there are at least m (m 1) subgraphs on s vertices with all items failed. Here ? denotes the set of all critical combinations, an element can be denoted as (k1; : : : ; ks), the indices of s vertices which span a particular connected subgraph. To get a family of dissociated random variables, we have to consider the family (Q( ); 2 ?) of subsets of the index set f1; 2; : : : ; ng consisting of s indices of s vertices which span a particular connected subgraph. De ne I = I (all the items of the s items of the subgraph fail). De ning ?vs we have the choice for the extend of overlap R: for 1 R s ? 1 we can de ne ?vs (3.5) (R) = f 2 ? n f g : j \ j = r; r = R; : : : ; s ? 1g: If the failure probabilities are equal q, one obtains for j?jqs?R+1 2 a bound of the order ? O const. qs?R+1 + exp(?const.j?jqs?R+1 ) ; where the constants depend on the graph, especially on jf : j \ j = rgj, r = 1; : : : ; s ? 1 and jf 2 ?0 (R) : j \ j = 0; j \ j 6= ; for some 2 ?vs (R)gj. For s ? details see [14]. The Kolmogorov distance has the same order if j?jq R+1 1. 3.5. Two dimensional consecutive-k-out-of-n system. This example is a particular connected-k2 example. We will discuss it brie y. The system consists of n2 components placed on a square grid of size n and it fails if and only if there exist at least m possibly overlapping square sub-grids of size k (1 < k < n) with all k2 components failed. The square subgrids of size k are the critical combinations. ? = f(r; s) : 1 r; s n?k+1g and for 2 ? denote by A = Ars the kk sub-grid with left lower-most component (r; s). We do not place it on a torus. Consequently, we have to take boundaries into account. De ne a family of subsets (Q( ); 2 ?) of the set f1; 2; : : : ; n2g consisting of the points of a k k sub-grid and identify each index with the left lower-mostPcomponent. De ne I = I (all items in A are failed) for each 2 ? and W = 2? I . To de ne the neighbourhood of very strong
COMPOUND POISSON APPROXIMATION FOR DISSOCIATED RANDOM VARIABLES 13
dependence, one has to decide how large R has to be (see (3.5)).. First of all we take R = k2 ? k. There are (n ? k +1)2 possible positions of the k k sub-grid, 4 positions 2 vs 2 where j?vs (k ? k)j = 2 (corners), 4(n ? k ? 1) positions where j? (k ? k)j = 3 2 (border), and (n ? k ? 1)2 more where j?vs (k ? k)j = 4. If2 the failure probability 2 is q for all items in the grid, we obtain E W = (n ? k + 1)2 qk and j?vs (k ? k)j 4 2 0 2 2 and j?vs (k ? k)j + j? (k ? k)j (2k + 1) ? 1. The calculations similar to those in [5, Section 4] and in [14, Section 3.7] lead to the following bound for (2.9): kX ?1 kX ?1 2 kX ?2 2 2 2 k k ? uv "1 (4k + 12k ? 3)q + 4 q + qk ?kv ; u=1 v=1 v=1 ? 2k?1 which is of order O q (n ? k + 1)2qk2 . Moreover, the i 's can be computed as
(n ? k + 1)2qk2
follows:
k2 q i = i 4 i ?2 1 qk(i?1) (1 ? qk )3?i + 4(n ? k ? 1) i ?3 1 qk(i?1) (1 ? qk )4?i 4 k ( i ? 1) k 5 ? i 2 (1 ? q ) + (n ? k ? 1) i ? 1 q
?
for i = 1; : : : ; 5. Again 1 > 0 and R() > 1. We obtain = O (n ? k + 1)2 qk2 . in Janson's inequality can be bounded by (2k + 1)2 ? 2 and we obtain in total variation distance that the upper bound is of order ? O q2k?1 + exp(?const. (n ? k + 1)2qk2 ) ; whenever (n ? k + 1)2 qk2 2. In Kolmogorov distance we obtain the same order if (n ? k + 1)2 ?qk2 1. The argument presented here provide an improvement 2 + 2 k ? 1 2 k on the order O q log ((n ? k + 1) q ) obtained in [14, Theorem 3.G]. The Poisson approximation is given in [5, Section 4]. Using the Poisson local approach we obtain an improvement on the order O(qk ) by a factor qk?1 . Moreover, the bound is applicable when E W ! 1 which is not the case for the bounds in [5] and in [14]. The extension of the above result to the case of unequal probabilities of failure of items can be carried out according to Remark 4.2 in [5]. The reliability of the system and bounds on the reliability of the system for m = 1 can be calculated as in [5]. One might be interested in taking R =? 1 (maximal clump) instead of R = k2 ? k. It would imply "1 to be of order O (n ? k + 1)2 qk2 qk2 . Thus the order of the upper bound would be O(qk2 ). However, in such a case the computation of the parameters i would be very laborious. 3.6. U -statistics. In our last example we study incomplete and complete U -statistics. We observe that for general nonnegative integer valued complete U -statistics the bound in total variation between the law of the statistic and an appropriate compound Poisson distribution introduced in Section 1 has the same order as the one in the Poisson case, provided we choose the neighbourhood of dependence
14
P. EICHELSBACHER AND M. ROOS
similar to (3.5). This is obvious, since the neighbourhoods are indeed not local. Suppose that ? is a collection of k-subsets = f 1; : : : ; k g of f1; 2; : : : ; ng and de ne X = (Y 1 ; : : : ; Y k ) for some nonnegative integer valued symmetric (in its arguments) functions , where Y1 ; : : : ; Yn are independent random elements P of some space X . Let W = 2? X . Given R 1, similarly to (3.5), the neighbourhoods of dependence can be de ned by ?vs = f : R j \ j k ? 1g: Thus, vs ?0 = f : j \ j 1 for some 2 f g [ ?vs (R)g n ff g [ ? (R)g:
Here R denotes the minimum of the number of items in common for and when 2 ?vs (R). There are two extreme cases of the above formulae. If one takes R = k, then ?vs = ; and one considers the Poisson case. The other extreme, R = 1, indicates that ?vs is the set of all 2 ? n f g which overlap with . The choice of R a ects the order of the term E (X U ) which is frequently that of the largest order. Now let us calculate "1 for W depending on R. Denote E (X ) = . For 1 < R < k the terms of (2.9) are of the following form: X
2?0 (R)
E (X X ) =
RX ?1
r=1 2?0 (R) j \ j=r
and X
2?vs (R)[?0 (R)
X
=
kX ?1
E (X X ) +
X
r=1 2?vs (R) j \ j=r
X
2?0 (R);j \ j=0
j \ j6=0; 2?vs (R)
+
X
2?0 (R);j \ j=0
:
j \ j6=0; 2?vs (R)
Consider the case = for all 2 ?. Let nr ( ) = jf : j \ j = rgj, r = 1; : : : ; k ? 1, m( ; R) = jf 2 ?0 (R) : j \ j = 0; j \ j 6= ; for some 2 ?vs (R)gj and r = E (X X ) for all pairs ; satisfying j \ j = r. Then we obtain RX ?1 kX ?1 2 2 "1 = + nr ( )r + m( ; R) + nr ( ) + m( ; R) 2 : r=1 r=1 2? X
In the language of reliability theory a similar formula was obtained in [14, Section 3.5]. In fact, the class considered there was a special class of incomplete U -statistics. ?n ?k?n?k Let us consider the case j?j = k . We obtain jf : j \ j = rgj = r k?r , which is of order O(nk?r ). The size of the?strong dependence region is of order O(nk?R ). ? ? P Moreover, it is obvious that j?0 j = nk ? kr=?R1 kr nk??kr , so ?w = ;. For simplicity we assume that the constants Cl (; ) in the total variation distance bound do not in uence the order of the bound (as we have observed in all our other examples)
COMPOUND POISSON APPROXIMATION FOR DISSOCIATED RANDOM VARIABLES 15
and that satis es the assumption R() > 1, 2 and ful lls Condition 2.5. Moreover, apply Chebychev's inequality to establish that P
?
? W 1=2(1 + c1 )m1 m2 1=2(1 ? c1 ) m1 ?1 = O(1=)
(use Bernstein-type or Hoe ding-type inequalities for U -statistics for more re nement, see for example [10, Section 2.1, 8.1]). Therefore, for complete U -statistics we nd that R ?1 ?1 X X ? ? k R n r r nk?r : dTV L(W ); CP (; ) const.() k + n ( ) = O n + r r=1 r=1
?1 r nk?r (see [7, Section The order for Poisson approximation is O nk?1 + kr=1 2.3, (3.2)]). Since in the compound Poisson case the order is in uenced by nk , the result is not an improvement in general. Consider the following example: k = 2 and (x; y) = (x; y) = x y and Y1 is an indicator with P(Y1 = 1) = p. Then the order of Poisson approximation is p2 n + p n, whereas the order of compound Poisson approximation is p n. In the incomplete case our result is applicable, provided the quantities nr ( ) and m( ; R) are uniformly bounded as ? grows. The smaller R is the better is the order of approximation. Examples 3.1 to 3.5 above are incomplete U -statistics. The choice of ?, the collection of k-subsets, represents the local character shared by all our examples. In the k-run example the k-subsets are k consecutive indices, thus j?j = n and the local clump has size k. In the connected s-system we have a xed set of all critical combinations. These observations lead to a natural question: \how incomplete" should a U -statistics be to get better results when approximating by a compound Poisson distribution rather than a Poisson one. ?
P
References 1. D. J. Aldous, Probability approximations via the Poisson clumping heuristic, Springer-Verlag, New York, 1989. 2. R. Arratia, L. Goldstein, and L. Gordon, Poisson approximation and the Chen-Stein method, Stat. Science 5 (1990), 403{434. 3. A. D. Barbour, L. H. Chen, and W.-L. Loh, Compound Poisson approximation for nonnegative random variables via Stein's method, Ann. Probab. 20 (1992), 1843{1866. 4. A. D. Barbour, O. Chryssaphinou, and M. Roos, Compound Poisson approximation in reliability theory, IEEE Transactions on Reliability 44 (1995), 398{402. , Compound Poisson approximation in systems reliability, Naval Research Logistics 5. 43 (1996), 251{264. 6. A. D. Barbour and G. K. Eagleson, Poisson convergence for dissociated statistics, Journal of the Royal statistical society B 46 (1984), 397{402. 7. A. D. Barbour, L. Holst, and S. Janson, Poisson Approximation, Oxford University Press, Oxford, 1992. 8. A. D. Barbour and S. Utev, Compound Poisson approximation in total variation, preprint, 1997.
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9. 10. 11. 12. 13. 14. 15. 16.
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, Solving the Stein equation in compound Poisson approximation, Advances in Applied Probability (1998), (to appear). Yu. V. Borovskikh, U -Statistics in Banach Spaces, VSP, Utrecht, 1996. A. J. Lee, U-Statistics: Theory and Practice, Marcel Dekker, New York, 1990. W. G. McGinley and R. Sibson, Dissociated random variables, Mathematical Proceedings of the Cambridge philosophical society 77 (1975), 185{188. M. Roos, Compound Poisson approximations for the numbers of extreme spacings, Adv. Appl. Probab. 25 (1993), 847{874. , Stein-Chen method for compound Poisson approximation, Ph.D. thesis, University of Zurich, Switzerland, 1993. , Stein-Chen method for compound Poisson approximation: the coupling approach, Probab. Theory and Math. Stat. 4 (1994), 645{660. , Stein's method for compound Poisson approximation: the local approach, Ann. Appl. Probab. 4 (1994), no. 4, 1177{1187. (Peter Eichelsbacher) Fakultat fur Mathematik, Universitat Bielefeld, Postfach, 10-
0131, D-33501 Bielefeld, Germany
E-mail address, Peter Eichelsbacher:
[email protected]
(Malgorzata Roos) Biostatistik, ISPM, Universitat Zurich, Sumatrastrasse 30, CH-8006
Zurich, Schweiz
E-mail address, Malgorzata Roos:
[email protected]
