РОССИЙСКАЯ АКАДЕМИЯ НАУК МАТЕМАТИЧЕСКИЙ ИНСТИТУТ ИМ. В.А. СТЕКЛОВА РОССИЙСКОЙ АКАДЕМИИ НАУК ВЫПУСК 3. Том 62 ИЮЛЬ АВГУСТ СЕНТЯБРЬ

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1 РОССИЙСКАЯ АКАДЕМИЯ НАУК МАТЕМАТИЧЕСКИЙ ИНСТИТУТ ИМ. В.А. СТЕКЛОВА РОССИЙСКОЙ АКАДЕМИИ НАУК ВЫПУСК 3 Том 62 ИЮЛЬ АВГУСТ СЕНТЯБРЬ Журнал основан А. Н. Колмогоровым в январе 956 года Выходит 4 раза в год МОСКВА 27

2 СОСТАВ РЕДКОЛЛЕГИИ: Ширяев А. Н. главный редактор), Боровков А. А., Булинский А. В., Ватутин В. А., Гущин А. А., Зубков А. М., Ибрагимов И. А., Кабанов Ю. М., Королев В. Ю., Лифшиц М. А., Синай Я. Г., Спокойный В. Г., Холево А. С. зам. главного редактора), Чибисов Д. М. зам. главного редактора), Яськов П. А. отв. секретарь) Журнал осуществляет свою деятельность под руководством Отделения математических наук РАН Адрес редакции: 999, Москва, ул. Губкина, 8, комн. 26; тел ) 94--8; Заведующая редакцией М. В. Хатунцева Учредители: Российская академия наук, Математический институт им. В. А. Стеклова Российской академии наук Издатель: Математический институт им. В. А. Стеклова Российской академии наук 999 Москва, ул. Губкина, 8 Отпечатано в ФГУП «Издательство «Наука» Типография «Наука»), 29 Москва, Шубинский пер., 6 Подписано к печати Формат бумаги 7 /6. Цифровая печать. Усл.печ.л. 7,. Усл.кр.-отт.,9 тыс. Бум.л. 6,6. Уч.-изд.л. 2,9. Тираж 2 экз. Зак Российская академия наук, 27 Математический институт им. В. А. Стеклова Российской академии наук, 27

3 A factorial moment distance and an application to the matching problem 67 c 27 г. AFENDRAS G., PAPADATOS N. A FACTORIAL MOMENT DISTANCE AND AN APPLICATION TO THE MATCHING PROBLEM В этой заметке мы вводим понятие расстояния факториальных моментов для неотрицательных целочисленных случайных величин и сравниваем его с расстоянием по вариации. Кроме этого, мы изучаем скорость сходимости в классической задаче о составлении пар и в обобщенной задаче о составлении пар. Ключевые слова и фразы: расстояние факториальных моментов, расстояние по вариации, задача о составлении пар. DOI: Introduction. Let π n = π n ),...,π n n)) be a random permutation of T n = {,...,n}, in the sense that π n is uniformly distributed over n! permutations of T n. A number j is a fixed point of π n if π n j) = j. Denote by Z n the total number of fixed points of π n, Z n = I{π n j) = j}, j= where I stands for the indicator function. The study of Z n corresponds to the famous matching problem, introduced by de Montmort in 78 [5]. Obviously, Z n can take the values,,...,n 2,n, and its exact distribution, using standard combinatorial arguments, is found to be PZ n = j) = j! n j k= ) k, j =,,...,n 2,n. It is obvious that Z n converges in law to Z, where Z is the standard Poisson distribution, Poi). Furthermore, the Poisson approximation is very accurate even for small n evidence of this may be found in []). Bounds on the error of the Poisson approximation in the matching problem, especially Department of Biostatistics, The State University of New York at Buffalo, Buffalo, NY, USA; gafendra@buffalo.edu Department of Mathematics, University of Athens, Panepistemiopolis, Greece; npapadat@math.uoa.gr ) This work was partially supported by the University of Athens Research Fund under grant 7/4/5637. Research has been co-financed by the European Union European Social Fund ESF) and Greek National Funds through the Operational Program «Education and Lifelong Learning» of the National Strategic Reference Framework NSRF) Research Funding Program: ARISTEIA, grant 4357.

4 68 Afendras G., Papadatos N. concerning the total variation distance, are also well known. Recall that the total variation distance of any two random variables X and X 2 is defined as d tv X,X 2 ) = sup PX A) PX 2 A), A BR) where BR) is the Borel σ-algebra of R. An appealing result is given by Diaconis [6], who proved that d tv Z n,z) 2 n /n!. This bound has been improved by DasGupta see [3], [4]): d tv Z n,z) 2 n n+)!..) It can be seen that d tv Z n,z) 2 n /n + )!, where a n b n means that lim n a n /b n ) = ; for a proof of a more general result see Theorem 3.2. Therefore, the bound.) is of the correct order. Consider now the sets of discrete random variables D n := {X: PX {,,...,n}) = }, D := {X: PX {,,...}) = }. Since the first n moments of Z n and Z are identical and Z n D n, Z D, one might think that inf {d tv X,Z)} d tv Z n,z) X D n However,.2) is not true. In fact, min {d tv X,Z)} = e X D n 2 n n+)!..2) j! e n+)!..3) Indeed, for any X,X 2 D with probability mass functions p and p 2, the total variation distance can be expressed as d tv X,X 2 ) = 2 p j) p 2 j) = p j) p 2 j)) +,.4) where x + = max{x,}. Thus, for any X D n so that p j) = for all j > n), we get d tv X,X 2 ) = p j) p 2 j)) + = p j) p 2 j)) p 2 j) = PX 2 > n),

5 A factorial moment distance and an application to the matching problem 69 with equality if and only if p j) p 2 j), j =,,...,n. Applying the preceding inequality to p 2 j) = PZ = j) = e /j! we get the equality in.3), and the minimum is attained by any random variable X D n with PX = j) e /j!, j =,,...,n. Furthermore, the well-known Cauchy remainder in the Taylor expansion reads as fx) f j) ) x j = j! n! x Applying.5) to fx) = e x we get the expression e j! = e e x y) n f n+) y)dy..5) ) = e j! n! y) n e y dy, and by the obvious inequalities < e y < +e )y, < y <, we have It follows that n+ < e n+)! < min X D n {d tv X,Z)} = e y) n e y dy < + e ). n+ n+2 j! < e + e ), n+)! n+2 and, therefore, min X Dn {d tv X,Z)} e /n+)!. In the present note we introduce and study a class of factorial moment distances, {d α,α > }. These metrics are designed to capture the discrepancy among discrete distributions with finite moment generating function in a neighborhood of zero and, in addition, they satisfy the desirable property min X Dn {d α X,Z)} = d α Z n,z). In Section 3 we study the rate of convergence in a generalized matching problem, and we present closed form expansions and sharp inequalities for the factorial moment distance and the variational distance. 2. The factorial moment distance. We start with the following observation: For the random variables Z and Z n, EZ) k = and EZ n ) k = I {k n}, k =,,..., 2.) where EX) k denotes the k-th order descending factorial moment of X for each x R, x) = and x) k = xx ) x k +), k =,2...). For a proof of a more general result see Lemma 3.. The factorial moment distance will be defined in a suitable sub-class of discrete random variables, as follows. For each t, we define X t) := {X D : there exists t > t such that P X +t ) < }, 2.2)

6 62 Afendras G., Papadatos N. where P X u) = Eu X is the probability generating function of X. Also, we define X ) := X t) = {X D : P X +t ) < for any t > }. 2.3) t [, ) Note that if X D n for some n, then X X t) for each t [, ]; therefore, each X t) is nonempty. For t < t 2, it is obvious that X t 2 ) X t ); that is, the family {X t), t } is decreasing in t. If X X ), then there exists a t > such that P X + t ) <, i.e., Ee θx <, where θ = ln + t ) >. Since X is nonnegative, Ee θx < implies that Ee ux < for all u θ,θ), which means that X has finite moment generating function at a neighborhood of zero. Therefore, X has finite moments of any order and its probability mass function is characterized by its moments; equivalently, X has finite descending factorial moment of any order and its probability mass function is characterized by these moments. This enables the following definition. Definition 2.. a) Let X,X 2 X ). For α >, we define the factorial moment distance of order α of X,X 2 by d α X,X 2 ) := k= α k EX ) k EX 2 ) k. 2.4) b) Let X X ) and {X n } n= X ). We say that X n converges in factorial moment distance of order α to X, in symbols X n α X, if d α X n,x) as n. One can easily check that the function d α : X ) X ) [, ] is a distance. Obviously, X n X implies that the moments of Xn converge to the α corresponding moments of X. Since every X X ) is characterized by its moments, it follows that d α convergence for any α > ) is stronger than the convergence in law; the later is equivalent to the convergence in total variation see Wang [8]). Of course, the converse is not true even in X ). For example, consider the random variable X with PX = ) =, and the sequence of random variables {X n } n=, where each X n has probability mass function p n j) = n, j =, n, j = n. It is obvious that {X,X,X 2,...} X ), and the total variation distance is d tv X n,x) = as n. n

7 A factorial moment distance and an application to the matching problem 62 Moreover, since EX) k = and EX n ) k = n ) k I{k n} for all k =,2,..., the d α distance does not converge to zero: d α X n,x) = k= α k n ) k I{k n} > α n )I{2 n} as n. 2 Remark 2.. Let X D n {X: X D n, X d = Z n }. It is obvious that EX) k = for all k > n, and we can find an index k {,...,n} such that EX) k. From 2.) and 2.4) we see that d α X,Z) > d α Z n,z). Hence, inf {d α X,Z)} = d α Z n,z) for all α >. X D n Proposition 2.. Let < α < α 2 and X,X 2 X ). Then a) d α X,X 2 ) d α2 X,X 2 ); b) we cannot find a constant C = Cα,α 2 ) < such that for all any random variables X,X 2 X ), d α X,X 2 ) Cd α2 X,X 2 ). Proof. a) is obvious. To see b), it suffices to consider X and X 2 with PX = ) = PX 2 = ) =. Then d α X,X 2 ) = for every α >. The proposition is proved. α From a) of the preceding proposition, X 2 α n X implies Xn X for every α < α 2. In the sequel we shall show that for any α 2, the inequality d tv X n,x) d α X n,x) holds true, provided {X,X,X 2,...} X ). To this end, we shall make use of the following «moment inversion» formula. Though this result is widely used see, e.g., [2, p.49]), we provide a short proof, specifying a condition under which the formula is valid. Lemma 2.. If X X ), then its probability mass function p can be written as pj) = ) k j k=j ) k EX) k, j =,, ) j Proof. By the assumption X X ), we can find a number t > such that E + t ) X = + t ) j pj) <. Since X is nonnegative, its probability generating function admits a Taylor expansion around with radius of convergence R +t > 2, i.e., Pu) = uj pj) R, u < R. It is well known that dk Pu) du k u= = EX) k, and since P admits a Taylor expansion around with radius of convergence R t >, we have Pu) = k= EX) k u ) k, u < R.

8 622 Afendras G., Papadatos N. Using the preceding expansion and the fact that R,+R ) we get pj) = j! d j du jpu) = u= j! = k=j ) k j j!k j)! EX) k, k=j u ) k j EX) k k j)! u= completing the proof. It should be noted that the condition X X t) for some t [,) is not sufficient for 2.5). As an example, consider the geometric random variable X with probability mass function pj) = 2 j, j =,,.... It is clear that X / X ), butx X t) for eacht [,). The factorial moments ofx are EX) k =, k =,,..., and the right-hand side of 2.5), k=j )k j k j), is a nonconvergent series. Theorem 2.. If X,X 2 X ), then d tv X,X 2 ) d α X,X 2 ) for each α 2. Proof. In view of Proposition 2., a), it is enough to prove the desired result for α = 2. By.4) and 2.5) we get d tv X,X 2 ) = 2 2 ) k j k=j k=j ) k j EX ) k EX 2 ) k ) ) k EX ) k EX 2 ) k. j Interchanging the order of summation according to Tonelli s theorem, we have d tv X,X 2 ) 2 = k k= k= 2 k ) k EX ) k EX 2 ) k j EX ) k EX 2 ) k. The proof is completed by the fact that EX ) = EX 2 ) =. Theorem 2. quantifies the fact that for any α 2, the d α convergence in X )) implies the convergence in total variation, and provides convenient bounds for the rate of the total variation convergence. However, we note that such convenient bounds do not hold for α < 2. In fact, for given α,2) and t, we cannot find a finite constant C = Cα,t) > such that d tv X,X 2 ) Cd α X,X 2 ) for all X,X 2 X t) see Remark 3.).

9 A factorial moment distance and an application to the matching problem An application to a generalized matching problem. Consider the classical matching problem where, now, we record only a proportion of the matches, due to a random censoring mechanism. The censoring mechanism decides independently to every individual match. Specifically, when a particular match occurs, the mechanism counts this match with probability λ, independently of the other matches, and ignores this match with probability λ, where < λ. We are now interested on the number Z n λ) of the counted matches. The case λ = corresponds to the classical matching problem, where all coincidences are recorded, so that Z n = Z n ). The probabilistic formulation is as follows: Let π n = π n ),...,π n n)) be a random permutation of {,...,n}, as in the introduction. Let also J λ),...,j n λ) be independent and identically distributed Bernoulliλ) random variables, independent of π n. The number Z n λ) of the recorded coincidences can be written as Z n λ) = J i λ)i{π n i) = i}. i= Let A i = {J i λ) = }, B i = {π n i) = i}, E i = A i B i, i =,...,n. Then Z n λ) presents the number of the events E s that will occur and, by standard combinatorial arguments, where PZ n λ) = j) = Pexactly j among E,...,E n occur) ) i = ) i j S i,n, j S,n =, S i,n = i=j k < <k i n PE k E ki ), Since the A s are independent of the B s, we have i =,...,n. PE k E ki ) = PA k A ki )PB k B ki ) = λ in i)!, n! so that S i,n = ) n λ in i)! = λi i n! i!, i =,,...,n. Therefore, the probability mass function of Z n λ) is given by p n;λ j) : = PZ n λ) = j) = j! = λj n j j! i= ) i j λ i i j)! i=j λ) i, j =,,...,n. 3.) i!

10 624 Afendras G., Papadatos N. The generalized matching distribution 3.) has been introduced by Niermann [7], who showed that p n;λ is a proper probability mass function for all λ, ]; however, Niermann did not give a probabilistic interpretation to the probability mass function p n;λ, and derived its properties analytically. Since n j λ) i lim = e λ n i! i= for any fixed j, we see that p n;λ converges pointwise to the probability mass function of Zλ), where Zλ) is a Poisson random variable with mean λ, Poiλ). Interestingly enough, the Poisson approximation is extremely accurate; numerical results are shown in Niermann s work. Also, Niermann proved that EZ n λ) = VarZ n λ) = λ for all n 2 and λ,]. In fact, the following general result shows that the first n moments of Z n λ) and Zλ) are identical, giving some light to the amazing accuracy of the Poisson approximation. Lemma 3.. For any λ,], E Z n λ) ) k = λk I{k n}, k =,2,.... Proof. For k > n the relation is obvious, since Z n λ) D n. For k =,...,n, E Z n λ) ) k = j=k n k = λ k λ j n j λ) i j k)! i! i= r= λ r n k) r λ) i r! i! i= n k = λ k p n k;λ r), and, sincep n k;λ is a probability mass function supported on{,,...,n k}, we get the desired result. For k = n, E Z n λ) ) n = n!p n;λn) = λ n, completing the proof. Corollary 3.. For any λ,] and α >, r= { )} inf dα X,Zλ) = dα Zn λ),zλ) ). X D n Thus, for λ,], Z n λ) minimizes the factorial moment distance from Zλ) over all random variables supported in a subset of {,,...,n}. Using 2.5) it is easily verified that Z n λ) is unique. Moreover, it is worth pointing out that for λ >, we cannot find a random variable X D n such that EX) k = λ k I{k n} for all k. Indeed, since D n X ) X ), assuming X D n and EX) k = λ k I{k n}, we get from 2.5) that PX = n ) = λn λ), n )! which implies that λ. Therefore, finding inf X Dn { dα X,Zλ) )} for λ > seems to be a rather difficult task.

11 A factorial moment distance and an application to the matching problem 625 We now evaluate some exact and asymptotic results for the factorial moment distance and the total variation distance between Z n λ) and Zλ) when λ,]. Theorem 3.. Fix α> and λ,] and let d α n) := d α Zn λ),zλ) ). Then, d α n) = αn λ n+ y) n e αλy dy. 3.2) n! Moreover, the following double inequality holds: + αλ n+2 + a 2 λ 2 n+2)n+3) < n+)! α n λ n+d αn) < + αλ n+2 + a 2 λ 2 e αλ n+2)n+3). Hence, as n, 3.3) d α n) αn λ n+ n+)! d α n) = αn λ n+ n+)! and, more precisely, + αλ )) n+2 +o. n 3.4) Proof. From the definition of d α and in view of.5) and Lemma 3., d α n) = α k=n+ αλ) k = α e αλ αλ) k ) k= = αn! αλ αλ x) n e x dx, and the substitution x = αλy leads to 3.2). Now 3.3) follows from the inequalities +αλy + 2 α2 λ 2 y 2 < e αλy < +αλy + 2 eαλ α 2 λ 2 y 2, < y <, while 3.4) is obvious from 3.3). Theorem 3. is proved. Theorems 2. and 3. give the following statement. Corollary 3.2. An upper bound for d tv Z n,z) is given by d tv Z n,z) < 2 n + 2 n+)! n+2 + 4e 2 ) n+2)n+3) 2 n n+)!. 3.5) The bound in 3.2) is of the correct order, and the same is true for the better result.), given by DasGupta [3], [4]. In contrast, the bound d tv Z n,z) 2 n /n!, given by Diaconis [6], is not asymptotically optimal, because 2 n /n + )! = o2 n /n!). Thus, it is of some interest to point out that the factorial distance d 2 provides an optimal rate upper bound for the variational distance in the matching problem. The situation is similar for the generalized matching distribution, as the following result shows.

12 626 Afendras G., Papadatos N. Theorem 3.2. For any λ,], let d tv n) := d tv Zn λ),zλ) ) be the variational distance between Z n λ) and Zλ). Then, d tv n) = λn+ 2n! Moreover, the following inequalities hold: d tv n) > 2n λ n+ 2λ n+)! n+2 2 n+ d tv n) < 2n λ n+ n+)! Hence, as n, 2λ n+2 2 n+ d tv n) 2n λ n+ n+)! d tv n) = 2n λ n+ n+)! [y n +2 y) n ]e λy dy. 3.6) )), ) + 4λ 2 n+3 n+2)n+3) 2 n+2 and, more precisely, 2λ )) n+2 +o. n )). 3.7) 3.8) Proof. Clearly, 3.8) is an immediate consequence of inequalities 3.7). Moreover, the inequalities 3.7) are obtained from 3.6) and the fact that λy < e λy < λy + 2 λ2 y 2, < y <. It remains to show 3.6). From.4) with p = p n;λ and p 2 the probability mass function of Poiλ), we get d tv n) = = λ j [ n j λ) i ] + e λ j! i! i= λ j [ ) n j λ + λ x) n j e dx] x, j! n j)! where the integral expansion is deduced by an application of.5) to the function fλ) = e λ. Thus, λ n k [ ) k λ ] + d tv n) = λ x) k e x dx n k)! k= = λ ) n e )λ x) x k λ n k dx. n! k Since keven keven ) n λ x) k λ n k = x k 2[ n +2λ x) n],

13 A factorial moment distance and an application to the matching problem 627 we obtain d tv n) = 2n! λ [x n +2λ x) n ]e x dx, and a final change of variables x = λy yields 3.6). Theorem 3.2 is proved. Remark 3.. Although the factorial moment distance d α dominates the variational distance when α 2, the situation for α,2) is completely different. To see this, assume that for some α,2) and t we can find a finite constant C = Cα, t) > such that d tv X,X 2 ) Cd α X,X 2 ) for all X,X 2 X t). 3.9) Obviously, Z and Z n, n =,2,..., lie in X ) X t). From Theorem 3.2 we know that n+)! lim n 2 n d tv Z n,z) =. On the other hand, from 3.3) with λ =, d α Z n,z) < αn + α n+)! n+2 + α 2 e α ), n+2)n+3) and, since α/2 <, this inequality contradicts 3.9): n+)! = lim n α C lim n 2 n d tv Z n,z) 2 ) n + α n+2 + α 2 e α n+2)n+3) )) =. Acknowledgment. The authors would like to thank an anonymous referee for the detail reading of the manuscript, leading to an improvement of its presentation. The first author would like to thank Dr. M. Markatou for financial support. REFERENCES. A. D. Barbour, L. Holst, S. Janson, Poisson approximation, Oxford Stud. Probab., 2, The Clarendon Press, Oxford Univ. Press, New York, 992, x+277 pp. 2. Ch. A. Charalambides, Combinatorial methods in discrete distributions, Wiley Ser. Probab. Stat., Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 25, xiv+45 pp. 3. A. DasGupta, The matching problem with random decks and the Poisson approximation, Technical report #99-, Purdue Univ., West Lafayette, IN, 999, 8 pp. 4. A. DasGupta, The matching, birthday and the strong birthday problem: a contemporary review, J. Statist. Plann. Inference, 3:-2 25),

14 628 Afendras G., Papadatos N. 5. P. R. de Montmort, Essay d analyse sur les jeux de hazard, Photographic reprint of the 2nd ed., Chelsea Publishing, New York, 98, xlii+46 pp.; st ed., 78; 2nd ed., Jacques Quillau, Paris, P. Diaconis, Application of the method of moments in probability and statistics, Moments in mathematics San Antonio, Tex., 987), Proc. Sympos. Appl. Math., 37, Amer. Math. Soc., Providence, RI, 987, S. Niermann, A generalization of the matching distribution, Statist. Papers, 4:2 999), Y. H. Wang, A compound Poisson convergence theorem, Ann. Probab., 9: 99), Поступила в редакцию 3.X.25 Исправленный вариант 7.III.26