A generalized allocation scheme


 Abner Hubbard
 1 years ago
 Views:
Transcription
1 Annales Mathematicae et Informaticae 39 (202) pp Proceedings of the Conference on Stochastic Models and their Applications Faculty of Informatics, University of Debrecen, Debrecen, Hungary, August 22 24, 20 A generalized allocation scheme István Fazekas a, Bettina Porvázsnyik b a University of Debrecen, Faculty of Informatics Debrecen, Hungary b University of Debrecen, Faculty of Science and Technology Debrecen, Hungary Dedicated to Mátyás Arató on his eightieth birthday Abstract The generalized allocation scheme was introduced by V.F. Kolchin []. Let ξ, ξ 2,..., ξ be independent identically distributed nonnegative integer valued nondegenerate random variables. Consider the random variables η,..., η with joint distribution Pη = k,..., η = k = P ξ = k,..., ξ = k ξi = n. Let ξ i have Poisson distribution, then (η,..., η ) has polynomial distribution. Therefore η = k,..., η = k means that the contents of the boxes are k,..., k after allocating n balls into boxes during the usual allocation procedure. Our aim is to study random variables η,..., η with joint distribution Pη = k,..., η = k = P ξ = k,..., ξ = k ξi n. It can be considered as a general allocation scheme when we place at least n balls into boxes. Let µ n denote the number of cases when η i = r. That is µ n is the number of boxes containing r balls. We shall prove limit theorems for Pµ n = k. Moreover, we shall consider the asymptotic behaviour of Pmax i η i r and Pmin i η i r. Keywords: generalized allocation scheme, conditional probability, law of large numbers, central limit theorem, Poisson distribution. MSC: 60C05, 60F05 Supported by the Hungarian Scientific Research Fund under Grant o. OTKA T07928/2009. Supported by the TÁMOP4.2.2/B0/ project. The project is cofinanced by the European Union and the European Social Fund. 57
2 58 I. Fazekas, B. Porvázsnyik. Introduction The usual allocation scheme is the following. Let n balls be placed successively and independently into boxes. At any allocation the ball can fall into each box with probability /. This model was widely studied. See the early papers Weiss [3], Rényi [2], Békéssy [] and the monograph KolchinSevast yanovchistyakov [8]. See also ChuprunovFazekas [2] for certain recent results. A generalization of the usual allocation scheme was introduced by V.F. Kolchin (see the monographs of Kolchin [7] and Pavlov [0]). Let η, η 2,..., η be nonnegative integervalued random variables. In Kolchin s generalized allocation scheme the joint distribution of η, η 2,..., η can be represented as Pη = k,..., η = k = P ξ = k,..., ξ = k ξ i = n, (.) where ξ, ξ 2,..., ξ are independent identically distributed nonnegative integer valued nondegenerate random variables and k, k 2,..., k are arbitrary nonnegative integers, k + k k = n. This scheme contains the usual allocation procedure, certain random forests, and several other models (see the monographs of Kolchin [7] and Pavlov [0]). The usual allocation scheme is obtained as follows. Let ξ i have Poisson distribution, i.e. P(ξ i = k) = λk k! e λ, k = 0,,.... Then ( ) n n! Pξ = k,..., ξ = k ξ + + ξ = n = k!... k! if k + + k = n. That is (η,..., η ) has polynomial distribution. ow η = k,..., η = k means that the cell contents are k,..., k after allocating n particles into cells considering the usual allocation procedure. The connection of the random forest and the generalized allocation scheme is the following. Let T n, denote the set of forests containing labelled roots and n labelled nonroot vertices. By Cayley s theorem, T n, has (n + ) n elements. Consider uniform distribution on T n,. Let η i denote the number of the nonroot vertices of the ith tree. Then Pη = k,..., η = k = n! (k + ) k... (k + ) k k!... k! ( + n) n. ow let ξ i have Borel distribution (see [5], [9]) P(ξ i = k) = λk (+k) k k! e (k+)λ, k = 0,,..., λ > 0. Then Pξ = k,..., ξ = k ξ + + ξ = n = n! (k + ) k... (k + ) k k!... k! ( + n) n if k + + k = n. See [7, 2, 0]. Therefore η,..., η satisfy (.).
3 A generalized allocation scheme 59 We can say that in the generalized allocation scheme we place n balls into boxes. In the framework of the generalized allocation scheme several asymptotic results can be obtained. Let µ r be the number of the random variables η, η 2,..., η being equal to r (r = 0,,..., n). Observe that µ r = µ rn = µ n = I η i =r (.2) can be considered as the number of boxes containing r balls. Here I A is the indicator of the set A, i.e. I A (x) = if x A and I A (x) = 0 if x A. (µ r, µ rn, and µ n are just different notations for the same quantity.) Limit results for µ r can be obtained in the following way. Let ξ 0 be a random variable with the same distribution as ξ. Let p r = Pξ 0 = r and Eξ 0 = a. Introduce notation S = ξ i. Denote by ξ (r) 0 a random variable with distribution Pξ (r) 0 = k = Pξ 0 = k ξ 0 r. (.3) The expectation and the second moment of ξ (r) 0 are the following a r = Eξ (r) 0 = a rp ( ) 2 r and E ξ (r) Eξ 0 = 0 2 r 2 p r. Let ξ (r) p r p,..., ξ(r) be independent copies of r ξ (r) 0. Let S(r) = ξ(r) i. Denote by C k the binomial coefficient Ck = ( ) k. V.F. Kolchin proved in [7] the following lemma. Lemma.. Let µ n and ξ(r) 0 be defined by (.2) and (.3), respectively. Then Pµ n = k = Cp k k PS(r) k k r( p r ) = n kr. (.4) PS = n Using this representation, normal and Poisson limit theorems were obtained (see [7], and [0]). In [4] a modification of the generalized allocation scheme was studied, that is in (.) the condition was changed for ξ i n. In this paper we introduce another scheme, i.e. we use in (.) condition of the form ξ i n. It can be considered as a general allocation scheme when we place at least n balls into boxes. Let µ n denote the number of cases when η i = r. That is µ n is the number of boxes containing r balls. We shall prove limit theorems for Pµ n = k. Moreover, we shall consider the asymptotic behaviour of Pmax i η i r and Pmin i η i r. In Section 2 Pµ n = k is studied. In sections 3 and 4 Pmax i η i r and Pmin i η i r are considered, respectively. 2. Another generalized allocation scheme Let ξ, ξ 2,..., ξ be independent identically distributed nonnegative integervalued nondegenerate random variables. Consider random variables η, η 2,..., η with
4 60 I. Fazekas, B. Porvázsnyik joint distribution Pη = k,..., η = k = P ξ = k,..., ξ = k ξ i n. (2.) In this case, we place at least n balls into boxes. Example 2.. Let ξ i have Poisson distribution, i.e. 0,,.... Then Pξ = k,..., ξ = k ξ + + ξ n = P(ξ i = k) = λk k! e λ, k = / k!... k! λk0 k=n (λ) k k! (2.2) if k + + k = k 0 n. ow, we place η (random number) balls into boxes. Assume that η n. Let η i denote the number of balls in the ith box. Then Pη = k,..., η = k = = P(η = k,..., η = k η = i)p(η = i) i=n k 0! k!... k! ( ) k0 P(η = k 0 ), (2.3) if k + + k = k 0 n. If we choose the a priori distribution of η as Poisson distribution truncated from below, i.e P(η = i) = (λ)i e λ/ i! k=n (λ) k e λ, i = n, n +,..., k! then we obtain (2.2). That is our scheme (2.) with ξ i having Poisson distribution describes the usual allocation when the number of balls is given by a truncated Poisson distribution. Let µ r = µ rn = µ n = I η i=r be the number of the boxes containing r balls. Then we have the following analogue of Kolchin s formula (.4) for our model. Recall that ξ (r) 0 is defined by (.3). Theorem 2.2. For all k = 0,, 2,..., we have Pµ n = k = Cp k k PS(r) k k r( p r ) n kr. (2.4) PS n Proof. (2.4) can be proved by a certain modification of the proof of Lemma.. Let A (r) k be the event that exactly k of the random variables ξ,..., ξ are equal to r. By (2.), we have Pµ n = k = P(A (r) k S n) = P(A(r) k, S n). P(S n)
5 A generalized allocation scheme 6 Furthermore, P(A (r) k, S n) = P(S n A (r) k )P(A(r) k ) = C k p k r( p r ) k P(S n ξ r,..., ξ k r, ξ k+ = r,..., ξ = r) = C k p k r( p r ) k P(S (r) k n kr). Here we have used that ξ,..., ξ are independent random variables and the event can occur C k different ways, moreover A (r) k P(S n A (r) k ) = P(S n, A (r) k ) P(A (r) k ) = P(ξ + + ξ n, ξ r,..., ξ k r, ξ k+ = r,..., ξ = r) P(ξ r,..., ξ k r, ξ k+ = r,..., ξ = r) = P(ξ + + ξ k n kr, ξ r,..., ξ k r, ξ k+ = r,..., ξ = r) P(ξ r,..., ξ k r, ξ k+ = r,..., ξ = r) = P(ξ + + ξ k n kr, ξ r,..., ξ k r). P(ξ r,..., ξ k r) The proofs of our limit theorems are based on representation (2.4). First we consider two theorems with normal limiting distribution. Let α n = n. Theorem 2.3. Let Eξ 0 = a be finite, Eξ (r) 0 = a r, s 2 r = p r ( p r ). () Let d < a. Then, uniformly for α n < d, we have Pµ n = k = 2πsr e u2 /2 ( + o()), (2.5) as n, and u = k pr s r belongs to an arbitrary bounded fixed interval. /2 (2) Suppose that a r < a. Let a r < d < d < a. If k belongs to a bounded interval, then we have lim Pµ n = k = 0. (2.6) n,, d <α n <d Proof. () By the MoivreLaplace Theorem we have C k p k r( p r ) k = as uniformly if u = k pr s r /2 s 2 r = p r ( p r ). lim P n,, α n <d 2πsr e u2 /2 ( + o()), (2.7) belongs to a bounded fixed interval, where As α n < d < a, applying Kolmogorov s law of large numbers, we obtain k lim P n,, α n <d ξ (r) i n kr ξ i n =. =, (2.8)
6 62 I. Fazekas, B. Porvázsnyik ow (2.4), (2.7) and (2.8) imply (2.5). (2) Let d < α n < d. By Kolmogorov s law of large numbers, we have lim P n,, d <α n <d k ξ (r) i n kr We obtain (2.6) from (2.4), if we apply (2.7) and (2.9). = 0. (2.9) Remark 2.4. It is easy to see that a < a r, a > a r and a = a r if and only if a > r, a < r and a = r, respectively. Let Φ denote the standard normal distribution function. Recall that a = Eξ 0, a r = Eξ (r) 0 and s 2 r = p r ( p r ). Theorem 2.5. Suppose that Eξ0 2 <. Denote by σ 2 the variance of ξ 0 and by σr 2 the variance of ξ (r) 0. Assume 0 < σ2, σr 2 <. Let C <. Then, as n, such that (α n a) C, we have ( ) a r C+us r Φ pr prσ r Pµ n = k = e u2 /2 2πsr Φ ( ) + o() C, (2.0) σ for u = k pr s r /2 belonging to any bounded fixed interval. Proof. As σ 2 = D 2 (ξ 0 ) < and σr 2 = D 2 (ξ (r) 0 ) <, by the central limit theorem, we obtain P ξ i n = P ξ i a (αn a) σ σ ( ) (αn a) = Φ + o(), (2.) σ and similarly we obtain P k ξ (r) i n kr = Φ a r (αn a) + us r p r + o(). (2.2) pr σ r Using (2.), (2.2), and (2.7), relation (2.4) implies the desired result. Using large deviation theorems we can describe the relation between µ n and µ n. Let X, X 2,... be independent identically distributed nonnegative nondegenerate random variables with lattice distribution (assume that the span of the
7 A generalized allocation scheme 63 distribution of X is ). Suppose that Cramér s condition is satisfied, that is Ee λ0x < for some λ 0 > 0. Let Z = X + + X. Introduce notation M(h) = Ee hx, a(h) = (ln(m(h))), v 2 (h) = a (h). As X is nondegenerate, therefore a (h) > 0, so a( ) is strictly increasing. We have the following lemma from []. Lemma 2.6. Let x be an integer number and let h = a ( x ). Then, as, we have P(Z = x) = P(Z x) = ( ( v(h) 2π M (h)e hx + O v(h) 2π M (h)e hx ( e h ) )), ( ( )) + O uniformly for x, with a(ε) x a(λ 0 ε), where ε is an arbitrary small positive number. In particular Introduce notation P(Z x) P(Z = x) = ( e h ) ( + o()). (2.3) L(λ) = Ee λξ0, L r (λ) = Ee λξ(r) 0 where we assume that there exist positive constants λ 0 > 0 and λ (r) 0 > 0 such that Ee λ0ξ0 < and Ee λ(r) 0 ξ(r) 0 < (Cramér s condition). Let m(λ) = (ln(l(λ))), σ 2 (λ) = m (λ), 0 λ λ 0, m r (λ) = (ln(l r (λ))), σ 2 r(λ) = m r(λ), 0 λ λ (r) 0. As ξ 0 is nondegenerate, therefore m(.) is strictly increasing. Assume that 0 < P(ξ 0 = 0) <. Moreover, if we additionally assume that r 0 and P(ξ 0 = 0) + P(ξ 0 = r) <, then ξ (r) 0 is nondegenerate, therefore similar property is valid for the function m r (.). Let h = m (α n ), h r = m r (α n ), and β(α n ) = e h e. hr Theorem 2.7. Assume r > 0, P(ξ 0 = 0) > 0, and P(ξ 0 = 0) + P(ξ 0 = r) <. Let maxa, a r < d < d 2 < minm(λ 0 ), m r (λ (r) 0 ). Then, as n,, we have uniformly for d < α n < d 2. Pµ n = k = Pµ n = kβ(α n )( + o()) (2.4) Proof. We obtain Theorem 2.7 from (2.4) and from Lemma., if we apply (2.3) both for ξ i and for ξ (r) i.
8 64 I. Fazekas, B. Porvázsnyik We shall use the so called power series distribution. Consider the random variable ξ 0 with the following distribution. Let b 0, b, b 2,... be a sequence of nonnegative numbers and let R denote the radius of convergence of the series B(θ) = k=0 b k θ k. k! Assume that R > 0. Let ξ 0 = ξ 0 (θ) have the following distribution p k = p k (θ) = Pξ 0 (θ) = k = Differentiating B(θ) for 0 θ < R, we obtain Eξ 0 (θ) = θb (θ) B(θ), D2 ξ 0 (θ) = θ2 B (θ) B(θ) b kθ k, k = 0,, 2,.... (2.5) k!b(θ) + Eξ 0 (θ) (Eξ 0 (θ)) 2 (see e.g. [7]). We will assume that the distribution of the random variable ξ 0 (θ) satisfies b 0 > 0, b > 0. (2.6) We emphasize that the distribution of ξ 0 = ξ 0 (θ) is not fixed, it depends on θ. We have the following Poisson limit theorem. Theorem 2.8. Suppose that the random variable ξ 0 = ξ 0 (θ) has distribution (2.5), n condition (2.6) is satisfied. Let θ K < R. Let r > and 0. Let r such that p r (θ) λ for some 0 < λ <. Then for all k we have Pµ n = k = λk e λ ( + o()). (2.7) k! Proof. Let k. By the Poisson limit theorem, one has Cp k k r( p r ) k = λk e λ ( + o()). (2.8) k! Relation p r (θ) λ implies that θ = o(), B(θ) = b 0 + o(), B (θ) = b + o() and B (θ) = b 2 + o(). Therefore θ = We know that Eξ 0 = θb (θ) B(θ). Therefore Eξ 0 = b b 0 ( r!(b0 λ + o()) b r ( ) r!(b0λ+o()) /r. b r ) /r ( ) /r ( + o()) = C ( + o()). (2.9) Here and in what follows C denotes an appropriate constant (its value can be different in different formulae). Similarly ( ) /r D 2 ξ 0 = C ( + o()). (2.20)
9 A generalized allocation scheme 65 ow applying condition (2.20), we obtain n r As Eξ (r) 0 = Eξ 0 rp r p r, so (2.9) and condition n r We have We obtain ow applying condition (2.24), we obtain 0, Chebishev s inequality and relations (2.9), P S n = ( + o()). (2.2) 0 imply that ( ) /r Eξ (r) 0 = C ( + o()). (2.22) D 2 ξ (r) 0 = Eξ2 0 a 2 p r ( p r ) 2 + 2arp r ( p r ) 2 r2 p r ( p r ) 2. (2.23) ( ) /r D 2 ξ (r) 0 = C ( + o()). (2.24) n r 0, Chebishev s inequality and relations (2.22), PS (r) k n kr = ( + o()). (2.25) Inserting (2.2), (2.25), and (2.8) into (2.4), we obtain (2.7). 3. Limit theorems for max i η i Let η () = max i η i. η () is the maximal number of balls contained by any of the boxes. Let ξ ( r) 0 be a random variable with distribution Pξ ( r) 0 = k = Pξ 0 = k ξ 0 r. Let ξ ( r) i, i =,...,, be independent copies of ξ ( r) 0. Let S ( r) = ξ( r) i and Eξ ( r) 0 = a r. We can see that a r a. Moreover, a r = a if and only if P(ξ 0 r) =, that is ξ 0 and ξ ( r) 0 have the same distribution. The following representation of η () is useful to obtain limit results. Theorem 3.. We have for all r where P r = Pξ 0 > r. Pη () r = ( P r ) PS( r) n PS n, (3.)
10 66 I. Fazekas, B. Porvázsnyik Proof. Pη () r = Pη r,..., η r = P ξ r,..., ξ r ξ i n P ξ r,..., ξ r, S n = P S n P S n ξ r,..., ξ r = (Pξ r) P S n = ( P r ) PS( r) n PS n. Theorem 3.2. () Let d < a r. Then for all fixed r, as n,, we have Pη () r = ( P r ) ( + o()) (3.2) uniformly for α n < d. (2) Suppose that a r < a and a r < d < d < a. Then for all fixed r, as n,, we have Pη () r = ( P r ) o() (3.3) uniformly for d > α n > d. Proof. () Apply Kolmogorov s law of large numbers for S and S ( r) in (3.). Then (3.2) follows. (2) If d < α n < d and we apply Kolmogorov s law of large numbers, then we obtain lim P S ( r) n,, d <α n <d n = 0. (3.4) Theorem 3.3. Suppose that Eξ0 2 < and let σ r 2 be the variance of ξ( r) 0. Let C <. Then, for all r, as n, such that (α n a) C, we have ( ) Pη () r = ( P r ) Φ C σ r Φ ( ) + o(), for a C r = a, (3.5) σ Pη () r = ( P r ) o(), for a r < a. (3.6) Proof. By the central limit theorem, we have P n = P ξ( r) i a r ξ ( r) i σ r (αn a r ) σ r
11 A generalized allocation scheme 67 ( ) (αn a r ) = Φ + o(). (3.7) σ r In relation (3.) apply (2.) and (3.7) to obtain (3.5) and (3.6). Let η () = max i η i be the maximum in the usual generalized allocation scheme (.). Using large deviation results, we can describe the relation of η () and η (). Introduce notation L r (λ) = Ee λξ( r) 0 where we assume that there exist a positive constant λ ( r) 0 > 0, such that Let Ee λ( r) 0 ξ ( r) 0 < (Cramér s condition). m r (λ) = (ln(l r (λ))), σ 2 r(λ) = m r(λ), 0 λ λ ( r) 0. Let r. If P(ξ 0 = 0) > 0 and P(ξ 0 r) > P(ξ 0 = 0), then ξ ( r) 0 is nondegenerate, therefore m r (.) is a strictly increasing function. Let h = m (α n ), h r = m r (α n ), and β r (α n ) = e h. e h r Theorem 3.4. Assume that r, P(ξ 0 = 0) > 0, and P(ξ 0 r) > P(ξ 0 = 0). Let maxa, a r < d < d 2 < minm(λ 0 ), m r (λ ( r) 0 ). Then, for all r as n,, we have uniformly for d < α n < d 2. Pη () r = Pη () rβ r (α n )( + o()) (3.8) Proof. For the usual generalized allocation scheme, V.F. Kolchin in [7] obtained that Pη () r = ( P r) PS( r) = n (3.9) PS = n for all r where P r = Pξ 0 > r. Using (3.9) and (3.) and applying (2.3) both for ξ i and for ξ ( r) i, the proof of Theorem 3.4 is complete. Theorem 3.5. Suppose that the random variable ξ = ξ(θ) has distribution (2.5), condition (2.6) is satisfied and θ K < R. Let r. Let θ = θ() be such that n p r+ (θ) λ where 0 < λ <. Then, as n, such that 0, we r/(r+) have Pη () = r = e λ + o(), (3.0) Pη () = r + = e λ + o(). (3.)
12 68 I. Fazekas, B. Porvázsnyik Proof. Relation n 0 implies that r/(r+) ( ) /(r+) (r + )!(b0 λ + o()) B(θ) = b 0 + o() and θ =. b r+ Using r + instead of r in the proof of Theorem 2.8, we obtain Let r. Then Moreover, P S n = ( + o()). (3.2) ( ( ) ) /(r+) k r Eξ ( r) k= k b k (r+)!(b0λ+o()) k!b 0 b r+ 0 = ( ( ) ) /(r+) k ( + o()) r b k (r+)!(b0λ+o()) k=0 k!b 0 b r+ ( ) /(r+) = C ( + o()). (3.3) ( ) /(r+) D 2 ξ ( r) 0 C ( + o()). (3.4) Using Chebishev s inequality, (3.3) and (3.4), we obtain Using relations θ 0 and p r+ (θ) λ, we obtain PS ( r) n = ( + o()). (3.5) ( P r ) = o(), ( P r ) = e λ + o(), ( P r+ ) = + o(). (3.6) Inserting (3.2), (3.5), and (3.6) into (3.), we obtain Pη () r = o(), Pη () r = e λ + o(), Pη () r + = + o(). These relations imply (3.0) and (3.). 4. Limit theorems for min i η i In this section we shall prove limit theorems for the minimal content of the boxes. Let η ( ) = min i η i. Let ξ ( r) 0 be a random variable with distribution Pξ ( r) 0 = k = Pξ 0 = k ξ 0 r. Let ξ ( r), i =,...,, be independent copies of ξ ( r) 0. Let S ( r) = ξ( r) i i and Eξ ( r) 0 = a r. One can see that Eξ ( r) 0 Eξ 0 and equality can happen if and only if ξ ( r) 0 = ξ 0. We start with an appropriate representation of η ( )
13 A generalized allocation scheme 69 Theorem 4.. We have for all r where Q r = Pξ 0 < r. Proof. Pη ( ) r = ( Q r ) PS( r) n PS n, (4.) Pη ( ) r = Pη r,..., η r = P ξ r,..., ξ r ξ i n P ξ r,..., ξ r, S n = P S n P S n ξ r,..., ξ r = (Pξ r) P S n = ( Q r ) PS( r) n PS n. Theorem 4.2. Let d < a. Then for all r, as n,, we have uniformly for α n < d. Pη ( ) r = ( Q r ) ( + o()) (4.2) Proof. We apply Kolmogorov s law of large numbers for S Then we obtain (4.2). and S ( r) in (4.). Theorem 4.3. Suppose that Eξ0 2 < and let σ r 2 be the variance of ξ( r) 0. Let C <. Then, for all r, as n, such that (α n a) C, we have ( ) Pη ( ) r = ( Q r ) Φ C σ r Φ ( ) + o(), for a C r = a, (4.3) σ ( ) Pη ( ) r = ( Q r ) Φ ( ) + o(), for a C r > a. (4.4) σ Proof. By the central limit theorem, we have P n = P ξ( r) i a r (αn a r ) ξ ( r) i σ r σ r ( ) (αn a r ) = Φ + o(). (4.5) In relation (4.) apply (2.) and (4.5). Then we obtain (4.3) and (4.4). σ r
14 70 I. Fazekas, B. Porvázsnyik References [] Békéssy, A. On classical occupancy problems. I. Magy. Tud. Akad. Mat. Kutató Int. Közl. 8 (963), o.2, [2] Chuprunov, A.. and Fazekas, I. Inequalities and strong laws of large numbers for random allocations. Acta Math. Hungar. 09 (2005), o.2, [3] Chuprunov, A.. and Fazekas, I. An inequality for moments and its applications to the generalized allocation scheme. Publ. Math. Debrecen, 76 (200), no. 3, [4] Chuprunov, A.. and Fazekas, I. One some analogue of the generalized allocation scheme. (Russian) Diskretnaya Matematika, 2(202), o.. [5] Johnson,. L., Kemp, A. W. and Kotz, S. Univariate Discrete Distributions. Third edition. Wiley Series in Probability and Statistics. WileyInterscience, Hoboken, J, [6] Kolchin, A. V. and Kolchin, V. F. On transition of distributions of sums of independent identically distributed random variables from one lattice to another in the generalised allocation scheme. (Russian) Diskret. Mat. 8 (2006), no. 4, 3 27; translation in Discrete Math. Appl. 6 (2006), no. 6, [7] Kolchin, V.F. Random Graphs. Cambridge University Press, Cambridge, 999. [8] Kolchin, V.F., Sevast yanov, B.A. and Chistyakov, V.P. Random allocations. V.H. Winston & Sons, Washington D. C [9] Lerner, B., Lone, A. and Rao, M. On generalized Poisson distributions. Probab. Math. Statist. 7 (997), no. 2, Acta Univ. Wratislav. o. 2029, [0] Pavlov, Yu. L. Random Forests. VSP, Utrecht, [] Rozovskii, L. V. Probabilities of large deviations for some classes of distributions satisfying the Cramér condition. (Russian) Zap. auchn. Sem. S.Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 298 (2003), Veroyatn. i Stat. 6, 6 85, 38 39; translation in J. Math. Sci. (. Y.) 28 (2005), no., [2] Rényi, A. Three new proofs and generalization of a theorem of Irving Weiss. Magy. Tud. Akad. Mat. Kutató Int. Közl. 7 (962), o.2, [3] Weiss, I. Limiting distributions in some occupancy problems. Ann. Math. Statist. 29 (958), o.3,
An example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationLess Hashing, Same Performance: Building a Better Bloom Filter
Less Hashing, Same Performance: Building a Better Bloom Filter Adam Kirsch and Michael Mitzenmacher Division of Engineering and Applied Sciences Harvard University, Cambridge, MA 02138 {kirsch, michaelm}@eecs.harvard.edu
More informationOn an antiramsey type result
On an antiramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider antiramsey type results. For a given coloring of the kelement subsets of an nelement set X, where two kelement
More informationA Perfect Example for The BFGS Method
A Perfect Example for The BFGS Method YuHong Dai Abstract Consider the BFGS quasinewton method applied to a general nonconvex function that has continuous second derivatives. This paper aims to construct
More informationRandom matchings which induce Hamilton cycles, and hamiltonian decompositions of random regular graphs
Random matchings which induce Hamilton cycles, and hamiltonian decompositions of random regular graphs Jeong Han Kim Microsoft Research One Microsoft Way Redmond, WA 9805 USA jehkim@microsoft.com Nicholas
More informationArithmetics on number systems with irrational bases
Arithmetics on number systems with irrational bases P. Ambrož C. Frougny Z. Masáková E. Pelantová Abstract For irrational β > 1 we consider the set Fin(β) of real numbers for which x has a finite number
More informationThe matching, birthday and the strong birthday problem: a contemporary review
Journal of Statistical Planning and Inference 130 (2005) 377 389 www.elsevier.com/locate/jspi The matching, birthday and the strong birthday problem: a contemporary review Anirban DasGupta Department of
More informationDecoding by Linear Programming
Decoding by Linear Programming Emmanuel Candes and Terence Tao Applied and Computational Mathematics, Caltech, Pasadena, CA 91125 Department of Mathematics, University of California, Los Angeles, CA 90095
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationSUMMING CURIOUS, SLOWLY CONVERGENT, HARMONIC SUBSERIES
SUMMING CURIOUS, SLOWLY CONVERGENT, HARMONIC SUBSERIES THOMAS SCHMELZER AND ROBERT BAILLIE 1 Abstract The harmonic series diverges But if we delete from it all terms whose denominators contain any string
More informationFinitetime Analysis of the Multiarmed Bandit Problem*
Machine Learning, 47, 35 56, 00 c 00 Kluwer Academic Publishers. Manufactured in The Netherlands. Finitetime Analysis of the Multiarmed Bandit Problem* PETER AUER University of Technology Graz, A8010
More informationThe Effect of Network Topology on the Spread of Epidemics
1 The Effect of Network Topology on the Spread of Epidemics A. Ganesh, L. Massoulié, D. Towsley Microsoft Research Dept. of Computer Science 7 J.J. Thomson Avenue University of Massachusetts CB3 0FB Cambridge,
More informationHow to Use Expert Advice
NICOLÒ CESABIANCHI Università di Milano, Milan, Italy YOAV FREUND AT&T Labs, Florham Park, New Jersey DAVID HAUSSLER AND DAVID P. HELMBOLD University of California, Santa Cruz, Santa Cruz, California
More informationThe Set Covering Machine
Journal of Machine Learning Research 3 (2002) 723746 Submitted 12/01; Published 12/02 The Set Covering Machine Mario Marchand School of Information Technology and Engineering University of Ottawa Ottawa,
More informationSMALE S 17TH PROBLEM: AVERAGE POLYNOMIAL TIME TO COMPUTE AFFINE AND PROJECTIVE SOLUTIONS.
SMALE S 17TH PROBLEM: AVERAGE POLYNOMIAL TIME TO COMPUTE AFFINE AND PROJECTIVE SOLUTIONS. CARLOS BELTRÁN AND LUIS MIGUEL PARDO Abstract. Smale s 17th Problem asks: Can a zero of n complex polynomial equations
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationRegular Languages are Testable with a Constant Number of Queries
Regular Languages are Testable with a Constant Number of Queries Noga Alon Michael Krivelevich Ilan Newman Mario Szegedy Abstract We continue the study of combinatorial property testing, initiated by Goldreich,
More informationONEDIMENSIONAL RANDOM WALKS 1. SIMPLE RANDOM WALK
ONEDIMENSIONAL RANDOM WALKS 1. SIMPLE RANDOM WALK Definition 1. A random walk on the integers with step distribution F and initial state x is a sequence S n of random variables whose increments are independent,
More informationHow to Gamble If You Must
How to Gamble If You Must Kyle Siegrist Department of Mathematical Sciences University of Alabama in Huntsville Abstract In red and black, a player bets, at even stakes, on a sequence of independent games
More informationTests in a case control design including relatives
Tests in a case control design including relatives Stefanie Biedermann i, Eva Nagel i, Axel Munk ii, Hajo Holzmann ii, Ansgar Steland i Abstract We present a new approach to handle dependent data arising
More informationWiener s test for superbrownian motion and the Brownian snake
Probab. Theory Relat. Fields 18, 13 129 (1997 Wiener s test for superbrownian motion and the Brownian snake JeanStephane Dhersin and JeanFrancois Le Gall Laboratoire de Probabilites, Universite Paris
More informationProbability: Theory and Examples. Rick Durrett. Edition 4.1, April 21, 2013
i Probability: Theory and Examples Rick Durrett Edition 4.1, April 21, 213 Typos corrected, three new sections in Chapter 8. Copyright 213, All rights reserved. 4th edition published by Cambridge University
More informationOn the numbertheoretic functions ν(n) and Ω(n)
ACTA ARITHMETICA LXXVIII.1 (1996) On the numbertheoretic functions ν(n) and Ω(n) by Jiahai Kan (Nanjing) 1. Introduction. Let d(n) denote the divisor function, ν(n) the number of distinct prime factors,
More informationSome Research Problems in Uncertainty Theory
Journal of Uncertain Systems Vol.3, No.1, pp.310, 2009 Online at: www.jus.org.uk Some Research Problems in Uncertainty Theory aoding Liu Uncertainty Theory Laboratory, Department of Mathematical Sciences
More informationSome Sharp Performance Bounds for Least Squares Regression with L 1 Regularization
Some Sharp Performance Bounds for Least Squares Regression with L 1 Regularization Tong Zhang Statistics Department Rutgers University, NJ tzhang@stat.rutgers.edu Abstract We derive sharp performance bounds
More informationMean Value Coordinates
Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its
More informationIntroduction to Queueing Theory and Stochastic Teletraffic Models
Introduction to Queueing Theory and Stochastic Teletraffic Models Moshe Zukerman EE Department, City University of Hong Kong Copyright M. Zukerman c 2000 2015 Preface The aim of this textbook is to provide
More informationmetric space, approximation algorithm, linear programming relaxation, graph
APPROXIMATION ALGORITHMS FOR THE 0EXTENSION PROBLEM GRUIA CALINESCU, HOWARD KARLOFF, AND YUVAL RABANI Abstract. In the 0extension problem, we are given a weighted graph with some nodes marked as terminals
More informationINTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES
INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the
More information