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Distinctness of compositions of an integer: A probabilistic analysis
 RANDOM STRUCTURES AND ALGORITHMS
, 2001
"... Compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. In this paper, we use as measure of distinctness the number of distinct parts (or components). We investigate, from a probabilistic point o ..."
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Cited by 32 (13 self)
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Compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. In this paper, we use as measure of distinctness the number of distinct parts (or components). We investigate, from a probabilistic point of view, the first empty part, the maximum part size and the distribution of the number of distinct part sizes. We obtain asymptotically, for the classical composition of an integer, the moments and an expression for a continuous distribution F, the (discrete) distribution of the number of distinct part sizes being computable from F. We next analyze another composition: the Carlitz one, where two successive parts are dierent. We use tools such as analytical depoissonization, Mellin transforms, Markov chain potential theory, limiting hitting times, singularity analysis and perturbation analysis.
Rounding of continuous random variables and oscillatory asymptotics
 Ann. Probab
"... We study the characteristic function and moments of the integervalued random variable ⌊X + α⌋, where X is a continuous random variables. The results can be regarded as exact versions of Sheppard’s correction. Rounded variables of this type often occur as subsequence limits of sequences of integerva ..."
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Cited by 19 (8 self)
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We study the characteristic function and moments of the integervalued random variable ⌊X + α⌋, where X is a continuous random variables. The results can be regarded as exact versions of Sheppard’s correction. Rounded variables of this type often occur as subsequence limits of sequences of integervalued random variables. This leads to oscillatory terms in asymptotics for these variables, something that has often been observed, for example in the analysis of several algorithms. We give some examples, including applications to tries, digital search trees and Patricia tries. 1. Introduction. Let
Local limit theorems for finite and infinite urn models
 Ann. Probab
, 2007
"... Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation. 1. Introduction. A classical theorem o ..."
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Cited by 17 (2 self)
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Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation. 1. Introduction. A classical theorem of Rényi [29] for the number of empty boxes, denoted by μ0(n, M), in a sequence of n random allocations of indistinguishable balls into M boxes with equal probability 1/M, can be stated as follows: If the variance of μ0(n, M) tends to infinity with n, then μ0(n, M) is asymptotically normally distributed. This result, seldom stated in this form in the literature,
Distribution of the Number of Factors in Random Ordered Factorizations of Integers
 J. Number Theory
, 1998
"... We study in detail the asymptotic behavior of the number of ordered factorizations with a given number of factors. Asymptotic formulae are derived for almost all possible values of interest. In particular, the distribution of the number of factors is asymptotically normal. Also we improve the error ..."
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Cited by 9 (2 self)
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We study in detail the asymptotic behavior of the number of ordered factorizations with a given number of factors. Asymptotic formulae are derived for almost all possible values of interest. In particular, the distribution of the number of factors is asymptotically normal. Also we improve the error term in Kalmar's problem of "factorisatio numerorum" and investigate the average number of distinct factors in a random ordered factorization.
Locally restricted compositions I. Restricted adjacent differences
 Elec. J. Combin
"... We study compositions of the integer n in which the first part, successive differences, and the last part are constrained to lie in prescribed sets L, D, R, respectively. A simple condition on D guarantees that the generating function f(x, L, D, R) has only a simple pole on its circle of convergence ..."
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Cited by 7 (3 self)
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We study compositions of the integer n in which the first part, successive differences, and the last part are constrained to lie in prescribed sets L, D, R, respectively. A simple condition on D guarantees that the generating function f(x, L, D, R) has only a simple pole on its circle of convergence and this at r(D), a function independent of L and R. Thus the number of compositions is asymptotic to Ar(D) −n for a suitable constant A = A(L, D, R). We prove a multivariate central and local limit theorem and apply it to various statistics of random locally restricted compositions of n, such as number of parts, numbers of parts of given sizes, and number of rises. The first and last parts are shown to have limiting distributions and to be asymptotically independent. If D has only finitely many positive elements D +, or finitely many negative elements D − , then the largest part and number of distinct part sizes are almost surely Θ((log n) 1/2). On the other hand, when both D + and D − have a positive asymptotic lower “local logdensity”, we prove that the largest part and number of distinct part sizes are almost surely Θ(log n), and we give sufficient
The number of distinct part sizes of some multiplicity in compositions of an Integer. A probabilistic Analysis
, 2003
"... Random compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. A possible measure of distinctness is the number X of distinct parts (or components). This parameter has been analyzed in several pa ..."
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Cited by 4 (2 self)
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Random compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. A possible measure of distinctness is the number X of distinct parts (or components). This parameter has been analyzed in several papers. In this article we consider a variant of the distinctness: the number X(m) of distinct parts of multiplicity m that we call the mdistinctness. A rst motivation is a question asked by Wilf for random compositions: what is the asymptotic value of the probability that a randomly chosen part size in a random composition of an integer has multiplicity m. This is related to E (X(m)), which has been analyzed by Hitczenko, Rousseau and Savage. Here, we investigate, from a probabilistic point of view, the rst full part, the maximum part size and the distribution of X(m). We obtain asymptotically, as !1, the moments and an expression for a continuous distribution ', the (discrete) distribution of X(m; ) being computable from '. We use tools such as Mellin transforms, urns models, Poissonization, saddle point method and generating functions.
Locally Restricted Compositions IV. Nearly Free Large Parts and GapFreeness
"... We define the notion of asymptotically free for locally restricted compositions, which means roughly that large parts can often be replaced by any larger parts. Two wellknown examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. Thi ..."
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Cited by 1 (0 self)
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We define the notion of asymptotically free for locally restricted compositions, which means roughly that large parts can often be replaced by any larger parts. Two wellknown examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part, number of distinct parts and number of parts of multiplicity k, all accurate to o(1). Dedicated to the memory of Herb Wilf. 1
multiplicity in compositions of an Integer. A probabilistic Analysis
"... Random compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. A possible measure of distinctness is the number X of distinct parts (or components). This parameter has been analyzed in several pa ..."
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Random compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. A possible measure of distinctness is the number X of distinct parts (or components). This parameter has been analyzed in several papers. In this article we consider a variant of the distinctness: the number X m ¡ of distinct parts of multiplicity m that we call the mdistinctness. A first motivation is a question asked by Wilf for random compositions: what is the asymptotic value of the probability that a randomly chosen part size in a random composition of an integer ν has multiplicity m. This is related to ¢ X m¡£ ¡ , which has been analyzed by Hitczenko, Rousseau and Savage. Here, we investigate, from a probabilistic point of view, the first full part, the maximum part size and the distribution of X m ¡. We obtain asymptotically, as ν ¤ ∞, the moments and an expression for a continuous distribution ϕ, the (discrete) distribution of X m ¥ ν ¡ being computable from ϕ.
urn models
, 2006
"... Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation. Key words and phrases: occupancy probl ..."
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Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation. Key words and phrases: occupancy problems, random allocations, local limit theo1 1