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38
Large deviations of combinatorial distributions II: Local limit theorems
, 1997
"... This paper is a sequel to our paper [17] where we derived a general central limit theorem for probabilities of large deviations applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit theorems in this paper. More precisely, given a seq ..."
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Cited by 32 (5 self)
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This paper is a sequel to our paper [17] where we derived a general central limit theorem for probabilities of large deviations applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit theorems in this paper. More precisely, given a sequence of integral random variables n#1 each of maximal span 1 (see below for definition), we are interested in the asymptotic behavior of the probabilities n = m} (m N, m = n x n # n , n := n , # n := n ), ##, where x n can tend to with n at a rate that is restricted to O(# n ). Our interest here is not to derive asymptotic expression for n = m} valid for the widest possible range of m, but to show that for m lying in the interval n O(# n ), very precise asymptotic formulae can be obtained. These formulae are in close connection with our results in [17]. Although local limit theorems receive a constant research interest [2, 3, 7, 14, 13, 24], our approach and results, especially Theorem 1, seem rarely discussed in a systematic manner. Recall that a lattice random variable X is said to be of maximal span h if X takes only values of the form b + hk, k Z, for some constants b and h > 0; and there does not exist b # and h # > h such that X takes only values of the form b # + h # k
Average State Complexity of Operations on Unary Automata
, 1999
"... . Define the complexity of a rational language as the number of states of its minimal automaton. Let A (respectively A 0 ) be an nstate (resp. n 0 state) deterministic and connected unary automaton. Our main results can be summarized as follows: 1. The probability that A is minimal tends t ..."
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Cited by 21 (3 self)
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. Define the complexity of a rational language as the number of states of its minimal automaton. Let A (respectively A 0 ) be an nstate (resp. n 0 state) deterministic and connected unary automaton. Our main results can be summarized as follows: 1. The probability that A is minimal tends toward 1=2 when n tends toward infinity, 2. The average complexity of L(A) is equivalent to n, 3. The average complexity of L(A) " L(A 0 ) is equivalent to 3i(3) 2 2 nn 0 , where i is the Riemann "zeta"function. 4. The average complexity of L(A) is bounded by a constant, 5. If n n 0 P (n), for some polynomial P , the average complexity of L(A)L(A 0 ) is bounded by a constant (depending on P ). Remark that results 3, 4 and 5 differ perceptibly from the corresponding worst case complexities, which are nn 0 for intersection, (n \Gamma 1) 2 + 1 for star and nn 0 for concatenation product. 1 Introduction This paper addresses a rather natural problem: find the averag...
Limit Theorems for the Number of Summands in Integer Partitions
, 2000
"... Central and local limit theorems are derived for the number of distinct summands in integer partitions, with or without repetitions, under a general scheme essentially due to Meinardus. The local limit theorems are of the form of Cramertype large deviations and are proved by Mellin transform and th ..."
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Cited by 14 (2 self)
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Central and local limit theorems are derived for the number of distinct summands in integer partitions, with or without repetitions, under a general scheme essentially due to Meinardus. The local limit theorems are of the form of Cramertype large deviations and are proved by Mellin transform and the twodimensional saddlepoint method. Applications of these results include partitions into positive integers, into powers of integers, into integers [j ], # > 1, into aj + b, etc.
Asymptotic Estimates of Elementary Probability Distributions
 Studies in Applied Mathematics
, 1996
"... Several new asymptotic estimates (with precise error bounds) are derived for Poisson and binomial distributions as the parameters tend to infinity. The analytic methods used are also applicable to other discrete distribution functions. ..."
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Cited by 12 (6 self)
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Several new asymptotic estimates (with precise error bounds) are derived for Poisson and binomial distributions as the parameters tend to infinity. The analytic methods used are also applicable to other discrete distribution functions.
Strong asymptotic freeness for Wigner and Wishart matrices preprint
"... For each n in N, let Xn = [(Xn)jk] n j,k=1 be a random Hermitian matrix such that the n2 random variables √ n(Xn)ii, √ √ 2nRe((Xn)ij)i
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Cited by 11 (1 self)
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For each n in N, let Xn = [(Xn)jk] n j,k=1 be a random Hermitian matrix such that the n2 random variables √ n(Xn)ii, √ √ 2nRe((Xn)ij)i<j, 2nIm((Xn)ij)i<j are independent identically distributed with common distribution µ on R. Let X (1) n,...,X (r) n be r independent copies of Xn and (x1,... xr) be a semicircular system in a C∗probability space. Assuming that µ is symmetric and satisfies a Poincaré inequality, we show that, almost everywhere, for any non commutative polynomial p in r variables, lim n −→+ ∞ p(X(1) n,...,X(r) n)  = p(x1,... xr) . (0.1) We follow the method of [9] and [15] which gave (0.1) in the Gaussian (complex, real or symplectic) case. We also get that (0.1) remains true when the X (i) n distributed. are Wishart matrices while the xi are MarchenkoPastur
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.
Transcendence of Formal Power Series With Rational Coefficients
, 1999
"... We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal po ..."
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Cited by 9 (2 self)
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We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal power series, binomial series, automatic sequences, pLucas sequences, ChomskySchutzenberger theorem. 1 Introduction Formal power series with integer coefficients often occur as generating series. Suppose that a set E contains exactly a n elements of "size" n for each integer n: the generating series of this set is the formal power series P n0 a n X n (this series belongs to Z[[X]] ae Q[[X]]). Properties of this formal power series reflect properties of its coefficients, and hence properties of the set E. Roughly speaking, algebraicity of the series over Q(X) means that the set has a strong structure. For example, the ChomskySchutzenberger theorem [16] asserts that the generating seri...