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theorem and domains of attraction, Probab. Theory Related Fields 102 (1995), 1-17. MR 96j :60033 On the almost sure central limit theorem and domains of attraction Probab. Theory Related Fields

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-Sklodowska, Lublin LVI 1 18 Fazekas, I. and Rychlik, Z., Almost sure central limit theorems for random fields, Math

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Abstract  

We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τ n = τ n(ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τ nc 0 n 2/3)/c 1 n 1/3 log1/2 n converges weakly, as n → ∞, to the standard normal distribution, where c 0 = ζ(2)/ [2ζ(3)]2/3, c 1 = √(1/3/) [2ζ(3)]1/3 and ζ(s) = Σj=1 j s.

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In a one-parameter model for evolution of random trees strong law of large numbers and central limit theorem are proved for the number of vertices with low degree. The proof is based on elementary martingale theory.

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Abstract  

By applying the Skorohod martingale embedding method, a strong approximation theorem for partial sums of asymptotically negatively dependent (AND) Gaussian sequences, under polynomial decay rates, is established. As applications, the law of the iterated logarithm, the Chung-type law of the iterated logarithm and the almost sure central limit theorem for AND Gaussian sequences are derived.

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Abstract  

Let Y s,n denote the number of part sizes ≧ s in a random and uniform partition of the positive integer n that are counted without multiplicity. For s = λ(6n)1/2/π + o(n 1/4), 0 ≦ λ < ∞, as n → ∞, we establish the weak convergence of Y s,n to a Gaussian distribution in the form of a central limit theorem. The mean and the standard deviation are also asymptotically determined.

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References [1] Berkes , I. , A central limit theorem for trigonometric series with small gaps, Z

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