An almost sure limit theorem for Wick powers of Gaussian differences quotients

In this paper, by using a new comparison inequality for order statistics of Gaussian variables, we proved an almost sure central limit theorem for extreme order statistics of stationary Gaussian sequences with covariance rn under the condition rn log n(log log n)1+? = O(1) for some ? > 0. A similar result on intermediate order statistics is also proved for stationary Gaussian sequences. The obtained results improve some of the existing results.

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Almost sure limit theorems for random allocations

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.42.2005.2.4 ◽

2005 ◽

Vol 42 (2) ◽

pp. 173-194

Author(s):

István Fazekas ◽

Alexey Chuprunov

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Limit Theorems ◽

Random Variables ◽

Central Domain ◽

Almost Sure Limit Theorem ◽

The Central Limit Theorem ◽

Almost Sure Limit Theorems ◽

Poisson Limit

Almost sure limit theorems are presented for random allocations. A general almost sure limit theorem is proved for arrays of random variables. It is applied to obtain almost sure versions of the central limit theorem for the number of empty boxes when the parameters are in the central domain. Almost sure versions of the Poisson limit theorem in the left domain are also proved.

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A note on the almost sure limit theorem for the product of partial sums

Applied Mathematics Letters ◽

10.1016/j.aml.2005.06.002 ◽

2006 ◽

Vol 19 (2) ◽

pp. 191-196 ◽

Cited By ~ 36

Author(s):

Khurelbaatar Gonchigdanzan ◽

Grzegorz A. Rempała

Keyword(s):

Limit Theorem ◽

Partial Sums ◽

Almost Sure Limit Theorem

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An Almost Sure Limit Theorem for Super-Brownian Motion

Journal of Theoretical Probability ◽

10.1007/s10959-008-0200-8 ◽

2008 ◽

Vol 23 (2) ◽

pp. 401-416 ◽

Cited By ~ 9

Author(s):

Li Wang

Keyword(s):

Brownian Motion ◽

Limit Theorem ◽

Almost Sure Limit Theorem ◽

Super Brownian Motion

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An almost sure limit theorem for mixtures of domains in random allocation

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.2007.1024 ◽

2007 ◽

Vol 44 (3) ◽

pp. 331-354

Author(s):

Peter Becker-Kern

Keyword(s):

Limit Theorem ◽

Asymptotic Normality ◽

Invariance Principle ◽

Equal Probability ◽

Random Numbers ◽

Random Allocation ◽

Almost Sure Limit Theorem ◽

Transfer Theorem ◽

Final Number ◽

Poisson Distributions

The problem of random allocation is that of placing n balls independently with equal probability to N boxes. For several domains of increasing numbers of balls and boxes, the final number of empty boxes is known to be asymptotically either normally or Poissonian distributed. In this paper we first derive a certain two-index transfer theorem for mixtures of the domains by considering random numbers of balls and boxes. As a consequence of a well known invariance principle this enables us to prove a corresponding general almost sure limit theorem. Both theorems inherit a mixture of normal and Poisson distributions in the limit. Applications of the general almost sure limit theorem for logarithmic weights complement and extend results of Fazekas and Chuprunov [10] and show that asymptotic normality dominates.

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