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Berkes, I., CÁki, E., Csörgö, S. and Megyesi, Z., Almost sure limit theorems for sums and maxima from the domain of geometrical partial attraction of semistable laws, in: Limit theorems in probability and

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Summary An integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable processes, moreover for processes being in the domain of attraction of a stable law or being in the domain of geometric partial attraction of a semistable or a max-semistable law.

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Summary In this note we prove an almost sure limit theorem for the products of U-statistics.

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Abstract  

A certain class of stochastic summability methods of mantissa type is introduced and its connection to almost sure limit theorems is discussed. The summability methods serve as suitable weights in almost sure limit theory, covering all relevant known examples for, e.g., normalized sums or maxima of i.i.d. random variables. In the context of semistable domains of attraction the methods lead to previously unknown versions of semistable almost sure limit theorems.

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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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Summary Some topics of our twenty some years of joint work is discussed. Just to name a few; joint behavior of the maximum of the Wiener process and its location, global and local almost sure limit theorems,  strong approximation of the planar local time difference, a general Strassen type theorem, maximal local time on subsets.

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Ibragimov, I. A. and Lifshits, M. A. , On almost sure limit theorems, Theory Probab. Appl. 44 (2000), 254–272. MR 2001g :60066 Lifshits M. A. On almost sure limit theorems

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Summary This article provides a glimpse of some of the highlights of the joint work of Endre Csáki and Pál Révész since 1979.  The topics of this short exploration of the rich stochastic milieu of this inspiring collaboration revolve around Brownian motion, random walks and their long excursions, local times and additive functionals, iterated processes, almost sure local and global central limit theorems, integral functionals of geometric stochastic processes, favourite sites--favourite values and jump sizes for random walk and Brownian motion, random walking in a random scenery, and large void zones and occupation times for coalescing random walks.

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). LANZINGER, H., An almost sure limit theorem for moving averages of random variables between the strong law of large numbers and Erdös-Rényi law, ESAIM Probab. & Statist . 2 (1998), 163-183. An almost sure limit theorem for moving

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IBRAGIMOV, I. A. and LIFSHITS, M. A., On the almost sure limit theorems, Theoret. Probab. Appl. 44 (1999), 328-350. On the almost sure limit theorems Theoret. Probab. Appl

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