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1
Content available remote The limiting behaviour of sums and maximums of iid random variables from the viewpoint of different observers
Krajka T. K., Rychlik Z.
Probability and Mathematical Statistics
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2014
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Vol. 34, Fasc. 2
237--252
EN
The limiting behaviour of observed and all random variables in the max limit schema was considered by Mladenović and Piterbarg (2006) and Krajka (2011). Here those results are generalised in two directions: we allow more than one observer and one superobserver; we consider the max limit schema as well as the sum limit schema.
2
Content available remote On the random functional central limit theorems with almost sure convergence for subsequences
Rychlik Z., Szuster K. S.
Demonstratio Mathematica
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2012
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Vol. 45, nr 2
283-296
EN
In this paper we present functional random sum central limit theorems with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related 'logarithmic 'limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random sum central limit theorems for subsequences.
3
Content available remote Limit theorems for products of sums of independent random variables
Krajka T. K., Rychlik Z.
Probability and Mathematical Statistics
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2010
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Vol. 30, Fasc. 1
73--85
EN
Let {Xn; n ≥ 1} be a sequence of independent random variables with finite second moments and {Nn; n ≥ 1} be a sequence of positive integer-valued random variables. Write Sn. =Σnk-1(Xk−EXk); n ≥1; and let N be a standard normal random variable. In the paper the convergences...[formula], are considered for some sequences {an} and {γn} of positive integer numbers such that Sn + an≥ 0 a.e. The case when γn are random variables is also considered. The main results generalize the main theorems presented by Pang et al. [3].
4
Content available remote The speed of convergence of random products of sums of independent random variables
Krajka T., Rychlik Z.
Journal of Mathematics and Applications
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2009
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Vol. 31
51-66
5
Content available remote On the random functional central limit theorems with almost sure convergence
Orzóg M., Rychlik Z.
Probability and Mathematical Statistics
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2007
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Vol. 27, Fasc. 1
125--138
EN
In this paper we present functional random-sum central limit theorems with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related ‘logarithmic’ limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random-sum central limit theorems.
6
Content available remote Almost sure limit theorems for semi-selfsimilar processes
Fazekas I., Rychlik Z.
Probability and Mathematical Statistics
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2005
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Vol. 25, Fasc. 2
241--255
EN
An integral analogue of the almost sure limit theorem is presented for semi-selfsimilar processes. 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 theorem is applied to obtain almost sure limit theorems for semistable processes. Discrete versions of the above theorems are proved. In particular, the almost sure functional limit theorem is obtained for semistable random variables.
7
Content available remote Some remarks on the almost sure central limit theorem for independent random variables
Rychlik Z., Szuster K. S.
Probability and Mathematical Statistics
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2003
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Vol. 23, Fasc. 2
241--249
EN
The purpose of this paper is the proof of an almost sure central limit theorem for subsequences. We obtain an almost sure convergence limit theorem for independent nonidentically distributed random variables. The presented results extend,to nonidentically distributed random variables, theorems given by Schatte [13].
8
Content available remote Marcinkiewicz-type strong law of large numbers for pairwise independent random fields
Czerebak-Mrozowicz E. B., Klesov O. I., Rychlik Z.
Probability and Mathematical Statistics
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2002
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Vol. 22, Fasc. 1
127--139
EN
We present the Marcinkiewicz-type strong law of large numbers for ran-dom fields{Xn, n ∈Zd+}of pairwise independent random variables, where Zd+, d≥1, is the set of positived-dimensional lattice points with coordinatewise partial ordering.
9
Content available remote Strong limit theorems for general renewal processes
Klesov O., Rychlik Z., Steinebach J.
Probability and Mathematical Statistics
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2001
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Vol. 21, Fasc. 2
329--349
EN
An approach is discussed to derive strong limit theorems for general renewal processes from the corresponding asymptotics of the underlying renewal sequence. Neither independence nor stationarity of increments is required. In certain situations, just the dualities between the renewal processes and their defining sequencesin combination with some regularity conditions on the normalizing constants are sufficient for the proofs. There are other cases, however, in which the duality argumentsdo not apply, and where other techniques have to be developed. Finally, there are also examples, in which an inversion of the limit theorems under consideration cannot work at all.
10
Content available remote Almost sure central limit theorems for weakly dependent random variables
Rodzik B., Rychlik Z.
Demonstratio Mathematica
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2001
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Vol. 34, nr 2
375-384
EN
We present almost sure central limit theorems for weakly dependnt random variables. The presented theorems generalize the results obtained by Peligrad and Shao (1995)
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