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Papers by Zdzislaw Rychlik

Research paper thumbnail of On Vervaat and Vervaat-error-type processes for partial sums and renewals

Journal of Statistical Planning and Inference, 2007

We study the asymptotic behaviour of stochastic processes that are generated by sums of partial s... more We study the asymptotic behaviour of stochastic processes that are generated by sums of partial sums of i.i.d. random variables and their renewals. We conclude that these processes cannot converge weakly to any nondegenerate random element of the space D[0, 1]. On the other hand we show that their properly normalized integrals as Vervaat-type stochastic processes converge weakly to a squared Wiener process. Moreover, we also deal with the asymptotic behaviour of the deviations of these processes, the so-called Vervaat-error type processes.

Research paper thumbnail of 1 Almost sure central limit theorems for random fields

Abstract. A general almost sure limit theorem is presented for random fields. It is applied to ob... more Abstract. A general almost sure limit theorem is presented for random fields. It is applied to obtain almost sure versions of some (functional) central limit theorems. AMS 2000 subject classification: 60F05 Central limit and other weak theorems, 60F17 Functional limit theorems; invariance principles, 60F15 Strong theorems. Key words and phrases. Almost sure central limit theorem, random field, multiindex, strong law of large numbers, functional limit theorem. 1. Introduction and

Research paper thumbnail of The order of approximation in the random central limit theorem

The order of approximation in the random central limit theorem

Lecture Notes in Mathematics

Research paper thumbnail of Inflated truncated negative binomial acceptance sampling plan

Applications of Mathematics

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of On the almost sure convergence of randomly indexed maximum of random variables

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

We prove an almost sure random version of a maximum limit theorem, using logarithmic means for \(... more We prove an almost sure random version of a maximum limit theorem, using logarithmic means for \(\max_{1\leq i\leq N_n} X_i\), where \(\{X_n, n \geq 1\}\) is a sequence of identically distributed random variables and \(\{N_n, n \geq 1\}\) is a sequence of positive integer random variables independent of \(\{X_n, n \geq 1\}\). Furthermore, we consider the almost sure random version of a limit theorem for \(k\)th order statistics.

Research paper thumbnail of Berry-esseen inequality for the sum of a random number of independent random variables

Berry-esseen inequality for the sum of a random number of independent random variables

Mathematical Notes of the Academy of Sciences of the USSR

Research paper thumbnail of A central limit theorem for sums of a random number of independent random variables

Research paper thumbnail of Marcinkiewicz-type strong law of large numbers for pairwise independent random fields

Marcinkiewicz-type strong law of large numbers for pairwise independent random fields

Probability and Mathematical Statistics, 2002

Abstract: We present the Marcinkiewicz-type strong law of large numbers for random fields {Xn, n∈... more Abstract: We present the Marcinkiewicz-type strong law of large numbers for random fields {Xn, n∈ Zd+} of pairwise independent random variables, where Zd+, d≥ 1, is the set of positive d-dimensional lattice points with coordinatewise partial ordering.

Research paper thumbnail of On the random functional central limit theorems with almost sure convergence

Probability and Mathematical Statistics, 2007

In this paper we present functional random-sum central limit theorems with almost sure convergenc... more 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.

Research paper thumbnail of Rates of convergence in the weak law for martingale random fields

Rates of convergence in the weak law for martingale random fields

Research paper thumbnail of On some inequalities for the concentration function of the sum of a random number of independent random variables

On some inequalities for the concentration function of the sum of a random number of independent random variables

Research paper thumbnail of On the rate of convergence for linear functionals of sums of martingale differences

On the rate of convergence for linear functionals of sums of martingale differences

Research paper thumbnail of Some remarks on stable sequences of random variables

Research paper thumbnail of Weak convergence of integral type functionals

Weak convergence of integral type functionals

Research paper thumbnail of Über die Konvergenzgeschwindigkeit in gewissen Grenzwertsätzen für die Summen einer zufälligen Anzahl zufälliger Summanden

Über die Konvergenzgeschwindigkeit in gewissen Grenzwertsätzen für die Summen einer zufälligen Anzahl zufälliger Summanden

Research paper thumbnail of Weak convergence of functionals of sums for martingale differences

Weak convergence of functionals of sums for martingale differences

Bulletin of the Polish Academy of Sciences Mathematics

Research paper thumbnail of On Barry-Esseen’s inequality for sums of a random number of independent random variables

On Barry-Esseen’s inequality for sums of a random number of independent random variables

Research paper thumbnail of On the rates of convergence in the generalized Anscombe theorem

On the rates of convergence in the generalized Anscombe theorem

Research paper thumbnail of Weak convergence of random reversed martingales with o-rates

Weak convergence of random reversed martingales with o-rates

Bulletin of the Polish Academy of Sciences Mathematics

Research paper thumbnail of Approximation of the distribution function of sums of a random number of random variables

Approximation of the distribution function of sums of a random number of random variables

Bulletin of the Polish Academy of Sciences Mathematics

On approxime la fonction de repartition d'un nombre aleatoire de variables aleatoires indepen... more On approxime la fonction de repartition d'un nombre aleatoire de variables aleatoires independantes par une fonction de repartition qui appartient a une classe plus large de distribution que la classe des lois stables

Research paper thumbnail of On Vervaat and Vervaat-error-type processes for partial sums and renewals

Journal of Statistical Planning and Inference, 2007

We study the asymptotic behaviour of stochastic processes that are generated by sums of partial s... more We study the asymptotic behaviour of stochastic processes that are generated by sums of partial sums of i.i.d. random variables and their renewals. We conclude that these processes cannot converge weakly to any nondegenerate random element of the space D[0, 1]. On the other hand we show that their properly normalized integrals as Vervaat-type stochastic processes converge weakly to a squared Wiener process. Moreover, we also deal with the asymptotic behaviour of the deviations of these processes, the so-called Vervaat-error type processes.

Research paper thumbnail of 1 Almost sure central limit theorems for random fields

Abstract. A general almost sure limit theorem is presented for random fields. It is applied to ob... more Abstract. A general almost sure limit theorem is presented for random fields. It is applied to obtain almost sure versions of some (functional) central limit theorems. AMS 2000 subject classification: 60F05 Central limit and other weak theorems, 60F17 Functional limit theorems; invariance principles, 60F15 Strong theorems. Key words and phrases. Almost sure central limit theorem, random field, multiindex, strong law of large numbers, functional limit theorem. 1. Introduction and

Research paper thumbnail of The order of approximation in the random central limit theorem

The order of approximation in the random central limit theorem

Lecture Notes in Mathematics

Research paper thumbnail of Inflated truncated negative binomial acceptance sampling plan

Applications of Mathematics

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of On the almost sure convergence of randomly indexed maximum of random variables

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

We prove an almost sure random version of a maximum limit theorem, using logarithmic means for \(... more We prove an almost sure random version of a maximum limit theorem, using logarithmic means for \(\max_{1\leq i\leq N_n} X_i\), where \(\{X_n, n \geq 1\}\) is a sequence of identically distributed random variables and \(\{N_n, n \geq 1\}\) is a sequence of positive integer random variables independent of \(\{X_n, n \geq 1\}\). Furthermore, we consider the almost sure random version of a limit theorem for \(k\)th order statistics.

Research paper thumbnail of Berry-esseen inequality for the sum of a random number of independent random variables

Berry-esseen inequality for the sum of a random number of independent random variables

Mathematical Notes of the Academy of Sciences of the USSR

Research paper thumbnail of A central limit theorem for sums of a random number of independent random variables

Research paper thumbnail of Marcinkiewicz-type strong law of large numbers for pairwise independent random fields

Marcinkiewicz-type strong law of large numbers for pairwise independent random fields

Probability and Mathematical Statistics, 2002

Abstract: We present the Marcinkiewicz-type strong law of large numbers for random fields {Xn, n∈... more Abstract: We present the Marcinkiewicz-type strong law of large numbers for random fields {Xn, n∈ Zd+} of pairwise independent random variables, where Zd+, d≥ 1, is the set of positive d-dimensional lattice points with coordinatewise partial ordering.

Research paper thumbnail of On the random functional central limit theorems with almost sure convergence

Probability and Mathematical Statistics, 2007

In this paper we present functional random-sum central limit theorems with almost sure convergenc... more 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.

Research paper thumbnail of Rates of convergence in the weak law for martingale random fields

Rates of convergence in the weak law for martingale random fields

Research paper thumbnail of On some inequalities for the concentration function of the sum of a random number of independent random variables

On some inequalities for the concentration function of the sum of a random number of independent random variables

Research paper thumbnail of On the rate of convergence for linear functionals of sums of martingale differences

On the rate of convergence for linear functionals of sums of martingale differences

Research paper thumbnail of Some remarks on stable sequences of random variables

Research paper thumbnail of Weak convergence of integral type functionals

Weak convergence of integral type functionals

Research paper thumbnail of Über die Konvergenzgeschwindigkeit in gewissen Grenzwertsätzen für die Summen einer zufälligen Anzahl zufälliger Summanden

Über die Konvergenzgeschwindigkeit in gewissen Grenzwertsätzen für die Summen einer zufälligen Anzahl zufälliger Summanden

Research paper thumbnail of Weak convergence of functionals of sums for martingale differences

Weak convergence of functionals of sums for martingale differences

Bulletin of the Polish Academy of Sciences Mathematics

Research paper thumbnail of On Barry-Esseen’s inequality for sums of a random number of independent random variables

On Barry-Esseen’s inequality for sums of a random number of independent random variables

Research paper thumbnail of On the rates of convergence in the generalized Anscombe theorem

On the rates of convergence in the generalized Anscombe theorem

Research paper thumbnail of Weak convergence of random reversed martingales with o-rates

Weak convergence of random reversed martingales with o-rates

Bulletin of the Polish Academy of Sciences Mathematics

Research paper thumbnail of Approximation of the distribution function of sums of a random number of random variables

Approximation of the distribution function of sums of a random number of random variables

Bulletin of the Polish Academy of Sciences Mathematics

On approxime la fonction de repartition d'un nombre aleatoire de variables aleatoires indepen... more On approxime la fonction de repartition d'un nombre aleatoire de variables aleatoires independantes par une fonction de repartition qui appartient a une classe plus large de distribution que la classe des lois stables