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Weak convergence of mutually independent $Xₙ^B$ and $Xₙ^A$ under weak convergence of $Xₙ ≡ Xₙ^B - Xₙ^A$

100%

Szczotka W.

Applicationes Mathematicae

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2006

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tom 33

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nr 1

41-49

EN

For each n ≥ 1, let ${v_{n,k}, k ≥ 1}$ and ${u_{n,k}, k ≥ 1}$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $Xₙ^B(t) = (1/cₙ) ∑_{j=1}^{[nt]}(v_{n,j} - v̅ₙ)$, $Xₙ^A(t) = (1/cₙ)∑_{j=1}^{[nt]}(u_{n,j}-u̅̅ₙ)$, t ≥ 0, and $Xₙ = Xₙ^B - Xₙ^A$. The main result gives conditions under which the weak convergence $Xₙ \mathrel{\mathop {\rightarrow}\limits^{𝓓}}X$, where X is a Lévy process, implies $Xₙ^B \mathrel{\mathop{\rightarrow}\limits^{𝓓}}X^B$ and $Xₙ^A\mathrel{\mathop{\rightarrow}\limits^{𝓓}}X^A$, where $X^B$ and $ X^A$ are mutually independent Lévy processes and $X = X^B - X^A$.

2

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Immediate service in a Beneš-type G/G/1 queueing system

75%

Szczotka W.

Applicationes Mathematicae

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1974-1975

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tom 14

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nr 3

357-363

3

On fully coupled continuous time random walks

64%

Szczotka W., Żebrowski P.

Applicationes Mathematicae

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2012

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tom 39

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nr 1

87-102

EN

Continuous time random walks with jump sizes equal to the corresponding waiting times for jumps are considered. Sufficient conditions for the weak convergence of such processes are established and the limiting processes are identified. Furthermore one-dimensional distributions of the limiting processes are given under an additional assumption.

4

Functional central limit theorems for seeds in a linear birth and growth model

64%

Dziwisz A., Szczotka W.

Applicationes Mathematicae

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2016

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tom 43

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nr 1

1-24

EN

A problem of heredity of mixing properties (α-mixing, β-mixing and ρ-mixing) from a stationary point process on ℝ × ℝ₊ to a sequence of some of its points called 'seeds' is considered. Next, using the mixing properties, several versions of functional central limit theorems for the distances between seeds and the process of the number of seeds are obtained.

5

Factorization of the distribution function of waiting time

63%

Szczotka W.

Applicationes Mathematicae

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1977-1980

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tom 16

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nr 3

395-403

6

M/G/1 queueing system with "fagging'' service channel

63%

Szczotka W.

Applicationes Mathematicae

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1972-1973

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tom 13

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nr 4

439-463

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6 Applicationes Mathematicae

6 Szczotka W.

1 Dziwisz A.

1 Żebrowski P.

1 2016

1 2012

1 2006

1 1979

1 1974

1 1973

Weak convergence of mutually independent $Xₙ^B$ and $Xₙ^A$ under weak convergence of $Xₙ ≡ Xₙ^B - Xₙ^A$

Immediate service in a Beneš-type G/G/1 queueing system

On fully coupled continuous time random walks

Functional central limit theorems for seeds in a linear birth and growth model

Factorization of the distribution function of waiting time

M/G/1 queueing system with "fagging'' service channel