Convergence in distribution for randomly stopped random fields

Author:
D. Silvestrov

Journal:
Theor. Probability and Math. Statist. **105** (2021), 137-149

MSC (2020):
Primary 60G60; Secondary 60F05, 60F99, 60G40

DOI:
https://doi.org/10.1090/tpms/1160

Published electronically:
December 7, 2021

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Additional Information

Abstract: Let $\mathbb {X}$ and $\mathbb {Y}$ be two complete, separable, metric spaces, $\xi _\varepsilon (x), x \in \mathbb {X}$ and $\nu _\varepsilon$ be, for every $\varepsilon \in [0, 1]$, respectively, a random field taking values in space $\mathbb {Y}$ and a random variable taking values in space $\mathbb {X}$. We present general conditions for convergence in distribution for random variables $\xi _\varepsilon (\nu _\varepsilon )$ that is the conditions insuring holding of relation, $\xi _\varepsilon (\nu _\varepsilon ) \stackrel {\mathsf {d}}{\longrightarrow } \xi _0(\nu _0)$ as $\varepsilon \to 0$.

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*Convergence of Probability Measures*, second edition, Wiley Series in Probability and Statistics, Wiley, New York, 2014. MR **2893652**
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*Random Summation. Limit Theorems and Applications*, CRC Press, Boca Raton, FL, 1996. MR **1387113**
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*Limit Distributions for Random Sequences with Random Indices and Their Applications*, Doctor of Science dissertation, Moscow State University, 1993.
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Additional Information

**D. Silvestrov**

Affiliation:
Department of Mathematics, Stockholm University, 106 81 Stockholm, Sweden

Email:
silvestrov@math.su.se

Keywords:
Random field,
random stopping,
convergence in distribution

Received by editor(s):
July 10, 2021

Published electronically:
December 7, 2021

Article copyright:
© Copyright 2021
Taras Shevchenko National University of Kyiv