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Teor. Veroyatnost. i Primenen., 2004, Volume 49, Issue 4, Pages 779–785 (Mi tvp193)  

Short Communications

A renewal equation in a multidimensional space

N. B. Engibaryan

Byurakan Astrophysical Observatory, National Academy of Sciences of Armenia

Abstract: The following renewal equation in a multidimensional space (REMS) is considered
$$ f(x)=g(x)+\int_{R^n}K(x-t) f(t) dt, $$
where $K$ is the density of a distribution in $R^n$. Assuming that $g\in L_1(R^n)$ and that the nonzero vector of the first moment of $K$ is finite we prove the existence and uniqueness of a solution of an REMS within a certain class of functions. The renewal density for the solution of this equation is constructed and its properties are investigated. We give a probabilistic interpretation for our results by means of an example from the theory of random walks in $R^n$.

Keywords: renewal, multidimensional space, solvability, joint motion.

DOI: https://doi.org/10.4213/tvp193

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English version:
Theory of Probability and its Applications, 2005, 49:4, 737–744

Bibliographic databases:

Received: 30.07.2002

Citation: N. B. Engibaryan, “A renewal equation in a multidimensional space”, Teor. Veroyatnost. i Primenen., 49:4 (2004), 779–785; Theory Probab. Appl., 49:4 (2005), 737–744

Citation in format AMSBIB
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\jour Theory Probab. Appl.
\yr 2005
\vol 49
\issue 4
\pages 737--744
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