Geodesic
A renewal equation in a multidimensional space
Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 4, pp. 779-785

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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
Mots-clés : joint motion. 
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     author = {N. B. Engibaryan},
     title = {A renewal equation in a multidimensional space},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {779--785},
     publisher = {mathdoc},
     volume = {49},
     number = {4},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2004_49_4_a8/}
}
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N. B. Engibaryan. A renewal equation in a multidimensional space. Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 4, pp. 779-785. http://geodesic.mathdoc.fr/item/TVP_2004_49_4_a8/