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 $R^n$.
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