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Homogeneous conservative Wiener-Hopf equation. (English. Russian original) Zbl 1143.45001
Sb. Math. 198, No. 9, 1341-1350 (2007); translation from Mat. Sb. 198, No. 9, 123-132 (2007).
A function $$S(x)$$ satisfying the equation
$S(x)= \int^\infty_0 k(x-y)S(y)\,dy,\;x> 0,\quad k(x)= k(-x)\geq 0,\quad \int^\infty_{-\infty} k(x)\,dx= 1,$ is said to be its $$P^*$$-solution if it is nondecreasing, right-continuous, non-trivial and $$S(x)= 0$$ for $$x< 0$$.
Main result: The renewal function $$u_+(x)$$ is a $$P^*$$-solution of the homogeneous generalized Wiener-Hopf equation
$S(x)= \int^x_{-\infty} S(x-y) F(dy),\quad x\geq 0,$ where $$F$$ is a distribution of the recurrent type, with the condition $$u_+(0+)= 1$$. Asymptotic properties of such solutions are also studied.

##### MSC:
 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 60G50 Sums of independent random variables; random walks
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