## A probabilistic look at the Wiener-Hopf equation.(English)Zbl 0926.60040

The Wiener-Hopf integral equation $Z(x)=z(x)+ \int^x_{-\infty} Z(x-y)F(dy), \quad x\geq 0,$ is analysed by purely probabilistic methods. The author investigates the nonnegative solutions $$Z$$ of the equation assuming $$z$$ to be nonnegative and bounded on finite intervals and $$F$$ to be a probability measure with existing mean $$\mu$$. He applies random walk techniques (ladder heights), supermartingales, coupling, $$0$$-1-laws, and exponential change of measure to characterize the set of solutions and to study their asymptotic properties, which strongly depend on the tails of the distribution $$F$$. The cases $$\mu>0$$ and $$\mu<0$$ are intrinsically different.

### MSC:

 60G50 Sums of independent random variables; random walks 60K25 Queueing theory (aspects of probability theory)
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