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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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