## Implicit renewal theory and tails of solutions of random equations.(English)Zbl 0724.60076

Let $$\Psi$$ be a random element of the space of Borel-measurable functions from $${\mathbb{R}}$$ to $${\mathbb{R}}$$. Let R be a random variable independent of $$\Psi$$. The author treats the random equation $$R=\Psi \circ R$$ where the equality denotes equality in probability law. After setting out this implicit renewal theory with corresponding rate results, more special equations are tackled. For example
(i) $$\Psi (t)=Q+MT$$, where Q and M are random and where a one-dimensional version of a result by H. Kesten [Acta Math. 131, 207-248 (1973; Zbl 0291.60029)] receives a new and simpler treatment;
(ii) $$\Psi (t)=Q+M \max (L,t)$$, where $$M\geq 0$$ a.s. and treated by G. Letac [Random matrices and their applications, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 50, 263-273 (1986; Zbl 0587.60057)];
(iii) $$\Psi (t)=\max (Q,Mt)$$ with applications on the G/G/1-queue.
Other examples are scattered throughout the paper. Proofs are gathered in a separate section, and show how classical renewal theory can be beautifully exploited to attack the more complicated implicit renewal theory. The paper ends with a discussion of the implications for extreme- value theory and an illustration from economics.

### MSC:

 60H25 Random operators and equations (aspects of stochastic analysis) 60K05 Renewal theory 60K25 Queueing theory (aspects of probability theory) 60K10 Applications of renewal theory (reliability, demand theory, etc.)

### Citations:

Zbl 0291.60029; Zbl 0587.60057
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