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

Goldie, Charles M.

Ann. Appl. Probab. 1, No. 1, 126-166 (1991).

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.

Reviewer: J.L.Teugels (Heverlee)

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.)

Keywords:

random equations; additive Markov process queues; renewal theory

Citations:

Zbl 0291.60029; Zbl 0587.60057