On a theorem of Breiman and a class of random difference equations. (English) Zbl 1141.60041

Denisov, Denis; Zwart, Bert

J. Appl. Probab. 44, No. 4, 1031-1046 (2007).

Consider the tail behavior of the product of two independent nonnegative random variables \(X\) and \(Y\). Assume that \(X\) is a regularly varying random variable with index \(\alpha\). The tail behavior of the product \(X Y\) is investigated when \(Y\) satisfies conditions which are weaker than \(E [Y^{\alpha+\varepsilon}] < +\infty\) for some \(\varepsilon > 0\) (called Breiman condition). This work continues studies of [L. Breiman, Theor. Probab. Appl. 10, 323–331 (1965; Zbl 0147.37004)] and also applies its results to random difference equations.
It would be interesting to extend the studies to the case of dependent random variables \(X\) and \(Y\).

Reviewer: Henri Schurz (Carbondale)

MSC:

60H25	Random operators and equations (aspects of stochastic analysis)
60F10	Large deviations

Keywords:

tail behavior; products of random variables; regularly varying random variables; subexponential distribution; Breiman’s Theorem

Citations:

Zbl 0147.37004