Iterated random functions and regularly varying tails

ABSTRACT

We consider solutions to so-called stochastic fixed point equation $R\stackrel{d}{=}\mathrm{\Psi }\left(R\right)$, where Ψ is a random Lipschitz function and R is a random variable independent of Ψ. Under the assumption that Ψ can be approximated by the function $x\mapsto Ax+B$, we show that the tail of R is comparable with the one of A, provided that the distribution of $\log (A\vee 1)$ is tail equivalent. In particular, we obtain new results for the random difference equation.

KEYWORDS:

• Stochastic recursions
• random difference equation
• stationary distribution
• convolution equivalent tails
• regularly varying distribution

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 60H25
• 60J10

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Plus some additional technical assumptions.

Additional information

Funding

The first author was partially supported by the Narodowe Centrum Nauki (NCN) Grant UMO-2014/15/B/ST1/00060. The second author was partially supported by the (Sonata Bis, grant number DEC-2014/14/E/ST1/00588).