On a stochastic difference equation and a representation of non–negative infinitely divisible random variables

Abstract

The present paper considers the stochastic difference equation Y n = A n Y n-1 + B n with i.i.d. random pairs (A n , B n ) and obtains conditions under which Y n converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A 1 A 2 ··· A n-1 B n . A second subject is the series ∑ C n f(T n ) with (C n ) a sequence of i.i.d. random variables, (T n ) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case A n and C n are independent, B n = A n C n , A n = U 1/α n with U n a uniform random variable, f(x) = e −x/α.

A probabilistic approach to scaling equations

“…However, the analysis in the proof above also applies in an abstract probability space which supports a sequence of independent, identically distributed random variables. We have subsequently noticed that our techniques establishing existence and almost everywhere convergence in that context produce results similar to those obtained by Vervaat [13] who studied a closely related problem. In addition, the methodology we use below to study L p convergence also applies in that setting and leads to L p results which include those in [13].…”

Tail-homogeneity of stationary measures for some multidimensional stochastic recursions

“…For a subadditive norm | | the proof is as in [24]. When C 0 in (D.1) is strictly bigger than 1 then we observe that there are C, l > 0 such that |x 1 · · · x n | ≤ n j=2 j l |x j | + C 2 0 |x 1 |.…”

“…Proof As in [24] we consider the series ∞ k=0 M 0 · · · M k−1 Q k and we observe that its general term converges exponentially fast to zero a.e.. Indeed, we have…”

Properties of residuals for spatial point processes

“…This equivalence is discussed by Vervaat (1979) with references to its prior occurrence. As shown in Appendix A, Eq.…”