Abstract
We consider a sequence of random variables (Rn) defined by the recurrence Rn=Qn+MnRn−1,n≥1, where R0 is arbitrary and (Qn,Mn),n≥1, are i.i.d. copies of a two-dimensional random vector (Q,M), and (Qn,Mn) is independent of Rn−1. It is well known that if E ln|M|<0 and E ln+|Q|<∞, then the sequence (Rn) converges in distribution to a random variable R given by Rd=∑∞k=1Qk∏k−1j=1Mj, and usually referred to as perpetuity. In this paper we consider a situation in which the sequence (Rn) itself does not converge. We assume that E ln|M| exists but that it is non-negative and we ask if in this situation the sequence (Rn), after suitable normalization, converges in distribution to a non-degenerate limit.
Citation
Paweł Hitczenko. Jacek Wesołowski. "Renorming divergent perpetuities." Bernoulli 17 (3) 880 - 894, August 2011. https://doi.org/10.3150/10-BEJ297
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