2015
DOI: 10.1214/ecp.v20-3915
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Functional limit theorems for divergent perpetuities in the contractive case

Abstract: Let M k , Q k k∈N be independent copies of an R 2 -valued random vector. It is known that if Y n := Q 1 + M 1 Q 2 + . . . + M 1 · . . . · M n−1 Q n converges a.s. to a random variable Y , then the law of Y satisfies the stochastic fixed-point equationIn the present paper we consider the situation when |Y n | diverges to ∞ in probability because |Q 1 | takes large values with high probability, whereas the multiplicative random walk with steps M k 's tends to zero a.s. Under a regular variation assumption we sho… Show more

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Cited by 10 publications
(11 citation statements)
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“…Among the few works which deal with recurrence versus transience of AR(1) processes are the unpublished preprint [Kel92, Part I] and [ZG04]. (For some recent work with deals with super-heavy tailed innovations see [BI15].) Both papers treat one-dimensional AR(1) processes with constant coefficient A 1 = a ∈ (0, 1).…”
Section: Introductionmentioning
confidence: 99%
“…Among the few works which deal with recurrence versus transience of AR(1) processes are the unpublished preprint [Kel92, Part I] and [ZG04]. (For some recent work with deals with super-heavy tailed innovations see [BI15].) Both papers treat one-dimensional AR(1) processes with constant coefficient A 1 = a ∈ (0, 1).…”
Section: Introductionmentioning
confidence: 99%
“…When the random series diverges, it is natural to investigate weak convergence on D of its partial sums, properly rescaled, as the number of summands becomes large. Some results of this flavor can be found in [4] and [9] (in these works many references to earlier one-dimensional results can be found). Here, we prove functional limit theorems in the situations that remained untouched.…”
Section: Introduction and Resultsmentioning
confidence: 94%
“…R 2 -valued random vectors independent of B 0 . Functional limit theorems for log + (X [n·] ) were obtained in [4] under the assumption that (1.1) holds with B 1 replacing J and that lim n→∞ (A 1 · . .…”
Section: Introduction and Main Resultsmentioning
confidence: 99%