Tail probabilities for infinite series of regularly varying random vectors

Details Tail probabilities for infinite series of regularly varying random vectors Tail probabilities for infinite series of regularly varying random vectors

2007-02-05

arxiv.org/abs/math/0702112v2

Bernoulli 2008, Vol. 14, No. 3, 838-864

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/08-BEJ125 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/08-BEJ125

A random vector $X$ with representation $X=\sum_{j\geq0}A_jZ_j$ is considered. Here, $(Z_j)$ is a sequence of independent and identically distributed random vectors and $(A_j)$ is a sequence of random matrices, `predictable' with respect to the sequence $(Z_j)$. The distribution of $Z_1$ is assumed to be multivariate regular varying. Moment conditions on the matrices $(A_j)$ are determined under which the distribution of $X$ is regularly varying and, in fact, `inherits' its regular variation from that of the $(Z_j)$'s. We compute the associated limiting measure. Examples include linear processes, random coefficient linear processes such as stochastic recurrence equations, random sums and stochastic integrals.

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