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Escape rates for multidimensional shift self-similar additive sequences. (English) Zbl 1354.60040

Let \(a>1\). A random sequence \(\{W(n):-\infty <n<\infty\}\) on \(\mathbb R^d\) is called a shift \(a\)-self-similar additive sequence if (i) the sequence is shift \(a\)-self-similar, that is, \(\{W(n+1): -\infty<n<\infty\} =\{a W(n):-\infty<n<\infty\}\) (where equality is in the sense of equality of the finite-dimensional distributions) and (ii) the sequence \(\{W(n)\}\) has independent increments, that is, for every \(n\) the set of random variables \(\{W(k): k \leq n\}\) and the random variable \(W(n+1)-W(n)\) are independent. The author studies the rate of escape (liminf behaviour) for shift self-similar additive sequences. Some applications to the laws of iterated logarithm for strictly stable Lévy processes on \(\mathbb R^d\) and independent Brownian motions are discussed.

MSC:

60G18 Self-similar stochastic processes
60G10 Stationary stochastic processes
60G51 Processes with independent increments; Lévy processes
60F15 Strong limit theorems
60J65 Brownian motion
60G52 Stable stochastic processes
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