Limit theorems related to a class of operator-self-similar processes

Abstract

An Rd-valued (d ≥ 1) stochastic process X = {X(t)}t≥0 is said to be operator-self-similar if there exists a linear operator D on Rd such that for each c > 0where means the equality for all finite-dimensional distributions and

The extremes of random walks in random sceneries

“…The case α = 1 is more difficult, since the underlying random walk then is only zero-recurrent. Beside the fundamental work of Kesten and Spitzer, a lot of refinements and generalisations in various directions were obtained by other authors (see Lang and Nguyen (1983), Shieh (1995), Maejima (1996), Arai (2001), Saigo and Takahashi (2005)). …”

Limit theorems for U-statistics indexed by a one dimensional random walk

“…Before we begin the proof of Proposition 3.1 we need to give details of one of the results of Maejima [14].…”

“…We suppose that the ξ x x∈Z belong to the domain of attraction of a Gaussian vector Z 1/2 . Then, Maejima [14] proved that for the linear operator…”

“…For an arbitrary transient random walk, n − 1 2 Z n is asymptotically normal (see [19], p. 53). Maejima [14] generalized the result of Kesten and Spitzer [12] In general, in the theory of U -statistics (of order 2), the classical object for study is the sequence…”

A self-similar process arising from a random walk with random environment in random scenery

“…There exist various generalizations of the results of Kesten and Spitzer (1979). We will only mention Shieh (1995), where the limiting process is generalized to higher dimensions, Lang and Nguyen (1983), which deals with multidimensional random walks and some special random scenery, Maejima (1996), where the random scenery belongs to the domain of attraction of an operator-stable distribution, Arai (2001), where the random scenery belongs to the domain of partial attraction of a semi-stable distribution, and Saigo and Takahashi (2005), where the random scenery and the random walk belong to the domain of partial attraction of semi-stable and operator semi-stable distributions.…”