On LIL behaviour for moving averages of some infinitely divisible random measures (Q1315404)

This paper examines to what extent the law of the iterated logarithm (LIL) holds for a class of Banach-valued moving average processes defined on infinitely divisible (i.d.) random measures on a metrizable locally compact separable Abelian topological group. The precise statement of the result requires considerable notation, and the following is but a first approximation. Let \(M\) denote an i.d. random measure on the group \(G\) which is generated by an i.d. law on \(R\) with Lévy measure \(\nu\). Let \(f\) be a map from \(G\) to a Banach space \(B\) which is almost surely bounded over compact subsets of \(G\) and which is nonzero on a nontrivial subset of \(G\) with respect to the Haar measure on \(G\). Define \(\xi(t)=\int_ Gf(t+r)dM(r)\). Under conditions on the variation of \(f\), the main theorem of this paper proves that for \(\psi:G\mapsto[-\infty,\infty]\), \[ \Bigl\{\omega\in\Omega:\biggl\{t\in G:\bigl\|\xi(\omega;t)\bigr\|>\psi(t)\biggr\}\text{ is not totally bounded}\Bigr\} \] has probability either 0 or 1. It is 0 if and only if there is a covering of \(G\) by open balls of suitable radius. The result is new even for moving averages of \(R\)-valued stable processes on the line. The proof is generally self-contained.