Visitations of ruled sums

Abstract

Let { X i } \{ {X_i}\} be a sequence of independent identically distributed random variables and for D ⊆ I + D \subseteq {I^ + } let S D = Σ i ∈ D X i {S_D} = {\Sigma _{i \in D}}{X_i} . A rule ( ) (\;) is a mapping I + → 2 I + : ∀ n | ( n ) | = n {I^ + } \to {2^{{I^ + }}}:\forall n|(n)| = n and S ( ) = { S ( n ) } {S_{(\;)}} = \{ {S_{(n)}}\} is its associated ruled sum. Ruled sums generalize ordinary sums S n {S_n} . Indeed, all a.e. results for S n {S_n} can be investigated for S ( n ) {S_{(n)}} frequently requiring different methods and sometimes yielding different conclusions. In a previous paper we studied strong laws of large numbers and the law of the iterated logarithm. In this paper we study infinite visitation. Under suitable hypotheses on the basic distribution function F of the X i {X_i} we show that, for all rules ( ) , S ( n ) (\;),{S_{(n)}} visits each integer infinitely often a.e. in the lattice case (or has all points of the real line as accumulation points in the nonlattice case). In fact we obtain a “rate of visitation.” There follows extensions of the Pólya theorem on encounters in the plane and 3-space from random walks to these ruled sums. Some equivalence relations and partial orderings on rules are defined. For normal variables this leads to an extension of the previously mentioned result for ruled sums of the type of the iterated logarithm law.