A strong law for a set-indexed partial sum process with applications to exchangeable and stationary sequences

Andrew Rosalsky

Stochastic Processes and their Applications, 1987, vol. 26, 277-287

Abstract: Consider a set-indexed partial sum process {[Sigma]j[epsilon]Jn Yj, n [greater-or-equal, slanted] 1} where {Yn, n [greater-or-equal, slanted] 1} are identically distributed random variables and {Jn, n [greater-or-equal, slanted] 1} is a nondecreasing sequence of finite sets of positive integers with j(n) = Card(Jn) --> [infinity]. A strong law of the form [Sigma]j[epsilon]Jn Yj/bj(n) --> 0 almost certainly is established where {bn, n [greater-or-equal, slanted] 1} are constants with bn/n --> [infinity]. As special cases, new results are obtained for exchangeable and stationary sequences. The result for stationary sequences strengthens a weak law proved by Maller [9] in that the convergence is shown to be almost certain.

Keywords: set-indexed; partial; sum; process; strong; law; of; large; numbers; almost; certain; convergence; identically; distributed; random; variables; exchangeable; sequences; U-statistics; stationary; sequences; pointwise; ergodic; theorem (search for similar items in EconPapers)
Date: 1987
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