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\(L^ 1\)-tightness and the law of large numbers in D(R). (English) Zbl 0673.60008

This article considers the weak and strong laws of large numbers (WLLN and SLLN) for random variables \(\{X_ i:i\geq 1\}\) taking their values in the space D (real-valued functions on the positive real line which are right-continuous and have left limits). This space has the Skorokhod metric topology. The essential condition of this paper is Condition T. This requires that for any \(\epsilon >0\), there exists a compact K in D such that for any \(q>0\) \[ \sup_{i}E(\sup \{| X_ i(t)|:0<t\leq q\}I[X_ i\not\in K])<\epsilon. \] Theorem 2.1 states that if the \(\{X_ i:i>1\}\) are pairwise-independent, centred, \(L^ 1\)-tight, and satisfy Condition T, then the partial sums of the \(X_ i\) are suitably tight in D. If the WLLN holds at each positive t, then the WLLN holds in D for \(\{X_ i:i>1\}.\)
Theorem 2.2 states a similar result for the SLLN. The \(X_ i\) are further required to be independent, and the SLLN must hold for \(\{X_ iI[X_ i\in K]\}\), K any compact set in D. Both the WLLN and SLLN are stated in terms of the uniform topology on D.
The proofs are based on the well-known results of Billingsley, and a lemma which bounds the oscillation of the sample mean by what happens on and off a compact set K.
Reviewer: A.R.Dabrowski

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F15 Strong limit theorems
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References:

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