XIII: 14, 174-198, LNM 721 (1979)
CAIROLI, Renzo; GABRIEL, Jean-Pierre
Arrêt de certaines suites multiples de variables aléatoires indépendantes (Several parameter processes, Independence)
Let $(X_n)$ be independent, identically distributed random variables. It is known that $X_T/T\in L^1$ for all stopping times $T$ (or the same with $S_n=X_1+...+X_n$ replacing $X_n$) if and only if $X\in L\log L$. The problem is to extend this to several dimensions, $N^d$ ($d>1$) replacing $N$. Then a stopping time $T$ becomes a stopping point, of which two definitions can be given (the past at time $n$ being defined either as the past rectangle, or the complement of the future rectangle), and $|T|$ being defined as the product of the coordinates). The appropriate space then is $L\log L$ or $L\log^d L$ depending on the kind of stopping times involved. Also the integrability of the supremum of the processes along random increasing paths is considered
Keywords: Stopping points, Random increasing paths
Nature: Original
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