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- W1991536141 abstract "Let ${X_i}$ be a sequence of independent, identically distributed non-degenerate random variables taking values in $mathbb{R}^d$ and $S_n = sum^n_{i = 1} X_i, M_n = max_{1leqq i leqq n} |S_i|$. Define for $x > 0, G(x) = P{| X_1 | > x}, K(x) = x^{-2}E(| X_1 |^2 1{| X_1 | leq x}), M(x) = x^{-1} |E(X_1 1{| X_1 | leq x})|,$ and $h(x) = G(x) + K(x) + M(x)$. Then if $beta = sup {alpha: lim sup x^alpha h(x) = 0}, delta = sup {alpha: lim inf x^alpha h(x) = 0}$, it is proved that $n^{-1/alpha}M_n rightarrow 0$ for $alpha delta$, while the $lim inf$ is 0 and the $lim sup$ is $infty$ for $beta < alpha < delta$. Some alternative characterizations of the indices $beta, delta$ are obtained as well as the analogous results for Levy processes." @default.
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- W1991536141 date "1981-12-01" @default.
- W1991536141 modified "2023-10-18" @default.
- W1991536141 title "The Growth of Random Walks and Levy Processes" @default.
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- W1991536141 doi "https://doi.org/10.1214/aop/1176994266" @default.
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