Inequalities for max|𝑆_{𝑘}|/𝑏_{𝑘} where 𝑘∈𝑁^{𝑟}

Shorack, Galen R.; Smythe, R. T.

doi:10.1090/S0002-9939-1976-0400386-4

Inequalities for $\max |S_k|/b_k$ where $k \in N^r$
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by Galen R. Shorack and R. T. Smythe PDF

Proc. Amer. Math. Soc. 54 (1976), 331-336 Request permission

Abstract:

Lemma $1$ presents a powerful general inequality for $\max |{S_{\mathbf {k}}}|/{b_{\mathbf {k}}}$. This is applied in multidimensional time to sums of independent random variables and martingales to yield both old and new inequalities of the Doob, Hájek-Rényi, Skorokhod and Marcinkiewicz-Zygmund types. A brief application is made to the partial sum process.

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