On the ratio of the expected maximum of a martingale and the L p-norm of its last term

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Abstract

For each p>1, the supremum, S, of the absolute value of a martingale terminating at a random variable X in L p, satisfies ES≦(Γ(q))1/q{norm of matrix}X{norm of matrix}p (q=p(p-1)-1).The maximum, M, of a mean-zero martingale which starts at zero and terminates at X, satisfies ES≦(Γ(q))1/q{norm of matrix}X{norm of matrix}p (q=p(p-1)-1), where σ q is the unique solution of the equation t = {norm of matrix}Z -t {norm of matrix} q for an exponentially distributed random variable Z with mean 1. σ p has other characterizations and satisfies lim p{norm of matrix} q - 1 σ q =c with c determined by ce c+1 = 1. Equalities in (1) and (2) are attainable by appropriate martingales which can be realized as stopped segments of Brownian motion. A presumably new property of the exponential distribution is obtained en route to inequality (2).

Original languageEnglish
Pages (from-to)270-280
Number of pages11
JournalIsrael Journal of Mathematics
Volume63
Issue number3
DOIs
StatePublished - Oct 1988

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