## 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 language | English |
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Pages (from-to) | 270-280 |

Number of pages | 11 |

Journal | Israel Journal of Mathematics |

Volume | 63 |

Issue number | 3 |

DOIs | |

State | Published - Oct 1988 |

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