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Weak convergence using higher-order cumulants

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Abstract

Denote byc j (F) thejth cumulant (or ‘semi-invariant’) of the distribution functionF. We say thatF is ‘specified by its higher-order cumulants’ if it is the unique distribution functionG having the following property: there exists a positive integerJ such thatc j (G)=c j (F) forj=1,2 andj≥J. Let (F n ∶n≥1) be a sequence of distribution functions, and suppose that there existsJ such thatc j (F n )→c j (F) asn→∞, forj=1,2 andj≥J. It is proved thatF n ⇒F so long asF is specified by its higher-order cumulants. It is an open problem to characterize the family of distributions which are specified by their higher-order cumulants.

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  1. School of Mathematics, University of Bristol, University Walk, BS8 1TW, Bristol, England

    Geoffrey Grimmett

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  1. Geoffrey Grimmett
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Grimmett, G. Weak convergence using higher-order cumulants. J Theor Probab 5, 767–773 (1992). https://doi.org/10.1007/BF01058728

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