Weak convergence using higher-order cumulants

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 andjJ. Let (F_nn1) be a sequence of distribution functions, and suppose that there existsJ such thatc_j(F_n)c_j(F) asn, forj=1,2 andjJ. It is proved thatF_nF 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.