On the relationship between convergence in distribution and convergence of expected extremes

Abstract:

It is well known that the expected values $\{ {M_k}(X)\} , k \geq 1$, of the k-maximal order statistics of an integrable random variable X uniquely determine the distribution of X. The main result in this paper is that if $\{ {X_n}\} , n \geq 1$, is a sequence of integrable random variables with ${\lim _{n \to \infty }}{M_k}({X_n}) = {\alpha _k}$ for all $k \geq 1$, then there exists a random variable X with ${M_k}(X) = {\alpha _k}$ for all $k \geq 1$ and ${X_n}\xrightarrow {\mathcal {L}}X$ if and only if ${\alpha _k} = o(k)$, in which case the collection $\{ {X_n}\}$ is also uniformly integrable. In addition, it is shown using a theorem of Müntz that any subsequence $\{ {M_{{k_j}}}(X)\} , j \geq 1$, satisfying $\sum 1/{k_j} = \infty$ uniquely determines the law of X.