A Normal Limit Theorem for Moment Sequences

Abstract

Let $\Lambda$ be the set of probability measures $\lambda$ on $\lbrack 0,1\rbrack$. Let $M_n = \{(c_1,\ldots,c_n)\mid\lambda \in \Lambda\}$, where $c_k = c_k(\lambda) = \int^1_0x^k d\lambda, k = 1,2,\ldots$ are the ordinary moments, and assign to the moment space $M_n$ the uniform probability measure $P_n$. We show that, as $n \rightarrow \infty$, the fixed section $(c_1,\ldots,c_k)$, properly normalized, is asymptotically normally distributed. That is, $\sqrt n\lbrack(c_1,\ldots,c_k) - (c^0_1,\ldots,c^0_k)\rbrack$ converges to $\mathrm{MVN}(0,\Sigma)$, where $c^0_i$ correspond to the arc sine law $\lambda_0$ on $\lbrack 0,1\rbrack$. Properties of the $k \times k$ matrix $\Sigma$ are given as well as some further discussion.