Sankhya: The Indian Journal of Statistics

1999, Volume 61, Series A, Pt. 3, 347-357

PARTIAL HAUSDORFF SEQUENCES AND SYMMETRIC PROBABILITIES ON FINITE PRODUCTS OF { 0, 1}

By

J.C. GUPTA, * Indian Statistical Institute, Calcutta*

SUMMARY. Let $\bh_n$ be the set of all partial Hausdorff sequences of order $n$, i.e., sequences $c_n(0), c_n(1), \ldots c_n(n), c_n(0)=1$, with $(-1)^m \btu^m c_n (k) \ge 0$ whenever $m+k \le n$. Further, let $\bp_n$ be the set of all symmetric probabilities on $\{0,1\}^n$. We study the interplay between the sets $\bh_n$ and $\bp_n$ to formulate and answer interesting questions about both. Assigning to $\bh_n$ the uniform probability measure we show that, as $n \rightarrow \infty$, the fixed section $(c_n(1), c_n(2), \ldots , c_n(k))$, properly centered and normalized, is asymptotically normally distributed. That is, $\sqrt{n} (c_n(1) - c_0(1), c_n(2)-c_0(2), \ldots , c_n(k) - c_0(k))$, converges weakly to MVN $(0, \Sigma)$, where $c_0(i)$ correspond to the moments of the uniform law $\lambda$ on $[0,1];$ the asymptotic covariances also depend on the moments of $\lambda$.

*AMS (1991) subject classification.* 60F05.

*Key words and phrases. *Partial Hausdorff sequences, symmetric
probabilities on finite products of {0,1}, normal limit.