Some Central Limit Theorems for Doubly Restricted Partitions

Citation

Skarda, Ralph Vencil, Jr. (1966) Some Central Limit Theorems for Doubly Restricted Partitions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/YH19-JH87. https://resolver.caltech.edu/CaltechTHESIS:10122015-160506112

Abstract

Let P_{K, L}(N) be the number of unordered partitions of a positive integer N into K or fewer positive integer parts, each part not exceeding L. A distribution of the form

Ʃ/N≤x P_K,L(N)

is considered first. For any fixed K, this distribution approaches a piecewise polynomial function as L increases to infinity. As both K and L approach infinity, this distribution is asymptotically normal. These results are proved by studying the convergence of the characteristic function.

The main result is the asymptotic behavior of P_K,K(N) itself, for certain large K and N. This is obtained by studying a contour integral of the generating function taken along the unit circle. The bulk of the estimate comes from integrating along a small arc near the point 1. Diophantine approximation is used to show that the integral along the rest of the circle is much smaller.

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