A Caltech Library Service

Some Central Limit Theorems for Doubly Restricted Partitions


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.


Let PK, 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 PK,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 PK,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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:(Mathematics)
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Apostol, Tom M.
Thesis Committee:
  • Unknown, Unknown
Defense Date:3 May 1965
Record Number:CaltechTHESIS:10122015-160506112
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9215
Deposited On:13 Oct 2015 15:06
Last Modified:08 Mar 2024 00:27

Thesis Files

PDF - Final Version
See Usage Policy.


Repository Staff Only: item control page