In this thesis, renewal theory is used to analyze patterns of outcomes for discrete random variables. We start by introducing the concept of a renewal process and by giving some examples. It is then shown that we may obtain the distribution of the number of renewals by time t and compute its expectation, the renewal function. We then proceed to state and illustrate the basic limit theorems for renewal processes and make use of Wald’s equation to give a proof of the Elementary Renewal Theorem. The concept of a delayed renewal process is introduced for the purpose of studying the patterns mentioned above. The mean and variance of the number of trials necessary to obtain the first occurrence of the desired pattern are found as well as how these quantities change in cases where a pattern may overlap itself. In addition to this, we explore the expected number of trials between consecutive occurrences of the pattern. If our attention is restricted to patterns from a given finite set, the winning pattern may be defined to be the one which occurs first and the game ends once a winner has been declared. We compute the probability that a given pattern is the winner as well as the expected length of the game. In concluding our discussion, we explore the expected waiting time until we obtain a complete run of distinct or increasing values.

Library of Congress Subject Headings

Renewal theory; Pattern recognition systems

Publication Date


Document Type


Student Type


Degree Name

Applied and Computational Mathematics (MS)

Department, Program, or Center

School of Mathematical Sciences (COS)


James Marengo

Advisor/Committee Member

Bernard Brooks

Advisor/Committee Member

Manuel Lopez


Physical copy available from RIT's Wallace Library at T57.33 .K54 2016


RIT – Main Campus