The study of success runs in Bernoulli trials has attracted indubitable attention of several researchers both for its inherent theoretical interest and intriguing applications in numerous scientific fields. In this PhD dissertation, we study T-contaminated head runs for the cases T=1,2 in which we present direct probablistic calculations.

We focus on the limiting distributional problems of run related random variables. This include Compound Poisson distribution as the limiting distribution of the number of the at most T-contaminated head run, exponential distribution as the limiting distribution of the first hitting time of a specified length of T-contaminated head run and accompanying distribution as asymptotic distribution for the length of the longest T-contaminated head run.

This dissertation consists of four chapters; Introduction, Limit theorems of T-contaminated runs of heads, Convergence rate for the longest T-contaminated head runs and Limit theorems for runs containing two types of contamination together with Summary and Appendix. It is based on three published papers with the candidate as author.

In chapter 1, we present some basic definitions and notations useful in the sequel. We introduce the theorems regarding the number, the waiting time of T-contaminated head run together with theorem of the accompanying distribution of the length of the longest T-contaminated head run. Simulation results are provided to reinforce our theoretical findings.

In chapter 2, we find the asymptotic distribution for the first hitting time of the T-contaminated run of heads having a specified length. Further to this, we concentrate on obtaining a limit theorem for the length of the longest T-contaminated head run. We give a proof
that the rate of convergence of our approximation of the accompanying distribution for the length of the longest T-contaminated head run performs exceedingly better than previous known results. We provide accompanying simulation results for the same.

In chapter 3, we study sequences of trials having three outcomes labelled; success, failure of type I and failure of type II. We obtain the limiting distribution of the first hitting time and the accompanying distribution for the length of the longest at most two-type contaminated run.

Besides the mathematical proofs, we provide simulation results supporting our theorems.

In chapter 4, we give a summary of chapters; 1, 2 and 3. Finally, we give in the Appendix, the main lemma of Csaki et al. We rewrite the proof for the non stationary case, finite form giving some additional explanation with a goal of precisely fixing the conditions of the lemma. We correct the misprints and omissions noted in the lemma which are important for our subsequent applications.