Limit theorems and convergence rate for longest contaminated runs of heads
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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
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
This dissertation consists of four chapters; Introduction, Limit theorems of
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
In chapter 2, we find the asymptotic distribution for the first hitting time of the
that the rate of convergence of our approximation of the accompanying distribution for the length of the longest
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.