Executive Summary : | The journey of runs has started by de Moivre in the year 1738 with the statement that "what is the probability of getting a run of length r or more in n trials". Later on, the theory was developed by Simpson (1740), Laplace (1812), Todhunter (1865) and Marbe (1934) among others. In the last decades, it became one of the most renowned topics in the literature due to its applicability in various fields of science such as molecular biology, DNA sequences, climatology, meteorology and agriculture, reliability theory, statistical testing and moving window detection problems, among many others. For instance, in molecular biology, DNA sequence matching models are studied to develop quantitative measures for assessing and interpreting genomic inhomogeneities for various species or units due to several chemical infections. In the beginning, the distributional properties of runs have been studied primarily for independent and identically distributed (iid) Bernoulli trials. In particular, in a sequence of Bernoulli trials (with outcomes success and failure), significant developments have been done with several applications for the following runs (i) consecutive k successes or failures which is called the distribution of order k or k-runs and (ii) consecutive k₁ failures followed by consecutive k2 successes which is known as the distribution of order (k₁, k2) or (k₁, k2)-runs. The (k₁, k2)-run was first introduced by Huang and Tsai (1991) and later on, many researchers work towards the development of the theory in various fields of probability and statistics. Note that k-runs and (k₁, k2)-runs considered only two states (0 and 1), however, the experiment can have more than two states in several real-life applications. For instance, if one is observing DNA sequences then a sequence of four alphabets should be considered. Therefore, the study of runs in a multi-state situation is an extremely significant work in pursuance of many applications but it is a big challenge to develop the theory for multi-state trials. However, Kong (2022) initiated the first research work on (k1, k2, ..., kl)-runs and a few works were done in this direction but only for iid trials. It is therefore interesting to study runs in multi-state trials with applications in various fields of probability and statistics. In this project, the principal investigator (P.I.) together with one scholar will focus on the following two key problems. (i) Identifying appropriate runs in the multi-state trials which are relevant in real-life applications and studying the distributional properties under iid and Markov-dependent trials. (ii) Investigating the long-run behaviour of the identified runs. The P.I. of the project is confident regarding the output of this research, which will surely be applied to DNA sequences in many real-life situations. |