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Fu, Wang and Lou（2002） present an exact and large deviation approximation for the distribution of the longest run in a sequence of multi-state Markov dependent trials with order k<=1 . In this thesis, we extend their results to general order k and, in addition, we derive the distribution of any pattern. As an application, we derived the distribution of the frequency of n -words in DNA sequences. The finite Markov chain imbedding technique by Fu and Koutras, (1994) is used to obtain the exact distributions. For k>=2 , numerical comparisons between the exact distributions and approximations of the frequency of n -words in DNA sequences are provided to illustrate the theoretical results. Furthermore, the distributions of the longest run statistics in DNA sequences are demonstrated through real data analysis. The results obtained in the thesis can be applied to protein sequences with minor modifications.

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