On the Normal Approximation for the Distribution of the Number of Simple or Compound Patterns in a Random Sequence of Multi-state Trials

Abstract

Distributions of numbers of runs and patterns in a sequence of multi-state trials have been successfully used in various areas of statistics and applied probability. For such distributions, there are many results on Poisson approximations, some results on large deviation approximations, but no general results on normal approximations. In this manuscript, using the finite Markov chain imbedding technique and renewal theory, we show that the number of simple or compound patterns, under overlap or non-overlap counting, in a sequence of multi-state trials follows a normal distribution. Poisson and large deviation approximations are briefly reviewed.