Waiting times and generalized Fibonacci sequences

Suppose we have a multinormal population with k possible outcomes E/sub 1/, E/sub 2/, ..., E/sub k/ and associated probabilities ..pi../sub 1/, ..pi../sub 2/, ..., ..pi../sub k/. At each of the independent trials, one of the outcomes is observed. One may be interested in the waiting time for the occurrence of a specified event, which consists of a succession of outcomes. In this paper, we consider the probability distribution of the waiting times associated with specified events, and show how they generalize the Fibonacci, Tribonacci, ..., sequences in different ways. This is possible, since the probability generating functions of the associated waiting time random variables can be utilized to derive the probability distributions.