Waiting for the pattern 123123 … 123 in independent trinomial observations

Classroom note: Waiting for a sequence pattern in independent multinomial observations

“…The results (2.5) and (3.5) agree with similar ones obtained in [1,4,5,8] when N ¼ 1, 2 or 3. Since f g (s) determines the distribution of X, other parameters such as the variance, V[X], and the probability P(X ¼ i) can be obtained from knowledge of the geometric transform.…”

“….123. . .N for any positive integer N. The terminology and method is the same as in [5,8]. It is clear from this last example that the sequence pattern will occur less often and thus the expected number of observations to produce a pattern will have a large value.…”

“…It is clear from this last example that the sequence pattern will occur less often and thus the expected number of observations to produce a pattern will have a large value. As in [5,8] we shall use the following notations. Let…”

“…This comment motivates this paper. We thus generalize the study in [8] by calculating the expected number of observations to produce the sequence pattern 123. . .N123.…”

“….HT. Subsequently, the author has in [8] adapted Rade's geometric transform approach to study the expected number of observations until the sequence pattern 123123. . .123 and commented that the approach could be extended to multinomial observations.…”