Waiting Time Random Variables: Upper Bounds

Using the Markov chain method and certain results from the Markov chain theory, we give (to illustrate our method) upper bounds for P(X = n) and P(X > n) when X = the waiting time of kth occurrence of s in an s-f sequence of trials (k >= 1; s - "success", f - "failure") and when X = the waiting time of a pattern Theta = a(i1) a(i2) ... a(ik) with a(i1) = a(i2) = = .... = a(ik) or a(i2), a(i3),...,a(ik) not equal a(i1) in an a(1)-a(2)-...-a(m) sequence of trials (m >= 2, k >= 1, i(1), i(2),...,i(k) is an element of < m >). Moreover, a more general case than the latter one, namely, X = the waiting time of a pattern Theta = a(i1) a(i2) ... a(i lambda) in an a(1)-a(2)-...-a(m) sequence of trials (m >= 2, k > 1, i(1), i(2),...,i(k) is an element of < m >) is considered our method works in each special case of this one. In this article, the trials are only independent they are or not identically distributed. On the other hand, we give two upper bounds for the expectation of a discrete waiting time random variable and, further, two applications of one of them.