The Fundamental Matrix of the General Random Walk with Absorbing Boundaries

Abstract

Abstract. The general random walk on the nonnegative integers with absorbing boundaries at 0 and n has the transition probabilities p0j = 0j , pnj = nj, pi;i 1 = pi, pi;i+1 = qi, and pii = ri, where pi + ri + qi = 1. The fundamental matrix B of this Markov chain is the inverse of matrix (I Q) where Q results from P by deleting the rows and columns 0 and n. Entry bij represents the expected number of occurrences of the transient state j prior to absorption if the random walk starts at state i. The absorption time as well as the absorption probabilities are easily derived once the fundamental matrix is known. Here, it is shown that the fundamental matrix can be determined in elementary manner via the adjugate of matrix (I Q).