Next: Example Up: Moments of inter-level passage Previous: Moments of inter-level passage   Contents

## Preliminaries

We first define the transition probability matrix, , such that its element, , represents the probability that (i) the sojourn time at state is AND (ii) the first transition out of state is to state . (Note that state denotes a state (), where . State is defined analogously.) By the following Lemma, which follows from the memoryless property of the exponential distribution, is also the product of the probabilities of events (i) and (ii).

Lemma 7   Let and be independent exponential random variables. Let . Then

Matrix has the same structural shape as , although its entries are different, and thus it can be represented in terms of submatrices, analogous to those in , indicating backward transitions, local transitions, and forward transitions, as follows:

Specifically, the element of is the probability that the sojourn time in state is AND the first transition out of state is to state . Likewise, the element of (respectively, ) is the probability that the sojourn time in state is AND the first transition out of state is to state (respectively, to state ).

Next, we define the -th moment of submatrices, , , , as follows:

for , and , where an integral of a matrix is a matrix of the integrals of the elements in . For the repeating part, we define , , and , omitting the superscript.

We now define the limits as of , , and as follows:

for . For the repeating part, we define , , and , omitting the superscript.

Subsections

Next: Example Up: Moments of inter-level passage Previous: Moments of inter-level passage   Contents
Takayuki Osogami 2005-07-19