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Unformatted text preview: ter the matrix into the computer as say
[A]. The selecting the NAMES we can enter [A] ^ 2 on the computation line to
get A2 . If we use this procedure to compute A20 we get a matrix with three
rows that agree in the ﬁrst six decimal places with
.468085 .340425 .191489 Later we will see that as n ! 1, pn converges to a matrix with all three rows
equal to (22/47, 16/47, 9/47).
To explain our interest in pm we will now prove:
Theorem 1.1. The m step transition probability P (Xn+m = j |Xn = i) is the
mth power of the transition matrix p.
The key ingredient in proving this is the Chapman–Kolmogorov equation
pm+n (i, j ) =
pm (i, k ) pn (k, j )
k Once this is proved, Theorem 1.1 follows, since taking n = 1 in (1.2), we see
pm+1 (i, j ) =
pm (i, k ) p(k, j )
k That is, the m +1 step transition probability is the m step transition probability
times p. Theorem 1.1 now follows.
Why is (1.2) true? To go from i to j in m + n steps, we have to go from i to
some state k in m steps and then from k to j in n steps. The Markov property
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
- Spring '10
- The Land