- #1
JohanL
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Homework Statement
Calculate the limit
$$lim_{s,t→∞} R_X(s, s+t) = lim_{s,t→∞}E(X(s)X(s+t))$$
for a continuous time Markov chain
$$(X(t) ; t ≥ 0)$$
with state space S and generator G given by
$$S = (0, 1)$$
$$ G=
\begin{pmatrix}
-\alpha & \alpha \\
\beta & -\beta\ \\
\end{pmatrix}
$$
respectively, where α, β > 0 are given constants.
Homework Equations
The Attempt at a Solution
[/B]
The chain is irreducible with stationary distribution $$π = ( \frac{\beta}{\alpha + \beta} \frac{\alpha}{\alpha + \beta})$$
(as this gives πG = 0).
This i figured out on my own and understand but the next part of the solution i don't get at all.
If anyone can give me an explanation or some hints or point me to an online or offline source
that covers this i would really appreciate it.
Noting that X(s)X(t) = 1 when both X(s) and X(t) are 1 while X(s)X(t) = 0 otherwise
it follows that
$$E(X(s)X(t)) = (\mu^{(s)})_{1} p_{11}(t) = (\mu^{(0)}P_{s})_{1} p_{11}(t) = (\mu^{(0)})_{0} p_{01}(s)+(\mu^{(0)})_{1} p_{11}(s) p_{11}(t) $$