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Consider the $Q$-matrix on $\mathbb{N}$ defined by $q(m,m+1)=-q(m,m)=\rho m$ and $q(m,m')=0$ if $|m-m'|>1$, where $\rho$ is a constant. Letting $(P_t)$ be the transition function of this process, to show is that for $k\leq \ell$ $$ P_t(k,\ell) = {\ell-1\choose \ell-k}e^{-k\rho t}(1-e^{-\rho t})^{\ell-k} $$ I have already determined that $p(m,m+1) = 1$ and each stay at state $m$ is $\sim \exp(\rho m)$.

Every try based on only calculations resulted in mess. I think we must use that something in this system is binomial, because the final expression looks like that. To come to there, we can think about the reserved process $(Y_t)$ of $(X_t)$ and transform the system a bit: if $Y_t$ is the number of particles at time $t$, then if we assume that each of the particles dies exponentially with rate $\rho$, then $(Y_t)$ is indeed the reversed process of $(X_t)$. This is because $p(m,m-1) = 1$ for $(Y_t)$ and $p(m,m+1)=1$ for $(X_t)$, and the lengths of periods at all states $m$ have the same distribution at both $(X_t)$ and $(Y_t)$. But I don’t know how to use this further. Bayes rule seems to be essential, but we need to know $P(X_s = k)/ P(X_{s+t} = \ell)$ for this.

Is it doable to calculate $P(X_s = k)/P(X_{s+t} = \ell)$, or would I yet better just calculate $P_t(k,\ell)$ without using the fact that something is binomial?

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