Birth-death master equation

For birth rates $b_n$, death rates $d_n$, and state probabilities $P_n$,

Since births and deaths change the state by $+1$ and $-1$,

For per-capita birth and death rates $B$ and $D$, the rates are $b_n=Bn$ and $d_n=Dn$. If $\mu =\mathbb{E}{N}_t$ and ${\sigma }^2=\mathrm{var}{N}_t$, thenThe expected value sees only the net growth rate $B-D$, whereas the variance also sees the total event rate $B+D$.

The $r$th falling factorial moment is $\mathbb{E}[(N{)}_r]$, where $(N{)}_r=N(N-1)\cdots (N-r+1)$.

For a nonnegative integer-valued random variable, the displayed inversion is valid when $\mathbb{E}(X{)}_s{s}^k/s!\to 0$. Indeed the gap between consecutive Jordan-Bonferroni exact-occurrence inequalities is $\mathbb{E}(X{)}_{k+m+1}/[k!(m+1)!]$, which tends to zero under this hypothesis. No exchange of an infinite signed sum with an expectation is needed.

If $N$ is Poisson with mean $\lambda$, then $\mathbb{E}[(N{)}_r]={\lambda }^r$; in particular, $\mathbb{E}N=\lambda$ and $\mathbb{E}[N(N-1)]={\lambda }^2$.

If nonnegative integer-valued random variables $X_n$ have all fixed factorial moments converging to ${\lambda }^k$, then their convergence in distribution is to the Poisson distribution with mean $\lambda$. The first-moment bound gives uniform tightness, and bounds on higher moments give uniform integrability of each fixed lower moment. Every subsequential limit therefore has the indicated factorial moments. For such a limit $X$ and $t\ge 0$, the nonnegative binomial expansion and the monotone convergence theorem give $\mathbb{E}(1+t{)}^X={\sum }_{k\ge 0}{t}^k{\lambda }^k/k!=e^{\lambda t}$. This identifies its moment-generating function as that of the Poisson distribution, uniquely determining the limit.