Uniformization truncation, mass loss, and arithmetic-error ledger

For any $\mu >0$, set $S_{\mu }=e^{\mu Q\mathrm{/}\nu }=e^{-\mu }{e}^{\mu P}$ and $R_{\mu ,K}={\sum }_{k=0}^K{w}_k(\mu )P^k$. By [F1], each $P^k$ is nonnegative and column stochastic; [F2] and [F3] give $S_{\mu }={\sum }_{k\ge 0}{w}_k(\mu )P^k$ with nonnegative weights summing to one. Thus $R_{\mu ,K}$ is nonnegative with common column sum $s_K:={\sum }_{k=0}^K{w}_k(\mu )\le 1$, and $S_{\mu }-R_{\mu ,K}$ is nonnegative with common column sum ${\tau }_K(\mu )$.

The induced norm formula [F6] gives $\mathrm{\parallel }{S}_{\mu }{\mathrm{\parallel }}_1=1$, $\mathrm{\parallel }{R}_{\mu ,K}{\mathrm{\parallel }}_1=s_K$, and $\mathrm{\parallel }{S}_{\mu }-R_{\mu ,K}{\mathrm{\parallel }}_1={\tau }_K(\mu )$. If $\lambda =\nu t=0$, only the $k=0$ Poisson weight is nonzero, so $p(t)=u_K(t)=p_0$ and the omitted mass is zero. If $\lambda >0$, apply step 1.1 with $\mu =\lambda$: $(S_{\lambda }-R_{\lambda ,K})p_0$ is nonnegative and its coordinate sum is ${\tau }_K(\lambda )$; by [F5], the exact one-series ${\mathrm{\ell }}^1$ omitted mass is therefore exactly the scalar Poisson tail.

In the time-split algorithm, $0<\mu \le 1$ and $K\ge 0$, so $\mu <K+2$ and [F4] gives ${\tau }_K(\mu )\le G_K(\mu )$. For $m$ substeps the telescoping identity $S_{\mu }^m-R_{\mu ,K}^m={\sum }_{j=0}^{m-1}{S}_{\mu }^{m-1-j}(S_{\mu }-R_{\mu ,K})R_{\mu ,K}^j$, [F5]–[F6], and step 2.1 give $\mathrm{\parallel }{S}_{\mu }^m-R_{\mu ,K}^m{\mathrm{\parallel }}_1\le m{\tau }_K(\mu )\le m{G}_K(\mu )$.

Every term in the telescoping identity is nonnegative, so the exact vector error on $p_0$ is nonnegative. Each column sum of $S_{\mu }^m$ is one and each column sum of $R_{\mu ,K}^m$ is $s_K^m$; hence its ${\mathrm{\ell }}^1$ mass is exactly $1-s_K^m=1-(1-{\tau }_K(\mu ){)}^m\le m{\tau }_K(\mu )$. This proves the exact time-split lost-mass identity and the preceding error bound.

Since $m\mu =\nu t$, [F8] gives $S_{\mu }^m=e^{tQ}$ and [F2] gives $S_{\mu }^m{p}_0=p(t)$, while [F7] identifies the exact truncated output as $R_{\mu ,K}^m{p}_0$. The triangle inequality for the vector $1$-norm [F5] therefore yields $\mathrm{\parallel }p(t)-\widehat{u}{\mathrm{\parallel }}_1\le \mathrm{\parallel }p(t)-R_{\mu ,K}^m{p}_0{\mathrm{\parallel }}_1+\mathrm{\parallel }{R}_{\mu ,K}^m{p}_0-\widehat{u}{\mathrm{\parallel }}_1\le m{G}_K(\mu )+E_{\mathrm{a}\mathrm{l}\mathrm{g}}\le T+E_{\mathrm{a}\mathrm{l}\mathrm{g}}$ when the validated upper bound $T$ is supplied. If $\nu t=0$, the algorithm returns $p_0$ and both error terms are zero.