Probability of the Good Event on the Synthetic Copy: Poisson Tail for the Cell Counts, the Window Discrepancy Bound for the Clock-Good Event, and the Bound on the Complement of the Good Event

Adopt the setting, hypotheses (OC), (X), (W) and notation of Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances (and hence of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record): the probability space $(\Omega,\mathcal{F},P)$ with the driving variables $K^{c}$, $V^{c}_i$, $U^{c,j}_i$, the natural numbers $J_c\ge1$, the event $\Omega^{U}_0$, the set $\mathcal{L}$ of transition labels (with $l(l-1)$ elements), the remaining data $N$, $l$, $B$, $T$ of that lemma and the real number $R>0$ with $R\ge NBT$, the cells $I_{c,j}$ of lengths $\mu_{c,j}$ indexed by $q=(c,j)\in\mathsf{L}$ (we write $\mu_q=\mu_{c,j}$; every $\mu_q$ is positive, the cell endpoints being strictly increasing), the cell-count vector $\mathsf{K}$ with coordinates $\mathsf{K}_q$, the copy clocks $\mathsf{P}^{\sharp}=(\mathsf{P}^{\sharp,c})_{c\in\mathcal{L}}$, the move size $\mathsf{m}$ (a natural number), the real numbers $L\ge0$ and $D\ge0$ and the clock-good event $G_{L,D}$. Let $G^{\mathsf{m}}$ be the set of $\omega\in G_{L,D}$ with $\mathsf{K}_q(\omega)\ge\mathsf{m}$ for every $q\in\mathsf{L}$ (the good event of Weighted Response and Pair-Exponent Quadratic Form on the Synthetic Copy: the Linearised Injection Equation, Comparison with the Profile Injection, and the Observation-Information Bound). Write $\exp$ for the exponential function and, for real $\mu\ge0$ and $x>0$, $\varpi_\mu(x)$ for the exponent of Series Formula, Exponential Moments, and Chernoff Tail Bounds for the Poisson Distribution ($\varpi_\mu(x)=\min(x^{2}/(4\mu),x/2)$ for $\mu>0$, $\varpi_0(x)=x/2$; this $\varpi$ is unrelated to the profile called $\varpi$ in Weighted Response and Pair-Exponent Quadratic Form on the Synthetic Copy: the Linearised Injection Equation, Comparison with the Profile Injection, and the Observation-Information Bound, which is not used here); the Poisson distribution is as in that definition.

1. (Poisson tail for the cell counts) For every $q\in\mathsf{L}$ the cell count $\mathsf{K}_q$ has the Poisson distribution with parameter $\mu_q$, and if $\mathsf{m}<\mu_q$ then $P(\mathsf{K}_q<\mathsf{m})\le\exp\bigl(-\varpi_{\mu_q}(\mu_q-\mathsf{m})\bigr).$

2. (The clock-good event) Suppose that $R$ is a natural number, that $M$ is a natural number with $L\le M$ (the window length; this $M$ is unrelated to the record majorant $M(r)$ of claim 5 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), and that $D>2$. Then $P(\Omega\setminus G_{L,D})\le2\,l(l-1)\,(R+1)(M+3)\,\exp\bigl(-\varpi_{M+2}(D-2)\bigr).$

3. (The good event) Under the hypotheses of claim 2, and if moreover $\mathsf{m}<\mu_q$ for every $q\in\mathsf{L}$, the set $G^{\mathsf{m}}$ belongs to $\mathcal{F}$ and $P(\Omega\setminus G^{\mathsf{m}})\le\sum_{q\in\mathsf{L}}\exp\bigl(-\varpi_{\mu_q}(\mu_q-\mathsf{m})\bigr)+2\,l(l-1)\,(R+1)(M+3)\,\exp\bigl(-\varpi_{M+2}(D-2)\bigr).$