The Poisson Removal Ratio for Moves of Several Points: Move Score, Mean, Exact Second Moment, Move Information, and Pointwise Bounds

Let $\mathbb{N}$ be the set of natural numbers, $\mathbb{N}_0=\mathbb{N}\cup\{0\}$, and let $\mathbb{R}$ be the set of real numbers, identified with the Euclidean space $\mathbb{R}^{1}$. Fix a real number $\mu>0$ and a natural number $\mathsf{m}\ge1$, and let $p_\mu$ be the Poisson probability mass function of Move Score and Move Information of the Poisson Probability Mass Function, $p_\mu(k)=\exp(-\mu)\mu^{k}/k!$ for $k\in\mathbb{N}_0$ and $p_\mu(x)=0$ for $x\in\mathbb{R}\setminus\mathbb{N}_0$, a discrete probability mass function on $\mathbb{R}$ with support $\mathbb{N}_0$ by claim 1 there; here $k!$ is the factorial with $0!=1$, $\exp$ the exponential function, and $x^{0}=1$. Sums over subsets of $\mathbb{N}_0$ of nonnegative functions are those of Sum of a Nonnegative Function over an Arbitrary Set, and finite products over an empty index range are $1$. For natural numbers $0\le i\le\mathsf{m}$ write $\binom{\mathsf{m}}{i}=\mathsf{m}!/(i!\,(\mathsf{m}-i)!)$.

Define the removal ratio $\varrho=\varrho^{\mathsf{m}}_\mu:\mathbb{N}_0\to[0,\infty)$ by $\varrho(k)=p_\mu(k-\mathsf{m})/p_\mu(k)$, so that

$$\varrho(k)=\prod_{i=1}^{\mathsf{m}}\frac{k-\mathsf{m}+i}{\mu}=\frac{k!}{(k-\mathsf{m})!\,\mu^{\mathsf{m}}}\quad(k\ge\mathsf{m}),\qquad \varrho(k)=0\quad(k<\mathsf{m}).$$

Consider the single move $a=(a_1)$ with $a_1=\mathsf{m}$ and a single weight $w=(w_1)$, $w_1\in\mathbb{R}$, in the sense of Move Score of a Discrete Probability Mass Function.

1. (Move score) The move score of $p_\mu$ for $a$ and $w$ is $\rho_{p_\mu,a,w}(k)=w_1\bigl(1-\varrho(k)\bigr)$ ($k\in\mathbb{N}_0$), and $k+a_1\in\mathbb{N}_0$ for every $k\in\mathbb{N}_0$.

2. (Mean) $\sum_{k\in\mathbb{N}_0}p_\mu(k)\varrho(k)=1$.

3. (Exact second moment and move information)

$$\sum_{k\in\mathbb{N}_0}p_\mu(k)\varrho(k)^{2}=\sum_{i=0}^{\mathsf{m}}\binom{\mathsf{m}}{i}^{2}\,i!\,\mu^{-i},$$

and consequently the move information of $p_\mu$ for $a$ and $w$ is

$$\mathsf{J}(p_\mu;a,w)=w_1^{2}\sum_{i=1}^{\mathsf{m}}\binom{\mathsf{m}}{i}^{2}\,i!\,\mu^{-i},\qquad\text{with}\qquad \frac{\mathsf{m}^{2}}{\mu}\le\frac{\mathsf{J}(p_\mu;a,w)}{w_1^{2}}\le\frac{\mathsf{m}^{2}}{\mu}+\frac{\mathsf{m}^{4}}{2\mu^{2}}\exp\Bigl(\frac{\mathsf{m}^{2}}{\mu}\Bigr)$$

when $w_1\neq0$ (for $w_1=0$ the move information is $0$).

4. (Pointwise bounds) For every $k\in\mathbb{N}_0$, $\varrho(k)\le(k/\mu)^{\mathsf{m}}$. If $x\ge0$ is real with $\mathsf{m}(x+\mathsf{m})\le\mu$, then every $k\in\mathbb{N}_0$ with $k\ge\mu-x$ satisfies

$$\varrho(k)\ge1-\frac{\mathsf{m}(x+\mathsf{m})}{\mu}.$$