Proof of Predictable-Window Moment Identities for the Homogeneous Poisson Process

Throughout, path properties refer to Counting Path and Its Jump Times: every path of $Y$ is nonnegative-integer-valued with $Y_0=0$ (property 1), nondecreasing in the level (property 2), and satisfies $Y_u=\inf\{Y_s:s>u\}$ (property 3).

Claim 1. For a random variable $w\ge0$ and a natural number $m$ let $w_m=2^{-m}\lceil2^{m}w\rceil$, so that $w\le w_m$, $w_m-w\le2^{-m}$, $w_{m+1}\le w_m$ (dyadic refinement), and $w_m$ takes countably many values $q$ with $\{w_m=q\}=\{q-2^{-m}<w\le q\}\in\mathcal{F}$. Hence $Y_{w_m}=\sum_{q}\mathbf{1}\{w_m=q\}\,Y_q$ is a random variable. By path monotonicity the sequence $(Y_{w_m})_m$ is nonincreasing and $Y_{w_m}\ge Y_w$; and for every $s>w$ one has $w_m<s$ for all large $m$, so $\inf_m Y_{w_m}\le Y_s$, whence $\inf_m Y_{w_m}\le\inf\{Y_s:s>w\}=Y_w$ by property 3. Thus $Y_w=\inf_m Y_{w_m}$ is a random variable, $\{\inf_mY_{w_m}<c\}=\bigcup_m\{Y_{w_m}<c\}$. For $0\le w\le w'$, the values $Y_w,Y_{w'}$ are nonnegative integers with $Y_w\le Y_{w'}$ (properties 1--2), so $Y_{w'}-Y_w$ is a nonnegative-integer-valued random variable. If moreover $w$ and every variable $Y_u$ ($u\ge0$) are measurable with respect to a $\sigma$-algebra $\mathcal{G}_0$, then so is every $Y_{w_m}$ (the events $\{w_m=q\}$ lie in $\sigma(w)\subseteq\mathcal{G}_0$ and the $Y_q$ are $\mathcal{G}_0$-measurable), hence so is $Y_w=\inf_mY_{w_m}$.

Claim 4. (i) Series form. By Poisson Distribution, $P_x(\{k\})=\exp(-x)x^k/k!$ for every nonnegative integer $k$, and $P_x(\mathbb{R}\setminus\mathbb{N}_0)=0$. The functions $f_K(t)=|t|^p\mathbf{1}_{[-K,K]}(t)$ increase pointwise to $|t|^p$ as the natural number $K$ increases, so by the Monotone Convergence Theorem $\mu_p(x)=\lim_K\int f_K\,dP_x$; and $f_K$ agrees $P_x$-almost everywhere with the simple function $\sum_{k=0}^{K}k^p\mathbf{1}_{\{k\}}$, whose integral is $\sum_{k=0}^{K}k^pP_x(\{k\})$ (two nonnegative measurable functions that agree $P_x$-almost everywhere have the same Lebesgue integral). The partial sums being nondecreasing, $\mu_p(x)=\exp(-x)\sum_{k\ge1}k^px^k/k!$ as claimed; in particular $\mu_p(0)=0$, only the vanishing terms remaining. (ii) Finiteness. By claim 2 of Image Measures, Measures with Densities, and Change of Variables, $\mu_p(x)=\mathbb{E}[K^p]$ for any random variable $K$ with distribution $P_x$, so finiteness follows from part (b) of Factorial Moments and Moments of Every Order of the Poisson Distribution; we record the elementary tail bound in order to fix the constant $c_p$. For $k>2p$ each of the $p$ factors of $k(k-1)\cdots(k-p+1)$ is at least $k-p\ge k/2$, so $k(k-1)\cdots(k-p+1)\ge(k/2)^p$ and hence $k^p\le 2^p\,k!/(k-p)!$. Therefore $\sum_{k>2p}k^px^k/k!\le2^p\sum_{k>2p}x^k/(k-p)!=2^p\sum_{j>p}x^{j+p}/j!\le2^px^p\exp(x)$, using the defining series of The Real Exponential Function; adding the finitely many terms with $k\le2p$ bounds the full series, and $x=1$ gives $c_p<\infty$. (iii) $\mu_1$. By Moments of the Poisson Distribution a random variable with the Poisson distribution $P_x$ has expectation $x$, and by claim 2 of Image Measures, Measures with Densities, and Change of Variables that expectation equals $\mu_1(x)$; so $\mu_1(x)=x$. (iv) Monotonicity. $\mu_p(0)=0\le\mu_p(x')$ for every $x'$, by (i). For $0<x\le x'$: by clauses 1 and 3 of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process (homogeneous case, mean function $\Lambda(t)=t$) together with $Y_0=0$, the variable $Y_x=Y_x-Y_0$ has distribution $P_x$, so $\mu_p(x)=\mathbb{E}[Y_x^{\,p}]$ by claim 2 of Image Measures, Measures with Densities, and Change of Variables. Path monotonicity gives $Y_x^{\,p}\le Y_{x'}^{\,p}$ pointwise, so $\mu_p(x)\le\mu_p(x')$. (v) Linear bound. For $x\in[0,1]$: $\exp(-x)\le1$ (its product with $\exp(x)\ge1$ equals $1$, and $\exp(x)\ge1$ from the defining series with nonnegative terms), so $\mu_p(x)\le\sum_{k\ge1}k^px^k/k!\le x\sum_{k\ge1}k^px^{k-1}/k!\le x\sum_{k\ge1}k^p/k!=c_px$, comparing partial sums termwise with $x^{k-1}\le1$. (vi) Continuity. Fix $R\ge0$ and write $S(y)=\sum_{k\ge1}k^py^k/k!$, finite on $[0,R]$ by (ii). For $y,y'\in[0,R]$ and a natural $K$: $|S(y)-S(y')|\le\sum_{k\le K}(k^p/k!)|y^k-y'^k|+2\sum_{k>K}k^pR^k/k!$. Given $\varepsilon>0$, the tail term is below $\varepsilon/2$ for some $K$ (tail of a convergent series), and $|y^k-y'^k|\le kR^{k-1}|y-y'|$ makes the finite sum below $\varepsilon/2$ for $|y-y'|$ small. So $S$ is continuous on every $[0,R]$, hence on $[0,\infty)$; $\exp(-\cdot)$ is continuous by Basic Properties of the Exponential Function; and $\mu_p=\exp(-\cdot)S$ is a product of continuous functions, hence continuous.

Claim 2. Reduction. If the identity holds for all bounded $\mathcal{G}$-measurable $Z\ge0$, then for general $Z:\Omega\to[0,\infty]$ apply it to $Z\wedge n$ and let $n\to\infty$: both sides converge to the corresponding expressions for $Z$ by the Monotone Convergence Theorem, the integrands being nondecreasing in $n$. So assume $0\le Z\le\zeta$ for a real $\zeta$. Grid. Fix a natural $m$, put $\delta=2^{-m}$ and $g_j=\theta+j\delta$ for $j=0,\dots,J_m$, with $J_m$ the least natural for which $g_{J_m}\ge\bar{V}+\bar{\Lambda}+1$. The vector $X_m=(Y_{g_1}-Y_{g_0},\dots,Y_{g_{J_m}}-Y_{g_{J_m-1}})$ is measurable with respect to $\sigma(Y_{\theta+s}-Y_\theta:s\ge0)$ (each coordinate is a difference of two increments beyond $\theta$), so $\sigma(X_m)$ is independent of $\mathcal{G}$: the defining product identities for the pair hold a fortiori on a sub-$\sigma$-algebra. Let $\mu_{X_m}$ be its image measure. Bracketing. Define $\alpha=\max\{j\ge0:g_j\le v\}$ and $\beta=\max\{j\ge0:g_j\le v+\lambda\}$; both are $\mathcal{G}$-measurable with finitely many values, $\alpha\le\beta\le J_m-1$, $g_\alpha\le v<g_{\alpha+1}$, and $g_\beta\le v+\lambda<g_{\beta+1}$. With $U_m=Y_{g_{\beta+1}}-Y_{g_\alpha}$, and $L_m=Y_{g_\beta}-Y_{g_{\alpha+1}}$ when $\alpha+1\le\beta$ and $L_m=0$ otherwise, path monotonicity gives $L_m\le Y_{v+\lambda}-Y_v\le U_m$. Exact evaluation of the brackets. Define $\Psi:\Omega\times\mathbb{R}^{J_m}\to[0,\infty)$ by $\Psi(\omega,x)=\sum_{0\le a\le b<J_m}\mathbf{1}\{\alpha=a,\beta=b\}(\omega)\,Z(\omega)\,\bigl((x_{a+1}+\dots+x_{b+1})^{+}\bigr)^p$, a finite sum of products of a $\mathcal{G}$-measurable factor and a Borel function of $x$, hence $\mathcal{G}\otimes\mathcal{B}_{J_m}$-measurable; and $\Psi(\omega,X_m(\omega))=Z\,U_m^{\,p}$, the coordinate block telescoping to $Y_{g_{\beta+1}}-Y_{g_\alpha}\ge0$. By claim 3 of Independence Fubini: Integration in an Independent Random Vector Given a Sub-Sigma-Algebra, $\mathbb{E}[Z\,U_m^{\,p}]=\mathbb{E}\bigl[\int\Psi(\cdot,x)\,d\mu_{X_m}(x)\bigr]$. For fixed $\omega$ with $\alpha=a,\beta=b$, the inner integral equals $Z(\omega)\,\mathbb{E}[(Y_{g_{b+1}}-Y_{g_a})^p]$ by claim 2 of Image Measures, Measures with Densities, and Change of Variables applied to the nonnegative Borel map $x\mapsto((x_{a+1}+\dots+x_{b+1})^{+})^p$; and $Y_{g_{b+1}}-Y_{g_a}$ has distribution $P_{g_{b+1}-g_a}$ (clause 3 of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process; here $g_{b+1}-g_a\ge\delta>0$), so this expectation is $\mu_p(g_{b+1}-g_a)$, again by claim 2 of Image Measures, Measures with Densities, and Change of Variables. Hence $\mathbb{E}[Z\,U_m^{\,p}]=\mathbb{E}[Z\,\mu_p(\ell^U_m)]$ with $\ell^U_m=g_{\beta+1}-g_\alpha\in[\lambda,\lambda+2\delta]$. The same argument with the block from $a+2$ to $b$ gives $\mathbb{E}[Z\,L_m^{\,p}]=\mathbb{E}[Z\,\mu_p(\ell^L_m)]$, where $\ell^L_m=g_\beta-g_{\alpha+1}$ when $\alpha+1\le\beta$ and $\ell^L_m=0$ otherwise; the degenerate cases need no appeal to clause 3: when $\alpha+1>\beta$ one has $L_m=0$ and $\ell^L_m=0$, and when $\alpha+1=\beta$ the block is empty and the increment $Y_{g_\beta}-Y_{g_{\alpha+1}}$ vanishes identically, so in both cases the identity reduces to $\mu_p(0)=0$ from claim 4 (i). In either case $|\ell^L_m-\lambda|\le2\delta$ (when $\beta=\alpha$ one has $\lambda<2\delta$). Limit. As $m\to\infty$, $\ell^U_m\to\lambda$ and $\ell^L_m\to\lambda$ pointwise, so by continuity of $\mu_p$ (claim 4), $\mu_p(\ell^{U}_m)\to\mu_p(\lambda)$ and $\mu_p(\ell^{L}_m)\to\mu_p(\lambda)$, with $0\le\mu_p(\ell_m)\le\mu_p(\bar{\Lambda}+2)<\infty$ by monotonicity; by the Dominated Convergence Theorem (constant dominating function $\zeta\mu_p(\bar{\Lambda}+2)$), $\mathbb{E}[Z\mu_p(\ell^{U}_m)]\to\mathbb{E}[Z\mu_p(\lambda)]$ and likewise for $L$. Since $\mathbb{E}[Z\mu_p(\ell^L_m)]=\mathbb{E}[ZL_m^p]\le\mathbb{E}[Z(Y_{v+\lambda}-Y_v)^p]\le\mathbb{E}[ZU_m^p]=\mathbb{E}[Z\mu_p(\ell^U_m)]$ for every $m$, the middle term, which does not depend on $m$, equals $\mathbb{E}[Z\mu_p(\lambda)]$.

Claim 3. The same construction with the map $t\mapsto\mathbf{1}\{t\ge1\}$ in place of $t\mapsto t^p$: pointwise $\mathbf{1}\{L_m\ge1\}\le\mathbf{1}\{Y_{v+\lambda}-Y_v\ge1\}\le\mathbf{1}\{U_m\ge1\}$; the inner integrals evaluate to $Z(\omega)\,P(Y_{g_{b+1}}-Y_{g_a}\ge1)=Z(\omega)(1-\exp(-(g_{b+1}-g_a)))$, since the increment is a nonnegative integer everywhere and $P_\ell(\{0\})=\exp(-\ell)$ by Poisson Distribution (degenerate blocks as in claim 2, with $1-\exp(0)=0$). Continuity of $\exp$ and domination by $\zeta$ give, in the limit, $\mathbb{E}[Z\mathbf{1}\{Y_{v+\lambda}-Y_v\ge1\}]=\mathbb{E}[Z(1-\exp(-\lambda))]$; the monotone reduction extends this to $[0,\infty]$-valued $Z$ as in claim 2. Finally $1-\exp(-\lambda)\le\lambda$ pointwise: for $\lambda\ge1$, $1-\exp(-\lambda)\le1\le\lambda$; for $0\le\lambda<1$, the defining series and the geometric partial-sum identity give $\exp(\lambda)=\sum_k\lambda^k/k!\le\sum_k\lambda^k\le1/(1-\lambda)$, so $\exp(-\lambda)=1/\exp(\lambda)\ge1-\lambda$. This yields the stated inequality.