Relative Perturbations of a Causal Intensity: Pair-Exponent Bound, the Pair Intensity as a Relative Perturbation, Replacement of the Pair Likelihood by the Base Likelihood, and the Weighted Pair-Covariance Sum

Let $\tilde{l}\ge1$ be a natural number, let $T>0$ be a real number, and let $(\mathbf{R},\mathcal{R},\rho)$ be the observation record space with horizon $T$ and $\tilde{l}$ channels, with channel set $V$. Causal intensities on $\mathbf{R}$, their total intensities and their likelihoods are as in those definitions; $\mathcal{R}$-measurability of real-valued functions on $\mathbf{R}$ is measurability with respect to $\mathcal{R}$ and the Borel $\sigma$-algebra of the real line, integrals over $\mathbf{R}$ of nonnegative measurable functions are those of Lebesgue Integral of a Nonnegative Measurable Function, integrable means integrable with respect to $\rho$, $\int_{[0,T]}\cdot\,ds$ is the Lebesgue integral over the compact interval $[0,T]$, $\exp$ is the real exponential function, $|\cdot|$ is the absolute value, and $t^{1/2}$ is the nonnegative square root of a real $t\ge0$.

Let $\mu=(\mu^\upsilon)_{\upsilon\in V}$ be a causal intensity on $\mathbf{R}$ with bound $\bar\mu$ for which there is a real number $\underline\mu>0$ with $\mu^\upsilon_s(r)\ge\underline\mu$ for all $s\in[0,T]$, $r\in\mathbf{R}$ and $\upsilon\in V$ (the base intensity), and write $\ell=\ell_\mu$ for its likelihood. Let $\mathsf{N}\in\mathcal{R}$ satisfy $\rho(\mathsf{N})=0$. For a real number $\vartheta>0$, a causal intensity $\nu=(\nu^\upsilon)_{\upsilon\in V}$ on $\mathbf{R}$ is called a relative $\vartheta$-perturbation of $\mu$ off $\mathsf{N}$ if $\bigl|\nu^\upsilon_s(r)-\mu^\upsilon_s(r)\bigr|\le\vartheta\,\mu^\upsilon_s(r)\qquad\text{for all }s\in[0,T],\ \upsilon\in V\text{ and }r\in\mathbf{R}\setminus\mathsf{N}$ (for $\mathsf{N}=\emptyset$ this is the relative $\vartheta$-perturbation of Chernoff Bound for the Under-Likelihood Set of a Relatively Perturbed Causal Intensity: Elementary Exponential Inequalities, the Tilted Power-Product Exponent, and the Markov Step). Let $\eta>0$ be a real number and put $\mathsf{E}_\eta=\tilde{l}\,T\,\bar\mu\,\eta^{2},\qquad \mathsf{e}_\eta=\tfrac12\,\mathsf{E}_\eta^{2}\exp(\mathsf{E}_\eta)+\mathsf{E}_\eta\bigl(\exp(9\,\mathsf{E}_\eta)-1\bigr)^{1/2}$ (the square root is defined since $\mathsf{E}_\eta\ge0$ gives $\exp(9\mathsf{E}_\eta)\ge1+9\mathsf{E}_\eta\ge1$ by claim 4 of Basic Properties of the Exponential Function).

For two causal intensities $\mu',\mu''$ on $\mathbf{R}$ with bounds $\bar\mu',\bar\mu''$, the pair intensity $\tilde\mu=(\mu'^\upsilon\mu''^\upsilon/\mu^\upsilon)_{\upsilon\in V}$, the pair exponent $E:\mathbf{R}\to\mathbb{R}$ and the pair covariance $C=\int_{\mathbf{R}}\ell\,(1-L')(1-L'')\,d\rho$ with the likelihood ratios $L'=\ell_{\mu'}/\ell$ and $L''=\ell_{\mu''}/\ell$ are those of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form for the base intensity $\mu$ and the perturbed intensities $(\mu',\mu'')$ (so $\ell>0$ on $\mathbf{R}$, $E(r)=\int_{[0,T]}\sum_{\upsilon\in V}(\mu'^\upsilon_s(r)-\mu^\upsilon_s(r))(\mu''^\upsilon_s(r)-\mu^\upsilon_s(r))/\mu^\upsilon_s(r)\,ds$, and $\tilde\mu$ is a causal intensity by claim 1 of that lemma); for a causal intensity $\nu$ on $\mathbf{R}$ with a bound, the ratio variance of that lemma is $\mathrm{Var}(\nu)=\int_{\mathbf{R}}\ell\,(\ell_\nu/\ell-1)^{2}\,d\rho$, the pair covariance for the pair $(\nu,\nu)$.

1. (Pair-exponent bound) Let $\mu',\mu''$ be relative $\eta$-perturbations of $\mu$ off $\mathsf{N}$ with bounds $\bar\mu',\bar\mu''$. Then $|E(r)|\le\mathsf{E}_\eta$ for every $r\in\mathbf{R}\setminus\mathsf{N}$; consequently $\ell_{\tilde\mu}E$ is integrable, $|C|\le\exp(\mathsf{E}_\eta)-1$ and $\Bigl|C-\int_{\mathbf{R}}\ell_{\tilde\mu}\,E\,d\rho\Bigr|\le\tfrac12\,\mathsf{E}_\eta^{2}\exp(\mathsf{E}_\eta).$

2. (The pair intensity as a relative perturbation) Assume $\eta\le1$ and let $\mu',\mu''$ be as in claim 1. Then the pair intensity $\tilde\mu$ is a causal intensity on $\mathbf{R}$ with bound $\bar\mu'\bar\mu''/\underline\mu$ which is a relative $3\eta$-perturbation of $\mu$ off $\mathsf{N}$; the function $\ell_{\tilde\mu}^{2}/\ell$ is integrable, and the ratio variance of $\tilde\mu$ satisfies $0\le\mathrm{Var}(\tilde\mu)=\int_{\mathbf{R}}\frac{\ell_{\tilde\mu}^{2}}{\ell}\,d\rho-1\le\exp(9\,\mathsf{E}_\eta)-1 .$

3. (Replacement of the pair likelihood by the base likelihood) Under the hypotheses of claim 2, $|\ell_{\tilde\mu}-\ell|$ and $\ell E$ are integrable, $\int_{\mathbf{R}}|\ell_{\tilde\mu}-\ell|\,d\rho\le\bigl(\exp(9\,\mathsf{E}_\eta)-1\bigr)^{1/2},\qquad\text{and}\qquad \Bigl|C-\int_{\mathbf{R}}\ell\,E\,d\rho\Bigr|\le\mathsf{e}_\eta .$

4. (Weighted pair-covariance sum) Assume $\eta\le1$, let $n\ge1$ be a natural number, let $\mu_1,\dots,\mu_n$ be relative $\eta$-perturbations of $\mu$ off $\mathsf{N}$ with bounds $\bar\mu_1,\dots,\bar\mu_n$, and let $w=(w_1,\dots,w_n)$ be a point of Euclidean space $\mathbb{R}^{n}$ with $\lVert w\rVert_1=\sum_{q=1}^{n}|w_q|$. Let $L_q=\ell_{\mu_q}/\ell$, and let $E_{qq'}$ and $C_{qq'}$ ($1\le q,q'\le n$) be the pair exponents and pair covariances of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form for the base intensity $\mu$ and the perturbed intensities $(\mu_1,\dots,\mu_n)$. Then $\mathcal{Q}=\sum_{q=1}^{n}\sum_{q'=1}^{n}w_qw_{q'}E_{qq'}$ is an $\mathcal{R}$-measurable bounded function on $\mathbf{R}$, $\ell\,\mathcal{Q}$ is integrable, and $0\le\int_{\mathbf{R}}\ell\Bigl(\sum_{q=1}^{n}w_q(1-L_q)\Bigr)^{2}d\rho=\sum_{q=1}^{n}\sum_{q'=1}^{n}w_qw_{q'}\,C_{qq'}\le\int_{\mathbf{R}}\ell\,\mathcal{Q}\,d\rho+\lVert w\rVert_1^{2}\,\mathsf{e}_\eta .$