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 $\tilde{l}\ge1$ 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, and for a causal intensity $\lambda$ with total intensity $\lambda^{\mathrm{tot}}$ the integrated total intensity $\int_{[0,T]}\lambda^{\mathrm{tot}}_s(r)\,ds$ is as in Integer Power Products of Causal-Intensity Likelihoods: Pathwise Identity, Integral Bounds, and the Pair Identity; integrals over $\mathbf{R}$ of nonnegative measurable functions are those of Lebesgue Integral of a Nonnegative Measurable Function, $\int_{[0,T]}\cdot\,ds$ is the Lebesgue integral over the compact interval $[0,T]$, $\exp$ is the real exponential function, $\mathbf{1}\{\cdot\}$ denotes the indicator of a subset of $\mathbf{R}$ (equal to $1$ on the set and to $0$ off it), and integer powers of positive real numbers are as in Integer Power Products of Causal-Intensity Likelihoods: Pathwise Identity, Integral Bounds, and the Pair Identity.

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 $s\in[0,T]$, $r\in\mathbf{R}$, $\upsilon\in V$ (the base intensity), let $\varepsilon$ be a real number with $0\le\varepsilon\le\tfrac12$, and let $\mu'=(\mu'^\upsilon)_{\upsilon\in V}$ be a causal intensity on $\mathbf{R}$ with bound $\bar\mu'$ which is a relative $\varepsilon$-perturbation of $\mu$: $|\mu'^\upsilon_s(r)-\mu^\upsilon_s(r)|\le\varepsilon\,\mu^\upsilon_s(r)\qquad\text{for all }s\in[0,T],\ r\in\mathbf{R},\ \upsilon\in V .$ Then $\mu'^\upsilon_s(r)\ge(1-\varepsilon)\mu^\upsilon_s(r)\ge\tfrac12\underline\mu>0$ everywhere. Write $\ell=\ell_\mu$, $\ell'=\ell_{\mu'}$ and $L=\ell'/\ell$ for the likelihood ratio. For an integer $\theta\ge1$ let $\tilde\mu_\theta$ be the causal intensity with components $\tilde\mu^\upsilon_\theta=(\mu^\upsilon)^{1+\theta}(\mu'^\upsilon)^{-\theta}$ and $E_\theta:\mathbf{R}\to\mathbb{R}$ the exponent function, $E_\theta(r)=\tilde\Lambda_{\theta,T}(r)-(1+\theta)\Lambda_T(r)+\theta\Lambda'_T(r)$ with $\Lambda_T$, $\Lambda'_T$ and $\tilde\Lambda_{\theta,T}$ the integrated total intensities of $\mu$, $\mu'$ and $\tilde\mu_\theta$, of claim 1 of Integer Power Products of Causal-Intensity Likelihoods: Pathwise Identity, Integral Bounds, and the Pair Identity applied to $(\mu,\mu')$ with the exponents $(1+\theta,-\theta)$, so that $\ell^{1+\theta}\ell'^{-\theta}=\ell_{\tilde\mu_\theta}\exp(E_\theta)$ on $\mathbf{R}$.

1. (Elementary inequalities; independent of the standing hypotheses) For every real $u$, $\exp(u)\ge1+u$, and for every real $z$, $|\exp(z)-1-z|\le\tfrac12z^{2}\exp(|z|)$. If $x$ is a real number and $\theta\ge1$ is an integer with $2\theta|x|\le1$, then $(1+x)^{-\theta}\le1-\theta x+8\,\theta^{2}x^{2}.$

2. (Tilted exponent) $\ell>0$ and $\ell'>0$ on $\mathbf{R}$. For every integer $\theta\ge1$ with $2\theta\varepsilon\le1$, $E_\theta(r)\le\bar{E}_\theta:=8\,\tilde{l}\,T\,\bar\mu\,\theta^{2}\varepsilon^{2}\quad\text{for every }r\in\mathbf{R},\qquad\text{and}\qquad \int_{\mathbf{R}}\ell^{1+\theta}\ell'^{-\theta}\,d\rho\le\exp(\bar{E}_\theta).$

3. (Chernoff bound) For every real number $\delta'$ with $0<\delta'<1$ and every integer $\theta\ge1$ with $2\theta\varepsilon\le1$, $\int_{\mathbf{R}}\ell\,\mathbf{1}\{L<1-\delta'\}\,d\rho\le(1-\delta')^{\theta}\exp(\bar{E}_\theta)\le\exp\bigl(-\theta\delta'+8\,\tilde{l}\,T\,\bar\mu\,\theta^{2}\varepsilon^{2}\bigr).$