Proof of 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

Throughout, measurability of real-valued functions on $\mathbf{R}$ is measurability with respect to $\mathcal{R}$ and the Borel $\sigma$-algebra of the real line, and the following two shorthands are used on an arbitrary measure space (below: $(\mathbf{R},\mathcal{R},\rho)$, and in Step 1 the interval $[0,T]$). Linearity refers to claim 2 of Linearity and Monotonicity of the Lebesgue Integral (finite linear combinations of integrable functions are integrable, by induction on the number of terms, the integral is linear, $|\int f\,d\rho|\le\int|f|\,d\rho$, and $\int f\,d\rho\le\int g\,d\rho$ when $f\le g$ pointwise), and domination refers to the following consequence of claims 1 and 2 of that theorem and of Integrable Function and the Lebesgue Integral (a measurable $f$ is integrable if and only if $\int|f|\,d\rho<\infty$): if $f$ is measurable and $|f|\le g$ pointwise with $g$ integrable, then $f$ is integrable and $|\int f\,d\rho|\le\int|f|\,d\rho\le\int g\,d\rho$. Sums, scalar multiples, products, absolute values and indicators of measurable functions are measurable by claims 1--4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, absolute-value arithmetic refers to claims 4 and 5 of Properties of the Absolute Value in an Ordered Field (multiplicativity $|xy|=|x||y|$ and the two-term triangle inequality), and triangle inequality means claim 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers ($|\sum_ka_k|\le\sum_k|a_k|$), used together with the comparison of claim 1 there. We write $\mathbf{1}_A$ for the indicator of $A\subseteq\mathbf{R}$ and $\mathsf{N}^{c}=\mathbf{R}\setminus\mathsf{N}$.

Step 0 (the null set). (a) For every measurable $h:\mathbf{R}\to[0,\infty)$, $\int_{\mathbf{R}}h\mathbf{1}_{\mathsf{N}}\,d\rho=0$. Indeed, for every natural number $k$ the function $\min(h,k)\mathbf{1}_{\mathsf{N}}$ is measurable, nonnegative and at most $k\mathbf{1}_{\mathsf{N}}$, whose integral is $0$ by Simple Function and Its Integral (it takes the value $k$ on $\mathsf{N}$ and $0$ on $\mathsf{N}^{c}$; in every case, including $k=0$, $\mathsf{N}=\emptyset$ and $\mathsf{N}=\mathbf{R}$, the standard representation of that definition and its convention $0\cdot\infty=0$ give the value $k\rho(\mathsf{N})+0\cdot\rho(\mathsf{N}^{c})=0$); so $0\le\int\min(h,k)\mathbf{1}_{\mathsf{N}}\,d\rho\le0$ by monotonicity (claim 1 of Linearity and Monotonicity of the Lebesgue Integral) and the nonnegativity of the integral of Lebesgue Integral of a Nonnegative Measurable Function, and Monotone Convergence Theorem applied to the nondecreasing sequence $(\min(h,k)\mathbf{1}_{\mathsf{N}})_{k}$, whose pointwise supremum is $h\mathbf{1}_{\mathsf{N}}$, gives the assertion. (b) (Almost-everywhere majorant) Let $f:\mathbf{R}\to\mathbb{R}$ be measurable, let $g:\mathbf{R}\to[0,\infty)$ be integrable, and let $c,M\ge0$ be real numbers with $|f|\le c\,g$ on $\mathsf{N}^{c}$ and $|f|\le M g$ on $\mathsf{N}$. Then $f$ is integrable and $|\int f\,d\rho|\le c\int g\,d\rho$. Indeed, $g_1=c\,g\mathbf{1}_{\mathsf{N}^{c}}+Mg\mathbf{1}_{\mathsf{N}}$ is measurable with $|f|\le g_1\le\max(c,M)\,g$ pointwise, and $\max(c,M)\,g$ is integrable by linearity, so $g_1$ and then $f$ are integrable by domination, and by linearity, monotonicity ($g\mathbf{1}_{\mathsf{N}^{c}}\le g$) and (a), $|\int f\,d\rho|\le\int g_1\,d\rho=c\int g\mathbf{1}_{\mathsf{N}^{c}}\,d\rho+M\int g\mathbf{1}_{\mathsf{N}}\,d\rho\le c\int g\,d\rho$.

Step 1 (claim 1, for an arbitrary parameter). Let $\vartheta>0$ be a real number and let $\mu',\mu''$ be relative $\vartheta$-perturbations of $\mu$ off $\mathsf{N}$ with bounds $\bar\mu',\bar\mu''$; we show $|E(r)|\le\tilde{l}T\bar\mu\vartheta^{2}$ for $r\in\mathsf{N}^{c}$, which for $\vartheta=\eta$ is the first assertion of claim 1. By claims 1 and 3 of Integer Power Products of Causal-Intensity Likelihoods: Pathwise Identity, Integral Bounds, and the Pair Identity applied to $(\mu,\mu',\mu'')$ with exponents $(-1,1,1)$, $\tilde\mu$ is a causal intensity and $E(r)=\tilde\Lambda_T(r)+\Lambda_T(r)-\Lambda'_T(r)-\Lambda''_T(r)$, where $\tilde\Lambda_T(r)$, $\Lambda_T(r)$, $\Lambda'_T(r)$, $\Lambda''_T(r)$ are the integrals over $[0,T]$ of the total intensities $s\mapsto\tilde\mu^{\mathrm{tot}}_s(r)$, $\mu^{\mathrm{tot}}_s(r)$, $\mu'^{\mathrm{tot}}_s(r)$, $\mu''^{\mathrm{tot}}_s(r)$; each of these four functions of $s$ is measurable with respect to the trace Borel $\sigma$-algebra $\mathcal{B}_{[0,T]}$ and bounded (Likelihood of a Causal Intensity on the Observation Record Space), hence integrable with respect to the measure $\lambda_{[0,T]}$ of claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval (its integral of $|\cdot|$ is at most a constant times $\lambda_{[0,T]}([0,T])=T$, by claim 1 of Linearity and Monotonicity of the Lebesgue Integral and Simple Function and Its Integral). Put $h_r(s)=\tilde\mu^{\mathrm{tot}}_s(r)+\mu^{\mathrm{tot}}_s(r)-\mu'^{\mathrm{tot}}_s(r)-\mu''^{\mathrm{tot}}_s(r)$; by linearity on $([0,T],\mathcal{B}_{[0,T]},\lambda_{[0,T]})$, $h_r$ is integrable with $E(r)=\int_{[0,T]}h_r(s)\,ds$, and pointwise, since $\tilde\mu^\upsilon+\mu^\upsilon-\mu'^\upsilon-\mu''^\upsilon=(\mu'^\upsilon\mu''^\upsilon-\mu^\upsilon\mu'^\upsilon-\mu^\upsilon\mu''^\upsilon+(\mu^\upsilon)^{2})/\mu^\upsilon$, $h_r(s)=\sum_{\upsilon\in V}\frac{(\mu'^\upsilon_s(r)-\mu^\upsilon_s(r))(\mu''^\upsilon_s(r)-\mu^\upsilon_s(r))}{\mu^\upsilon_s(r)} .$ Fix $r\in\mathsf{N}^{c}$. For all $s\in[0,T]$ and $\upsilon\in V$, the relative-perturbation bounds, absolute-value arithmetic and $\mu^\upsilon_s(r)>0$ give $\Bigl|\frac{(\mu'^\upsilon_s(r)-\mu^\upsilon_s(r))(\mu''^\upsilon_s(r)-\mu^\upsilon_s(r))}{\mu^\upsilon_s(r)}\Bigr|\le\frac{\vartheta\mu^\upsilon_s(r)\cdot\vartheta\mu^\upsilon_s(r)}{\mu^\upsilon_s(r)}=\vartheta^{2}\mu^\upsilon_s(r)\le\vartheta^{2}\bar\mu ,$ so $|h_r(s)|\le\tilde{l}\bar\mu\vartheta^{2}$ for every $s\in[0,T]$ by the triangle inequality. The constant function $\tilde{l}\bar\mu\vartheta^{2}$ on $[0,T]$ is a simple function with integral $\tilde{l}\bar\mu\vartheta^{2}\lambda_{[0,T]}([0,T])=\tilde{l}T\bar\mu\vartheta^{2}$ (Simple Function and Its Integral and claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval), hence integrable, and $|h_r|$ is integrable and at most this constant, so by linearity $|E(r)|\le\int_{[0,T]}|h_r(s)|\,ds\le\tilde{l}T\bar\mu\vartheta^{2}$. For $\vartheta=\eta$ this is $|E|\le\mathsf{E}_\eta$ on $\mathsf{N}^{c}$, and the remaining assertions of claim 1 (integrability of $\ell_{\tilde\mu}E$ and the two inequalities) are claim 3 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form, applied with $n=2$, $\mu_1=\mu'$, $\mu_2=\mu''$, $(q,q')=(1,2)$, $\bar{E}=\mathsf{E}_\eta$ and the null set $\mathsf{N}$.

Step 2 (claim 2). Assume $\eta\le1$. By claim 1 of Integer Power Products of Causal-Intensity Likelihoods: Pathwise Identity, Integral Bounds, and the Pair Identity applied to $(\mu,\mu',\mu'')$ with exponents $(-1,1,1)$ (as in claim 3 of that lemma, the lower bound $\underline\mu$ serving for $\mu$), $\tilde\mu$ is a causal intensity on $\mathbf{R}$ with bound $\bar\mu'\bar\mu''/\underline\mu$. Fix $r\in\mathsf{N}^{c}$, $s\in[0,T]$ and $\upsilon\in V$, write $m=\mu^\upsilon_s(r)$, $m'=\mu'^\upsilon_s(r)$, $m''=\mu''^\upsilon_s(r)$ and $\tilde m=\tilde\mu^\upsilon_s(r)=m'm''/m$, and put $a=(m'-m)/m$ and $b=(m''-m)/m$ (recall $m\ge\underline\mu>0$), so that $|a|\le\eta$, $|b|\le\eta$, $m'=m(1+a)$ and $m''=m(1+b)$. Then $\tilde m-m=\frac{m'm''}{m}-m=m\bigl((1+a)(1+b)-1\bigr)=m\,(a+b+ab),\qquad |a+b+ab|\le|a|+|b|+|a||b|\le\eta+\eta+\eta^{2}\le3\eta ,$ using absolute-value arithmetic and $\eta\le1$; hence $|\tilde\mu^\upsilon_s(r)-\mu^\upsilon_s(r)|\le3\eta\,\mu^\upsilon_s(r)$, i.e. $\tilde\mu$ is a relative $3\eta$-perturbation of $\mu$ off $\mathsf{N}$. Now apply Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form with base intensity $\mu$, $n=1$ and the single perturbed intensity $\mu_1=\tilde\mu$ (bound $\bar\mu'\bar\mu''/\underline\mu$), writing $\tilde{L}=\ell_{\tilde\mu}/\ell$. Claim 1 there gives that $\ell\tilde{L}\tilde{L}=\ell_{\tilde\mu}^{2}/\ell$ (pointwise, as $\ell>0$) is integrable with $\int_{\mathbf{R}}\ell_{\tilde\mu}^{2}/\ell\,d\rho=1+C_{11}$, where $C_{11}=\mathrm{Var}(\tilde\mu)=\int_{\mathbf{R}}\ell(\tilde{L}-1)^{2}\,d\rho$ is finite and nonnegative; this is the integrability, the identity and the lower bound in claim 2. For the upper bound, the pair exponent $E_{11}$ of that application is the pair exponent of the present lemma for the pair $(\mu',\mu'')$ replaced by $(\tilde\mu,\tilde\mu)$ (both are defined by the same formula), and $\tilde\mu$ is a relative $3\eta$-perturbation of $\mu$ off $\mathsf{N}$ with bound $\bar\mu'\bar\mu''/\underline\mu$; so Step 1 with $\vartheta=3\eta$ and $(\tilde\mu,\tilde\mu)$ in place of $(\mu',\mu'')$ gives $|E_{11}(r)|\le\tilde{l}T\bar\mu(3\eta)^{2}=9\mathsf{E}_\eta$ for every $r\in\mathsf{N}^{c}$, and claim 3 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form (with $\bar{E}=9\mathsf{E}_\eta$ and the null set $\mathsf{N}$) gives $C_{11}\le|C_{11}|\le\exp(9\mathsf{E}_\eta)-1$.

Step 3 (claim 3). Keep the hypotheses of claim 2. The likelihoods $\ell$ and $\ell_{\tilde\mu}$ are measurable (claim 2 of Survival Identity, Measurability, Event-Count Recursion, and Normalization of the Likelihood of a Causal Intensity), $\ell$ is integrable with $\int_{\mathbf{R}}\ell\,d\rho=1$ and $\ell>0$ (claim 1 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form), and $\alpha=|\ell_{\tilde\mu}-\ell|$ is measurable. Apply the weighted Cauchy-Schwarz inequality, claim 1 of Weighted Cauchy-Schwarz Inequality on a Measure Space and the Symmetrised Score Functional: Bounds and Averaging, on the measure space $(\mathbf{R},\mathcal{R},\rho)$ with $\beta=\ell$ and this $\alpha$ (the requirement that $\alpha$ vanish where $\beta$ vanishes is vacuous, as $\ell>0$). The auxiliary function called $q$ in that claim, here written $\mathsf{q}$, equals $\alpha^{2}/\ell=(\ell_{\tilde\mu}-\ell)^{2}/\ell=\ell(\tilde{L}-1)^{2}$ on all of $\mathbf{R}$, whose integral is $\mathrm{Var}(\tilde\mu)<\infty$ by Step 2. Hence $\alpha$ is integrable and $(\int_{\mathbf{R}}\alpha\,d\rho)^{2}\le1\cdot\mathrm{Var}(\tilde\mu)\le\exp(9\mathsf{E}_\eta)-1$. Since $\int\alpha\,d\rho\ge0$ and $(\exp(9\mathsf{E}_\eta)-1)^{1/2}\ge0$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\int_{\mathbf{R}}|\ell_{\tilde\mu}-\ell|\,d\rho\le(\exp(9\mathsf{E}_\eta)-1)^{1/2}$.

Next, $E$ is measurable and bounded on $\mathbf{R}$ (claim 1 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form); let $M\ge0$ be a real number with $|E|\le M$ on $\mathbf{R}$. Then $\ell E$ is measurable with $|\ell E|\le M\ell$, hence integrable by domination, and $\ell_{\tilde\mu}E$ is integrable by claim 3 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form (applicable by Step 1). The measurable function $f=(\ell_{\tilde\mu}-\ell)E=\ell_{\tilde\mu}E-\ell E$ satisfies $|f|\le\mathsf{E}_\eta\,\alpha$ on $\mathsf{N}^{c}$ (Step 1) and $|f|\le M\alpha$ on $\mathsf{N}$, with $\alpha$ integrable, so by Step 0(b) and linearity $\Bigl|\int_{\mathbf{R}}\ell_{\tilde\mu}E\,d\rho-\int_{\mathbf{R}}\ell E\,d\rho\Bigr|=\Bigl|\int_{\mathbf{R}}f\,d\rho\Bigr|\le\mathsf{E}_\eta\int_{\mathbf{R}}\alpha\,d\rho\le\mathsf{E}_\eta\bigl(\exp(9\mathsf{E}_\eta)-1\bigr)^{1/2}.$ Combining with the second inequality of claim 1 by the triangle inequality, $|C-\int\ell E\,d\rho|\le|C-\int\ell_{\tilde\mu}E\,d\rho|+|\int\ell_{\tilde\mu}E\,d\rho-\int\ell E\,d\rho|\le\tfrac12\mathsf{E}_\eta^{2}\exp(\mathsf{E}_\eta)+\mathsf{E}_\eta(\exp(9\mathsf{E}_\eta)-1)^{1/2}=\mathsf{e}_\eta$.

Step 4 (claim 4). Assume $\eta\le1$ and let $\mu_1,\dots,\mu_n$, $w$, $L_q$, $E_{qq'}$ and $C_{qq'}$ be as in claim 4. The pair exponent and the pair covariance of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form at an index pair $(q,q')$ depend only on $\mu$, $\mu_q$ and $\mu_{q'}$, and coincide with the pair exponent $E$ and the pair covariance $C$ of the present lemma for $(\mu',\mu'')=(\mu_q,\mu_{q'})$, which are relative $\eta$-perturbations of $\mu$ off $\mathsf{N}$ with bounds $\bar\mu_q,\bar\mu_{q'}$. Hence, by Step 3, $\ell E_{qq'}$ is integrable and $|C_{qq'}-\int_{\mathbf{R}}\ell E_{qq'}\,d\rho|\le\mathsf{e}_\eta$ for all $q,q'$. Each $E_{qq'}$ is measurable and bounded (claim 1 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form), so the finite linear combination $\mathcal{Q}=\sum_{q,q'}w_qw_{q'}E_{qq'}$ is measurable and bounded, and $\ell\mathcal{Q}=\sum_{q,q'}w_qw_{q'}\,\ell E_{qq'}$ is integrable by linearity with $\int_{\mathbf{R}}\ell\mathcal{Q}\,d\rho=\sum_{q,q'}w_qw_{q'}\int_{\mathbf{R}}\ell E_{qq'}\,d\rho$. Therefore, by the triangle inequality for finite sums and $\sum_{q,q'}|w_q||w_{q'}|=\lVert w\rVert_1^{2}$, $\Bigl|\sum_{q,q'}w_qw_{q'}C_{qq'}-\int_{\mathbf{R}}\ell\mathcal{Q}\,d\rho\Bigr|=\Bigl|\sum_{q,q'}w_qw_{q'}\Bigl(C_{qq'}-\int_{\mathbf{R}}\ell E_{qq'}\,d\rho\Bigr)\Bigr|\le\sum_{q,q'}|w_q||w_{q'}|\,\mathsf{e}_\eta=\lVert w\rVert_1^{2}\,\mathsf{e}_\eta ,$ which gives the right-hand inequality of claim 4. For the identity, claim 2 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form with the ratio factors $\mathsf{r}_1=\dots=\mathsf{r}_n=1$ states that $\ell(\sum_qw_q(1-L_q))^{2}$ is integrable with integral $(\sum_qw_q(1-1))^{2}+\sum_{q,q'}w_qw_{q'}C_{qq'}=\sum_{q,q'}w_qw_{q'}C_{qq'}$. Finally this integrand is nonnegative pointwise, and the zero function is integrable with integral $0$ (linearity with both coefficients $0$), so monotonicity gives $\sum_{q,q'}w_qw_{q'}C_{qq'}\ge0$. $\blacksquare$