Proof of Integer Power Products of Causal-Intensity Likelihoods: Pathwise Identity, Integral Bounds, and the Pair Identity

Preliminaries. Sums, products and pointwise limits of measurable real functions are measurable by Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and compositions with sequentially continuous maps by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable. By Basic Properties of the Exponential Function, $\exp$ is positive and increasing, $\exp(0)=1$, and $\exp(a+b)=\exp(a)\exp(b)$; hence $\exp(-a)=1/\exp(a)$, and for every integer $\theta$ and real $a$, $\exp(a)^{\theta}=\exp(\theta a)$ (for $\theta\ge0$ by repeated use of the functional equation, and for $\theta<0$ from $\exp(a)^{\theta}=1/\exp(a)^{-\theta}=1/\exp(-\theta a)=\exp(\theta a)$). The finite products of real numbers are the finite products of Finite Product Notation in a Field in the field of real numbers (both are defined by the same recursion), so Properties of Finite Products applies to them. Two elementary rules are used. (R1) For reals $x,y$ and an integer $\theta$, $(xy)^{\theta}=x^{\theta}y^{\theta}$ whenever the powers are defined (for $\theta\ge1$ this is claim 2 of Properties of Finite Products with constant factors; for $\theta=0$ both sides are $1$; for $\theta\le-1$ and $x,y>0$ take reciprocals of the case $-\theta$). (R2) For reals $y_{i,m}$ ($1\le i\le n$, $1\le m\le k$), $\prod_{i=1}^{n}\prod_{m=1}^{k}y_{i,m}=\prod_{m=1}^{k}\prod_{i=1}^{n}y_{i,m}$, by induction on $n$: the case $n=1$ is trivial, and $\prod_{m}\prod_{i\le n}y_{i,m}=\prod_m\bigl(\prod_{i<n}y_{i,m}\bigr)y_{n,m}=\bigl(\prod_m\prod_{i<n}y_{i,m}\bigr)\bigl(\prod_my_{n,m}\bigr)$ by the recursion (claim 1) and multiplicativity (claim 2) of Properties of Finite Products. Applied with $y_{i,m}=x_{i,m}^{\theta_i}$, (R1) and (R2) give $\prod_i\bigl(\prod_mx_{i,m}\bigr)^{\theta_i}=\prod_i\prod_mx_{i,m}^{\theta_i}=\prod_m\prod_ix_{i,m}^{\theta_i}$ for reals $x_{i,m}$ that are positive whenever $\theta_i<0$ and nonnegative otherwise (the powers with $\theta_i\ge0$ are then defined for all such $x_{i,m}$, and (R1) with $\theta_i\ge0$ needs no positivity). Integrals of nonnegative functions are handled with claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and integrable functions with claim 2 there and Integrable Function and the Lebesgue Integral.

Step 1: claim 1. $\tilde\mu$ is a causal intensity. Fix $\upsilon$. For $i$ with $\theta_i\ge0$ the map $(\mu^\upsilon_i)^{\theta_i}$ is a finite product of copies of the measurable map $\mu^\upsilon_i$ (or the constant $1$), hence measurable with values in $[0,\bar\mu_i^{\theta_i}]$; for $i$ with $\theta_i<0$, $\mu^\upsilon_i$ takes values in $[\underline\mu,\bar\mu_i]$, so $(\mu^\upsilon_i)^{\theta_i}=1/(\mu^\upsilon_i)^{-\theta_i}$ is the composition of the sequentially continuous map $x\mapsto1/x^{-\theta_i}$ on $[\underline\mu,\infty)$ with $\mu^\upsilon_i$, hence measurable with values in $(0,\underline\mu^{\theta_i}]$. The product $\tilde\mu^\upsilon$ is therefore measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$, with values in $[0,\bar{\tilde\mu}]$ (factorwise monotonicity of products of nonnegative reals, claim 5 of Properties of Finite Products, the omitted factors of the subset products being $1$), which is condition (i) of Causal Intensity on the Observation Record Space with the bound $\bar{\tilde\mu}$. Condition (ii) holds because each factor satisfies it: $\tilde\mu^\upsilon_s(r)=\prod_i\mu^\upsilon_{i,s}(r)^{\theta_i}=\prod_i\mu^\upsilon_{i,s}(\pi_{s-}(r))^{\theta_i}=\tilde\mu^\upsilon_s(\pi_{s-}(r))$.

$E$ is measurable and bounded. $E=\tilde\Lambda_T-\sum_i\theta_i\Lambda_{i,T}$ is a linear combination of the $\mathcal{R}$-measurable functions $\tilde\Lambda_T$ and $\Lambda_{i,T}$ (claim 2 of Survival Identity, Measurability, Event-Count Recursion, and Normalization of the Likelihood of a Causal Intensity, applied to $\tilde\mu$ and to each $\mu_i$), hence measurable; and since $0\le\tilde\Lambda_T\le\tilde{l}\bar{\tilde\mu}T$ and $0\le\Lambda_{i,T}\le\tilde{l}\bar\mu_iT$ by monotonicity of the integral (each total intensity being bounded by $\tilde{l}$ times the bound), $|E|\le\tilde{l}T(\bar{\tilde\mu}+\sum_i|\theta_i|\bar\mu_i)$.

Positivity. For $i$ with $\theta_i<0$ and $r=(k,t,v)$, every factor $\mu^{v_m}_{i,t_m}(r)\ge\underline\mu>0$ and $\exp(-\Lambda_{i,T}(r))>0$, so $\ell_{\mu_i}(r)>0$ by Likelihood of a Causal Intensity on the Observation Record Space; thus the powers $\ell_{\mu_i}(r)^{\theta_i}$ are defined and positive.

Pathwise identity. Fix $r=(k,t,v)$ and write $x_{i,m}=\mu^{v_m}_{i,t_m}(r)$ for $1\le m\le k$; these are nonnegative, and positive for every $i$ with $\theta_i<0$, so the rules of the preliminaries apply. By Likelihood of a Causal Intensity on the Observation Record Space, (R1), (R2) and $\exp(a)^{\theta}=\exp(\theta a)$,

while $\ell_{\tilde\mu}(r)\exp(E(r))=\bigl(\prod_{m}\tilde\mu^{v_m}_{t_m}(r)\bigr)\exp(-\tilde\Lambda_T(r))\exp(\tilde\Lambda_T(r)-\sum_i\theta_i\Lambda_{i,T}(r))$, and the functional equation makes the two exponential factors agree. (For $k=0$ both products are $1$.) This proves claim 1.

Step 2: claim 2. By claim 1, $\prod_i\ell_{\mu_i}^{\theta_i}=\ell_{\tilde\mu}\exp(E)$ pointwise; $\ell_{\tilde\mu}$ is $\mathcal{R}$-measurable and nonnegative (claim 2 of Survival Identity, Measurability, Event-Count Recursion, and Normalization of the Likelihood of a Causal Intensity) and $\exp(E)$ is measurable and positive, so the product is measurable and nonnegative. If $\underline E\le E\le\bar E$ then, $\exp$ being increasing, $\exp(\underline E)\ell_{\tilde\mu}\le\ell_{\tilde\mu}\exp(E)\le\exp(\bar E)\ell_{\tilde\mu}$ pointwise, and monotonicity and homogeneity of the integral together with $\int_{\mathbf{R}}\ell_{\tilde\mu}\,d\rho=1$ (claim 4 of Survival Identity, Measurability, Event-Count Recursion, and Normalization of the Likelihood of a Causal Intensity, applied to the causal intensity $\tilde\mu$) give the two bounds; if $E\equiv E_0$, take $\underline E=\bar E=E_0$.

Step 3: claim 3. The hypotheses of claim 1 hold for $(\mu_1,\mu_2,\mu_3)=(\mu,\mu',\mu'')$ and $(\theta_1,\theta_2,\theta_3)=(-1,1,1)$, the lower bound being required only for $\mu$. Then $\tilde\mu^\upsilon=\mu'^\upsilon\mu''^\upsilon/\mu^\upsilon$, and pointwise in $(s,r)$, with the channel index suppressed,

so $E(r)=\int_{[0,T]}\sum_\upsilon(\tilde\mu^\upsilon_s-\sum_i\theta_i\mu^\upsilon_{i,s})(r)\,ds$ (linearity of the integral, the integrands being bounded and measurable) is the displayed integral. Positivity of $\ell_\mu$ is the positivity part of claim 1, so $L'$ and $L''$ are well defined and measurable (quotients of measurable functions with positive denominator, via the continuous map $(a,b)\mapsto a/b$ on $\mathbb{R}\times(0,\infty)$). By claim 1 and the boundedness of $E$, $\ell_{\mu'}\ell_{\mu''}/\ell_\mu=\ell_{\tilde\mu}\exp(E)$ is nonnegative and measurable with $\int\ell_{\mu'}\ell_{\mu''}/\ell_\mu\,d\rho\le\exp(\bar E)<\infty$ by claim 2 (with $\bar E=\tilde{l}T(\bar{\tilde\mu}+\bar\mu+\bar\mu'+\bar\mu'')$), hence integrable; $\ell_\mu$, $\ell_{\mu'}$, $\ell_{\mu''}$ are nonnegative, measurable and integrate to $1$ (claim 4 of Survival Identity, Measurability, Event-Count Recursion, and Normalization of the Likelihood of a Causal Intensity), hence integrable. Expanding pointwise, $\ell_\mu(1-L')(1-L'')=\ell_\mu-\ell_{\mu'}-\ell_{\mu''}+\ell_{\mu'}\ell_{\mu''}/\ell_\mu$ is a linear combination of integrable functions, hence integrable, and linearity of the integral gives $\int\ell_\mu(1-L')(1-L'')\,d\rho=1-1-1+\int\ell_{\mu'}\ell_{\mu''}/\ell_\mu\,d\rho$, which is the asserted identity. $\blacksquare$