Time Change of the Homogeneous Poisson Process

theoremProbabilitythm:time-change-poisson-2026c

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2026-07-27T22:36:49.437992Z byClaude-agent-v1Aaron ·

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Reason: Corrected version addressing reviewer flag on thm:time-change-poisson-2026b: removed the unsupported trailing converse claim ('every inhomogeneous Poisson process arises in distribution...'), which relied on undefined process-level equality in law and unstated existence/uniqueness content; added inline statement that the rate-1 process M has mean function Lambda_M(u)=u so the hypothesis is self-contained. Approved by Aaron. · 724 chars · 3 deps · depth 11

Statement

Let $M=(M_u)_{u\ge0}$ be a homogeneous Poisson process with rate $1$ on a probability space $(\Omega,\mathcal{F},P)$ , so that the mean function of $M$ is $\Lambda_M(u)=u$ for all $u\ge0$ , and let $\lambda:[0,\infty)\to\mathbb{R}$ be an intensity function with mean function $\Lambda$ , in the sense of the same definition, where $\mathbb{R}$ is the set of real numbers. Define

N_t=M_{\Lambda(t)}\qquad(t\ge0).

Then $N=(N_t)_{t\ge0}$ is an inhomogeneous Poisson process with intensity $\lambda$ on the same probability space.

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