Lagmental Vicfred

Poisson Increments Depend Only on Interval Length by Vicfred

tags: mathematics, stochastic processes, order statistics, information theory

A rate-lambda Poisson process counts independent arrivals with Poisson-distributed increments. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

The mathematical object

Continuous-time processes are described by finite-dimensional distributions plus path regularity. A Poisson process \(N_t\) has independent increments, while Brownian motion \(B_t\) has Gaussian increments.

$$ N_t-N_s\sim\operatorname{Poisson}(\lambda(t-s))\qquad(0\le s<t) $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ \mathbf P(N_t=k)=e^{-\lambda t}\frac{(\lambda t)^k}{k!} $$

One explicit computation

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \mathbf E[z^{N_t}]=\exp(\lambda t(z-1)),\qquad\mathbf E[N_t]=\operatorname{Var}(N_t)=\lambda t $$

Why the identity matters

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad N_t-N_s\sim\operatorname{Poisson}(\lambda(t-s))\qquad(0\le s<t),\\[5pt] \mathsf{C}\;&:\quad \mathbf P(N_t=k)=e^{-\lambda t}\frac{(\lambda t)^k}{k!}. \end{aligned} $$

Where it can fail

Matching means and variances does not identify a distribution. Independence, increment laws, and sample-path properties are separate ingredients.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathbf P(N_t=k)=e^{-\lambda t}\frac{(\lambda t)^k}{k!} \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Wed 14 October 2015. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.