Let \(X\) be a Poisson random variable with rate \(\lambda\), and let \(Y\) be an independent Poisson random variable with rate \(\mu\). Then, simple calculations show that \(X+Y\) is a Poisson random variable with rate \(\lambda + \mu\). This is known as the superposition property of the Poisson distribution. In the opposite direction, suppose \(X\) is a Poisson random variable with rate \(\lambda\), and let \(Z_1,Z_2,\dots\) be an independent sequence of i.i.d. Bernoulli random variables with probability \(p\). Then, \(\sum_{i=1}^\infty Z_i \mathbb{I}_{\{i \le X\}}\) is Poisson distributed with rate \(\lambda p\). This is known as the thinning property. It turns out that this latter property is far more general, and the goal of this post is to illustrate thinning for general Poisson random measures.
For a given \(\sigma\)-finite measure space \((E,\mathcal{E},\mu)\), the Poisson random measure (PRM) with intensity \(\mu\) is a random measure \(\omega\mapsto N_\omega(\cdot)\) defined on some probability space \((\Omega,\mathcal{F},\mathbb{P})\) such that:
- for all \(\omega\in\Omega\), \(N_\omega(\cdot)\) is a measure on \((E,\mathcal{E})\);
- for all \(A \in \mathcal{E}\), the random variable \(\omega\mapsto N_\omega(A)\) is Poisson distributed with rate \(\mu(A)\);
- if \(A_1,\dots, A_k \in \mathcal{E}\) do not intersect, then \(N(A_1),\dots,N(A_k)\) are mutually independent.
(In the remainder of this post, I’ll omit \(\omega\)).
Example. Suppose \(E=\mathbb{R}_+\), let \(\mathcal{E}\) be the Borel $\sigma$-algebra, and \(\mu(dx) = \lambda \,dx\), where \(dx\) is the Lebesgue measure. Then, the process \(t \mapsto N([0,t])\) is a Poisson process with rate \(\lambda\).
More generally, (integrals with respect to) Poisson random measures offer a convenient way to describe and represent stochastic processes with jumps. Compare to the setting of processes with continuous paths, where the Ito integral plays the corresponding role.
One approach to proving large deviations results for stochastic processes is to characterize certain changes of measure as “control” problems for random paths. In the continuous setting, the Girsanov theorem indicates that such changes of measure are achieved by imposing drift via adapted processes. In the jump setting, such “control” is achieved through a generalization of the “thinning” mechanism described above.
Instead of getting into details, I’d like to just visualize this thinning mechanism in a very simple case: with constant “control”. Consider the plot below, which displays the outcome of a Poisson random measure \(N\) on \([0,40] \times [0,1]\), with intensity the Lebesgue measure. The blue and grey points together represent the outcome of the Poisson random measure \(N\). The blue dots alone represent a thinning of \(N\); that is, the outcome of the PRM \(\mathbb{I}_{[0,c]}(x) N(dt\, dx)\), where in the plot below, I have chosen \(c=0.65\). In the second plot are the associated homogeneous Poisson processes \(t\mapsto N([0,t]\times [0,1])\), again with grey representing the total process (with rate 1) and blue representing the thinned process (with rate 0.65). In particular, every jump of the blue path (respectively, grey path) corresponds to a blue point (respectively, a blue or grey point). Lastly, the dashed lines represent the “average” behavior of the associated Poisson processes.
For reference, the plot was produced in Python with matplotlib, and then ported to the web with mpld3. To zoom in and pan, use the icons in the lower left of the figure. The code can be found in my Github repo.