Poisson laws are the universal limits of many independent rare events. Passing from a count to a random measure preserves where those events occur and leads to Poisson processes, point processes, thinning, superposition, and compensated jump martingales.
The Poisson law
Definition. A random variable \(N\) has the Poisson law with parameter \(\lambda\ge0\) if \[ P(N=k)=e^{-\lambda}\frac{\lambda^k}{k!}, \qquad k=0,1,2,\ldots. \] Its probability generating function, characteristic function, mean, and variance are \[ \mathbb Ez^N=e^{\lambda(z-1)},\qquad \mathbb Ee^{itN}=e^{\lambda(e^{it}-1)},\qquad \mathbb EN=\operatorname{Var}(N)=\lambda. \]
Proposition (additivity). If \(N_1\sim\operatorname{Poisson}(\lambda_1)\) and \(N_2\sim\operatorname{Poisson}(\lambda_2)\) are independent, then \(N_1+N_2\sim\operatorname{Poisson}(\lambda_1+\lambda_2)\).
Proof. Multiply their generating functions: \[ \mathbb Ez^{N_1+N_2} =e^{\lambda_1(z-1)}e^{\lambda_2(z-1)} =e^{(\lambda_1+\lambda_2)(z-1)}. \]
The law of small numbers
Theorem (Poisson convergence). Let \(X_{n,k}\) be row-wise independent Bernoulli variables with \(P(X_{n,k}=1)=p_{n,k}\). If \[ \max_{k\le k_n}p_{n,k}\to0, \qquad \sum_{k=1}^{k_n}p_{n,k}\to\lambda, \] then \[ \sum_{k=1}^{k_n}X_{n,k} \Rightarrow\operatorname{Poisson}(\lambda). \]
Proof. For \(z\in[0,1]\), \[ \mathbb Ez^{\sum_kX_{n,k}} =\prod_k\{1+p_{n,k}(z-1)\}. \] Since \[ \sum_kp_{n,k}^2 \le\left(\max_kp_{n,k}\right)\sum_kp_{n,k}\to0, \] the expansion \(\log(1+u)=u+O(u^2)\), uniformly for the relevant \(u\), gives \[ \log\mathbb Ez^{\sum_kX_{n,k}} =(z-1)\sum_kp_{n,k}+o(1) \to\lambda(z-1). \] The generating functions converge to that of \(\operatorname{Poisson}(\lambda)\), which determines the law on \(\mathbb N_0\).
Theorem (Le Cam inequality). If \(W=\sum_{k=1}^nX_k\) is a sum of independent Bernoulli\((p_k)\) variables and \(Z\sim\operatorname{Poisson}(\lambda)\), \(\lambda=\sum_kp_k\), then \[ d_{\mathrm{TV}}(\mathcal L(W),\mathcal L(Z)) \le 2\sum_{k=1}^np_k^2. \]
Proof outline. Couple each Bernoulli \(X_k\) to an independent Poisson variable \(Z_k\sim\operatorname{Poisson}(p_k)\). Direct comparison of their masses gives a coupling with \(P(X_k\ne Z_k)\le 2p_k^2\). Under the product coupling, \(Z=\sum_kZ_k\sim\operatorname{Poisson}(\sum_kp_k)\), and \[ d_{\mathrm{TV}}(\mathcal L(W),\mathcal L(Z)) \le P(W\ne Z) \le\sum_kP(X_k\ne Z_k) \le2\sum_kp_k^2. \]
Poisson processes and waiting times
Definition. A homogeneous Poisson process with rate \(\lambda>0\) is a right-continuous counting process \((N_t)_{t\ge0}\) such that \(N_0=0\), its increments over disjoint intervals are independent, and \[ N_t-N_s\sim\operatorname{Poisson}(\lambda(t-s)), \qquad0\le s<t. \]
Theorem (arrival-time construction). Let \(E_1,E_2,\ldots\) be i.i.d. exponential\((\lambda)\), put \(T_n=\sum_{k=1}^nE_k\), and define \[ N_t=\max\{n:T_n\le t\}. \] Then \(N\) is a Poisson process of rate \(\lambda\). Conversely, the interarrival times of a Poisson process are i.i.d. exponential\((\lambda)\).
Proof. Since \(T_n\) has the gamma density \[ f_{T_n}(u)=\frac{\lambda^nu^{n-1}e^{-\lambda u}}{(n-1)!}, \quad u>0, \] integration over \(T_n\le t<T_{n+1}\) gives \[ P(N_t=n)=e^{-\lambda t}\frac{(\lambda t)^n}{n!}. \] The memoryless property of the exponential law says that, conditional on the history through time \(s\), the residual waiting time to the next arrival is again exponential and independent of that history. Repeating this argument gives independent stationary increments. Conversely, \[ P(T_1>t)=P(N_t=0)=e^{-\lambda t}, \] and the independent-increment property restarts the process after each arrival, producing independent exponential gaps.
Proposition (compensation). For a rate-\(\lambda\) Poisson process with natural filtration, \[ M_t=N_t-\lambda t \] is a square-integrable martingale, and \(\mathbb E(M_t-M_s)^2=\lambda(t-s)\).
Proof. Independent increments give \[ \mathbb E[N_t-N_s\mid\mathcal F_s]=\lambda(t-s), \] hence \(\mathbb E[M_t\mid\mathcal F_s]=M_s\). The variance identity follows from \(\operatorname{Var}(N_t-N_s)=\lambda(t-s)\).
Poisson random measures
Definition. Let \((E,\mathcal E)\) carry a sigma-finite measure \(\nu\). A Poisson random measure \(N\) with intensity \(\nu\) is an integer-valued random measure such that: \[ N(A)\sim\operatorname{Poisson}(\nu(A)) \quad\text{when }\nu(A)<\infty, \] and \(N(A_1),\ldots,N(A_m)\) are independent for pairwise disjoint \(A_1,\ldots,A_m\).
Theorem (Laplace functional). For every measurable \(f:E\to[0,\infty)\), \[ \mathbb E\exp\!\left\{-\int_E f(x)N(\mathrm{d} x)\right\} = \exp\!\left\{-\int_E(1-e^{-f(x)})\nu(\mathrm{d} x)\right\}. \] This identity determines the law of \(N\).
Proof. First take \(f=\sum_{j=1}^ma_j1_{A_j}\) on disjoint sets of finite intensity. Independence and the Poisson generating function give \[ \prod_{j=1}^m \mathbb Ee^{-a_jN(A_j)} = \prod_{j=1}^m \exp\{\nu(A_j)(e^{-a_j}-1)\}, \] which is the displayed formula. Approximate a general nonnegative \(f\) increasingly by simple functions and use monotone convergence on both sides.
Proposition (thinning and superposition). Independently retain each point of a Poisson random measure of intensity \(\nu\) with location-dependent probability \(q(x)\). The retained and deleted processes are independent Poisson random measures with intensities \(q\nu\) and \((1-q)\nu\). The sum of independent Poisson random measures with intensities \(\nu_1,\nu_2\) is Poisson with intensity \(\nu_1+\nu_2\).
Proof. Conditional on \(N(A)=m\), independent marking splits the count binomially. Evaluating the joint generating function of retained and deleted counts yields the product of Poisson generating functions. Apply this on disjoint simple-function cells and then use the Laplace-functional characterization. Superposition follows by multiplying the two Laplace functionals.
Rare exceedances as a point process
Example. Let \(X_1,X_2,\ldots\) be i.i.d. and choose levels \(u_n\) such that \(nP(X_1>u_n)\to\tau\). Then \[ \sum_{i=1}^n1_{\{X_i>u_n\}} \Rightarrow\operatorname{Poisson}(\tau), \] and therefore \[ P\!\left(\max_{1\le i\le n}X_i\le u_n\right) =P(\text{no exceedance})\longrightarrow e^{-\tau}. \] Recording rescaled exceedance locations as \(\sum_i1_{\{X_i>u_n\}}\delta_{i/n}\) retains more information and leads to Poisson point-process limits in extreme-value theory.