List of Probability Distributions and Processes

Introduction

This article lists the probability distributions and the stochastic processes the corpus introduces. A probability distribution is a law on a measurable space, and a process is a family of random variables whose finite-dimensional laws are distributions; a row below names one, records its law, its moments and the convergence theorem attached to it, and points to the article that introduces it. Every row points to an article; this article introduces nothing and proves nothing. The objects are grouped by the layer that introduces them: the discrete distributions, the absolutely continuous and singular distributions, the general theory of laws and their characteristic functions, the processes built on the distributions, and the limit theorems that govern them. The rows that record a failure — a distribution with no mean, a limit theorem that needs a hypothesis it does not have, a stationary process that is not ergodic — stand beside them as non-examples.

The Discrete Distributions

Object	Its law, moments and characteristic function	Introduced in
the Bernoulli distribution $\mathrm{Bern}(p)$	$p\delta_1+(1-p)\delta_0$ on $\{0,1\}$; mean $p$, variance $p(1-p)$; $\varphi(t)=1-p+pe^{it}$	Measure-Theoretic Probability
the binomial distribution $\mathrm{Bin}(n,p)$	$\sum_k\binom nk p^k(1-p)^{n-k}\delta_k$; mean $np$, variance $np(1-p)$; the sum of $n$ independent Bernoulli laws	Measure-Theoretic Probability
the Poisson distribution $\mathrm{Pois}(\lambda)$	$e^{-\lambda}\sum_k\lambda^k\delta_k/k!$ on $\mathbb Z_{\geq0}$; mean and variance $\lambda$; $\varphi(t)=e^{\lambda(e^{it}-1)}$	Measure-Theoretic Probability
the point mass $\delta_x$	the law of the constant random variable; mean $x$, variance $0$; a degenerate distribution	Measure-Theoretic Probability
the uniform law on a finite set	equal mass on each point; the counting measure normalised	Measure-Theoretic Probability

The Absolutely Continuous and Singular Distributions

Object	Its density, moments and characteristic function	Introduced in
the uniform distribution on $[0,1]$	density $\mathbf 1_{[0,1]}$; mean $1/2$, variance $1/12$; the universal law of the quantile construction	Measure-Theoretic Probability
the normal distribution $N(\mu,\sigma^2)$	density $(2\pi\sigma^2)^{-1/2}e^{-(x-\mu)^2/(2\sigma^2)}$; mean $\mu$, variance $\sigma^2$; $\varphi(t)=e^{i\mu t-\sigma^2t^2/2}$	Measure-Theoretic Probability; Laws of Large Numbers and the Central Limit Theorem
the standard normal $N(0,1)$	the limit law of the central limit theorem; all moments finite, the cumulants vanishing above the second	Laws of Large Numbers and the Central Limit Theorem
the Cauchy distribution	density $1/(\pi(1+x^2))$; no mean and no variance; $\varphi(t)=e^{-\lvert t\rvert}$	Measure-Theoretic Probability; Laws of Large Numbers and the Central Limit Theorem
the Cantor distribution	singular continuous, supported on the Cantor set; distribution function the Cantor function; no density	Measure-Theoretic Probability
the stable and infinitely divisible laws	the class closed under convolution and scaling; their characteristic functions given by the Lévy–Khintchine representation	Laws of Large Numbers and the Central Limit Theorem
the Gaussian measure on a Banach space	the cylinder measure of a Gaussian process; its characteristic function is $e^{-\lVert x\rVert^2/2}$	Stochastic Partial Differential Equations
the Wiener measure	the law of Brownian motion on the space of continuous paths	Brownian Motion and Stochastic Calculus
the equilibrium and harmonic measures	the distribution of the exit point of a Brownian path from a domain	Potential Theory; Brownian Motion and Stochastic Calculus

The Theory of Laws and Their Transforms

Object	What it carries	Introduced in
the distribution function $F_X(x)=\mathbb P(X\leq x)$	a right-continuous nondecreasing function determining the law	Measure-Theoretic Probability
the law $\mu_X=\mathbb P\circ X^{-1}$	the pushforward of the probability measure; a probability measure on the range	Measure-Theoretic Probability
the Lebesgue decomposition of a law	the split into an absolutely continuous, a discrete and a singular continuous part	Measure-Theoretic Probability
the quantile function $F^{-1}$ and the Skorokhod representation	every law on $\mathbb R$ is $F^{-1}(U)$ for $U$ uniform, so $[0,1]$ is universal	Measure-Theoretic Probability
the expectation, variance and covariance	the moments as integrals, with the monotone and dominated convergence theorems	Measure-Theoretic Probability
the characteristic function $\varphi_X(t)=\mathbb E[e^{itX}]$	the Fourier transform of the law; it determines the law and inverts it	Measure-Theoretic Probability; Laws of Large Numbers and the Central Limit Theorem
the moment generating function and the cumulants	the exponential moments where they exist, and the cumulant expansion	Measure-Theoretic Probability
the Lévy–Khintchine representation	the characteristic function of an infinitely divisible law as an exponential of a Lévy measure	Laws of Large Numbers and the Central Limit Theorem
the Markov operator and the transition kernel	the action of a Markov chain on functions and on measures	Markov Chains and Processes

The Processes

Object	Its definition, and the theorem attached to it	Introduced in
a sequence of independent random variables	the product law; the setting of the law of large numbers and the central limit theorem	Measure-Theoretic Probability
the simple random walk	the sum of independent Bernoulli steps; recurrent in dimension one and two, transient from three	Markov Chains and Processes; Random Walks on Groups
a Markov chain	the Markov property and the transition operator; recurrence, transience and the ergodic theorem	Markov Chains and Processes
a Markov process in continuous time and its semigroup	the transition semigroup and its generator; the martingale problem	Markov Chains and Processes
the Poisson process	the counting process with independent stationary increments of Poisson law; its generator is the difference operator	Markov Chains and Processes
Brownian motion	continuous paths, independent stationary Gaussian increments; quadratic variation $\langle B\rangle_t=t$ and Lévy's characterisation	Brownian Motion and Stochastic Calculus
a martingale	the conditional-expectation identity $\mathbb E[M_{n+1}\mid\mathcal F_n]=M_n$; optional stopping and the convergence theorem	Martingales
a stationary process	the law invariant under the shift; the ergodic theorem and its mean-square form	Laws of Large Numbers and the Central Limit Theorem; Ergodic Theory
a Gaussian process	every finite-dimensional law is Gaussian; determined by its mean and covariance	Stochastic Partial Differential Equations
white noise	the distributional derivative of Brownian motion; the Gaussian field with delta covariance	Stochastic Partial Differential Equations
the Ornstein–Uhlenbeck process	the Gaussian stationary Markov process with an exponential covariance; the Brownian motion in a quadratic potential	Brownian Motion and Stochastic Calculus; Ergodic Theory
the solution of a stochastic differential equation	the Itô and Stratonovich integrals; existence, uniqueness and the Markov property	Stochastic Differential Equations

The Convergence Theorems

Object	What it asserts	Introduced in
the weak law of large numbers	the empirical mean converges in probability to the mean; the Chebyshev case needs a finite variance, Khinchin's theorem does not	Laws of Large Numbers and the Central Limit Theorem
the strong law of large numbers	the empirical mean converges almost surely; Kolmogorov's theorem under the first moment	Laws of Large Numbers and the Central Limit Theorem
the central limit theorem	the standardised sum converges in distribution to $N(0,1)$; the Lindeberg–Lévy theorem	Laws of Large Numbers and the Central Limit Theorem
the Lindeberg–Feller theorem	the central limit theorem for triangular arrays under the Lindeberg condition	Laws of Large Numbers and the Central Limit Theorem
the Berry–Esseen theorem	the rate $O(n^{-1/2})$ of the convergence, with a constant depending on a third moment	Laws of Large Numbers and the Central Limit Theorem
the local limit theorem	the density converges pointwise to the Gaussian density	Laws of Large Numbers and the Central Limit Theorem
large deviations	the exponential rate of decay of the probability of a deviation; the Cramér rate function	Laws of Large Numbers and the Central Limit Theorem
the invariance principle (Donsker)	the rescaled random walk converges in distribution to Brownian motion	Laws of Large Numbers and the Central Limit Theorem; Brownian Motion and Stochastic Calculus
the martingale convergence theorem	a bounded submartingale converges almost surely	Martingales
the optional stopping theorem	the expectation identity preserved at a stopping time	Martingales
the ergodic theorem	time averages converge to space averages; Birkhoff's almost-sure and von Neumann's mean-square forms	Ergodic Theory

Distributions and Processes That Fail a Hypothesis

Object	The failure	Introduced in
the Cauchy distribution	has no mean and no variance, so the law of large numbers and the central limit theorem do not apply; the sample mean has the same law	Measure-Theoretic Probability
a law with an infinite second moment	the central limit theorem fails; the limit is a stable law with index $\alpha<2$	Laws of Large Numbers and the Central Limit Theorem
a distribution that is not determined by its moments	the moment sequence does not determine the law, so the moments do not give the characteristic function	Measure-Theoretic Probability
a stationary but not ergodic process	the time averages need not converge to the space average	Ergodic Theory
the empirical measure of a non-ergodic system	converges to a mixture, not to a single distribution	Ergodic Theory
a Markov chain that is not irreducible	has no unique stationary distribution, and the convergence to stationarity fails	Markov Chains and Processes
the sample paths of Brownian motion	are almost surely nowhere differentiable and of infinite variation, so the classical integral does not exist	Brownian Motion and Stochastic Calculus

Summary

This list gathers the probability distributions and processes of the corpus: the Bernoulli, binomial, Poisson and degenerate discrete laws, the uniform, normal and Cauchy laws with the Cantor distribution and the stable and infinitely divisible family, the theory of the law, the distribution function and the characteristic function, the Markov chains, Poisson process, Brownian motion, martingales, stationary and Gaussian processes and white noise, and the limit theorems — the laws of large numbers, the central limit theorem and its refinements, the invariance principle, the martingale convergence theorem and the ergodic theorem. The closing table records the distributions and processes that fail a hypothesis of the theory.

Summary of Notation

The objects are named by their standard symbols; the tables use the following.

Symbol	Meaning
$\mathrm{Bern}(p)$, $\mathrm{Bin}(n,p)$, $\mathrm{Pois}(\lambda)$	Bernoulli, binomial, Poisson laws
$N(\mu,\sigma^2)$, $N(0,1)$	normal and standard normal laws
$\mu_X=\mathbb P\circ X^{-1}$	the law of $X$
$F_X$, $F^{-1}$, $\varphi_X$	distribution function, quantile function, characteristic function
$\mathbb E$, $\operatorname{Var}$, $\operatorname{Cov}$	expectation, variance, covariance
$M_n$, $\mathcal F_n$	martingale and its filtration
$B_t$	Brownian motion
$T_t$, $L$	transition semigroup and its generator