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The Birkhoff ergodic theorem states: for a measure-preserving transformation $\theta$ of a finite measure space and $f\in L^1$, the averages $A_{n}f=n^{-1}{\sum }_{j=0}^{n-1}f\circ {\theta }^j$ converge almost everywhere and in $L^1$ to an invariant function $f^{\ast }$. On a probability space,where $\mathcal{I}$ is the invariant sigma-algebra modulo null sets. In particular the integral is preserved, and if $\theta$ is ergodic then $f^{\ast }=\int fd\mu$ almost everywhere; for a finite space of nonunit mass, this constant is $\mu (E{)}^{-1}\int fd\mu$ when $\mu (E)>0$.

The maximal ergodic lemma says that for real integrable $f$, with $S_{k}f={\sum }_{j=0}^{k-1}f\circ {\theta }^j$ and $E_N=\{{\max }_{1\le k\le N}{S}_{k}f>0\}$,The same inequality holds on $E_{\infty }=\{{\sup }_{k\ge 1}{S}_{k}f>0\}$, by taking the increasing limit and dominated convergence. These statements do not require the transformation to be invertible or the space to be ergodic.

A measure-preserving transformation $T$ on a probability space $(X,\mathcal{A},m)$ is ergodic if every invariant event $A\in \mathcal{A}$, meaningup to an $m$-null set, satisfies $m(A)\in \{0,1\}$. Thus the invariant sigma-algebra is trivial modulo null sets.

On the probability space $([0,1],\mathcal{B},\lambda )$, letFor every $x\in [0,1]$, eventually $x\notin (0,1/n)$, so $f_n(x)\to 0$. Nevertheless,for every $n$. ConsequentlyThis gives strict inequality in Fatou lemma.

The Dambis-Dubins-Schwarz theorem states that if $M$ is a continuous local martingale with $M_0=0$ and $[M{]}_{\infty }=\infty$, then, forthe process $W_s=M_{{\tau }_s}$ is a standard Brownian motion and $M_t=W_{[M{]}_t}$. If $[M{]}_{\infty }<\infty$, one obtains the same representation after enlarging the probability space and continuing $W$ independently beyond $[M{]}_{\infty }$.

A discrete random variable on a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ is a real-valued measurable function $X:\Omega \to \mathbb{R}$ whose probability distribution is concentrated on some finite or countable set $S$:Measurability means $\{\omega :X(\omega )\le t\}\in \mathcal{F}$ for every real $t$. Its probability mass function is $p_X(x)=\mathbb{P}(X=x)$, and ${\sum }_{x\in S}{p}_X(x)=1$. Requiring a countable image everywhere, rather than outside a null set, is an equivalent convention after harmless modification on a null set.

Use the ordered outcomes of the two rolls as the sample space:Thus each elementary outcome has probability $1/36$. This is the usual probability space for two independent fair dice.

An idealized underlying probability space iswith the product Borel sigma-algebra. Its 27 independent random coordinates are $\mu$, $\sigma$, twenty past standard-normal noises, and five future standard-normal noises. Define $T_i(\omega )=\mu +\sigma {\epsilon }_i$. Sampling an unknown variance does not mean drawing a new variance on each day; a single draw of ${\sigma }^2$ applies to all durations in a trace.

Let $A$ be the event that the first twenty durations satisfy their corresponding observed inequalities. The observations are the twenty late/early labels, conditional on the fixed departure thresholds. They are not exact values of the latent $T_i$. The posterior probability measure on traces isHere $\mathbb{P}(A)>0$: at every interior parameter pair each strict normal-tail probability lies in $(0,1)$, so their finite product has positive prior integral.

The returned outcome isThus the underlying sample space has dimension 27, the successful output space is $\{0,1\}$, and the observations define the conditioning event $A$. The desired output probability distribution is the pushforward measure of ${\mathbb{P}}_A$ under $Y$. Rejection is an implementation device, not a third scientific answer: None is discarded before forming the conditional output law.

Convergence in distribution means that the cumulative distribution functions satisfyat every $x$ where the cumulative distribution function of $X$ is continuous. Values at its discontinuities need not converge. This definition uses only the probability distributions, so the random variables need not be defined on the same probability space.

The cumulative distribution function of the uniform distribution on $[0,1/n]$ isThe constant zero has cumulative distribution function $F(t)=1_{[0,\infty )}(t)$. Its continuity points are precisely $t\ne 0$. For fixed $t<0$, $F_n(t)=0=F(t)$; for fixed $t>0$, $F_n(t)=1=F(t)$ eventually. This proves convergence in distribution to zero directly from the definition.

Also, $\mathbb{P}(0\le U_n\le 1/n)=1$. For every $\epsilon >0$ and $n>1/\epsilon$, $\mathbb{P}(|U_n|>\epsilon )=0$. HenceThe convergence in probability conclusion holds for any realization of these random variables on a common probability space; no independence of random variables is needed.

First take integers $1\le k<n$. For independent copies $X_1,\dots ,X_n$ of $X$, defineThe two blocks involve disjoint families of independent random variables, so $A$ and $B$ are independent. The assumed strictly stable distribution gives $A\stackrel{d}{=}X$, $B\stackrel{d}{=}X$, andTherefore, on any fixed probability space carrying $X$ and an independent copy $X^{\prime }$, their joint probability distribution givesNow fix an arbitrary $0<c<1$ and choose rationals $r_m=k_m/n_m\in (0,1)$ tending to $c^{\alpha }$. Continuity of the real powers gives, on that fixed probability space,The left sides all have the probability distribution of $X$. Since almost sure convergence implies convergence in distribution, their limit has that same probability distribution. HenceThis proves strict stability implies self-decomposability. The limit argument uses no moment-generating function or finite moments, which is essential for general stable distributions.

Using the Cauchy distribution from the preceding part gives one common probability space for all starting points:As $(x,y)\to (0,0)$ with $y>0$, $x+yC\to 0$ almost surely, since $C$ is a finite real random variable almost surely. Continuity of $f$ at zero gives $f(x+yC)\to f(0)$ almost surely, and boundedness supplies an integrable constant bound. Dominated convergence theorem therefore provesThe argument applies to every sequence approaching zero, including tangential approaches. Along the real boundary the extension is already $f$, which is continuous. Hence the extension is continuous at zero in the closed half-plane. No finite moment of the Cauchy distribution is required.

The probability measure $\mathbb{P}$ maps the sigma-algebra $\mathcal{F}$ of events into $[0,1]$. Its probability axioms are nonnegativity, normalization $\mathbb{P}(\Omega )=1$, and countable additivity: for every pairwise disjoint sequence of events $A_1,A_2,\dots$,The sample space is $\Omega$, and $\mathcal{F}$ specifies which subsets are events. Requiring countable additivity, rather than only its finite version, is part of the probability space definition.

Take an independent and identically distributed sequence with $X_i\sim \mathrm{Bernoulli}(r)$, independent of $N$. Conditional on $N=n$, the random sum ${\sum }_{i=1}^n{X}_i$ has binomial distribution with parameters $n,r$, including the empty sum at $n=0$. This construction therefore has exactly the prescribed joint law of $(N,Z)$ and represents $Z$ as the required random sum.

For a literal representation that retains an already given pair $(N,Z)$, enlarge the probability space by auxiliary randomness. Conditional on $N=n,Z=z$, choose uniformly a size-$z$ subset of $\{1,\dots ,n\}$, put ones at those indices and zeros at the others, and append independent Bernoulli variables of parameter $r$. Given $N=n$, a specified length-$n$ pattern with $z$ ones has probability $\left(\genfrac{}{}{0ex}{}{n}{z}\right)r^z(1-r{)}^{n-z}/(\genfrac{}{}{0ex}{}{n}{z})=r^z(1-r{)}^{n-z}$. Thus the entire sequence is independent and identically distributed and independent of $N$, while its first $N$ terms sum to the original $Z$.

A point process records random locations by a random measure that counts them. Examples include arrival times of a queue, positions of stars, and the finite eigenvalue configuration of a random matrix. The count in an observation window is a random variable; its expected value is supplied by the intensity measure of a point process, while its fluctuations and relationships with other windows require more information.

For a precise spatial convention, take $E={\mathbb{R}}^d$ with its Borel sigma-algebra and a probability space $(\Omega ,\mathcal{F},\mathbb{P})$. A locally finite point process $\Pi$ is a measurable function from $\Omega$ to the space ${\mathcal{N}}_{\mathrm{lf}}(E)$ of locally finite point measures, equipped with the sigma-algebra generated by evaluations $\nu \mapsto \nu (A)$ for Borel sets $A$. Each realization has the formwhere $J(\omega )$ is finite or countable and repeated locations are allowed; ${\delta }_x$ is a Dirac measure. Here a point measure is a counting measure with possible multiplicities, rather than the counting measure of a set of distinct locations.^[ref]

Local finiteness means ${\Pi }_{\omega }(B)<\infty$ for every bounded Borel set $B$, on a single almost sure event. In ${\mathbb{R}}^d$ it suffices to check the countable collection of balls of integer radius centered at the origin, and this is equivalent to finiteness on every compact set. On a general measurable space there is no notion of boundedness without additional structure: state a metric or an explicit family of observation windows. Another convention on locally compact spaces requires finiteness on compact sets; this need not coincide with finiteness on all bounded sets outside ${\mathbb{R}}^d$. Local finiteness does not require finitely many points in all of $E$.^[ref]

Let $Y$ be the discounted gains of finitely many financial assets on any probability space. Tilting the physical probability measure by a normalized multiple of $(1+\Vert Y\Vert {)}^{-1}$ makes $Y$ integrable without changing null events. The positive-weight expectation cone alternative then says either a bounded strictly positive weight makes the weighted gain expectation zero, or a deterministic risky holding gives an arbitrage. Normalizing the weight yields an equivalent martingale measure. Conversely, under such a measure a nonnegative terminal gain with zero initial price has zero expectation, and must vanish almost surely. This proves the one-period fundamental theorem of asset pricing without assuming physical integrability of asset gains.

On a probability space, the same inequality readsIf the displayed moments are finite, the product is an integrable random variable. The random variables can be dependent: the estimate controls their possible overlap without assuming a factorization of the expected value. This is useful when a calculation needs the product to be integrable but the joint probability distribution is complicated.

A mathematical account of random outcomes using a probability space and specified random variables. It determines probabilities for events or observed data. A statistical model commonly supplies a family of such probability distributions with unknown parameters; choosing a model and estimating its parameters are distinct tasks.

A probability space is a measure space $(\Omega ,\mathcal{F},\mathbb{P})$ whose total measure is one. The sample space $\Omega$ contains the elementary outcomes; the sigma-algebra $\mathcal{F}$ contains the events to which the model assigns probabilities; the probability measure $\mathbb{P}$ supplies those values. The sigma-algebra lets us combine observable questions using complements and countable unions while remaining within the model.

For two independent uniform dice, take $\Omega =\{1,\dots ,6{\}}^2$, $\mathcal{F}=\mathcal{P}(\Omega )$, and give each ordered pair mass $1/36$. A random variable might record the sum, the larger result, or whether both results are even integers. These are different functions on the same sample space, rather than different elementary experiments. A probability distribution records the mass of the values of one such random variable and can forget information retained by the full probability space.

Probability theory describes uncertainty through a probability space: possible outcomes form a sample space, observable questions are events, and a probability measure assigns them numbers between zero and one. The Kolmogorov axioms make those assignments consistent. They let us combine information about several events, update it through conditional probability, and study random variables without listing every possible outcome separately.

For example, roll two dice whose results are independent random variables, each with a discrete uniform distribution on $\{1,\dots ,6\}$. There are 36 equally likely ordered outcomes. Only $(1,6),(2,5),\dots ,(6,1)$ have sum seven, so that event has probability $6/36=1/6$. The sum itself is not uniformly distributed: sum two has only one outcome. Choosing a probability distribution for the elementary outcomes is part of the model; equal likelihood does not follow merely from counting the named possibilities.

The subject becomes especially useful when complete outcome lists are enormous or infinite. Independence turns a joint calculation into simpler pieces, expected value summarizes a random variable, and the law of large numbers explains how averages can become predictable even when individual observations are uncertain. The central limit theorem describes the scale and shape of many remaining fluctuations. These are distinct claims: a small variance of an average does not make every observation close to its expected value.

There is a satisfying unity here: dice, hidden causes, and Brownian motion all use the same rules for events and conditional probability. A model still needs justification. Its exact deductions describe what its assumptions imply; comparison with observations belongs to statistical inference.

For finitely many probability spaces $(X_i,{\mathcal{F}}_i,{\mu }_i)$, their product probability space has underlying set ${\prod }_i{X}_i$, the product sigma-algebra and product measure ${⨂}_i{\mu }_i$. The coordinate projections are jointly independent: for measurable $B_i\subseteq X_i$, the probability of all coordinate conditions is ${\prod }_i{\mu }_i(B_i)$. Conversely a family of independent coordinates has this product joint distribution. Product structure permits coordinate exposure and independent resampling in concentration arguments.