Probability and statistics

For probability densities $p$ and $q$, the Kullback-Leibler divergence from $q$ to $p$ is

For probability distributions $p$ and $q$ on the same finite set,with equality exactly when $p=q$ on the support of $p$.

A probability space is a measure space whose total measure is one. Its measurable subsets are events, and its measure $\mathbb{P}$ is their probability.

An event is a measurable subset of a probability space.

For a point uniformly distributed in the unit disk, its polar coordinates are independent with densities

For three independent points from any rotationally symmetric planar distribution with nonzero radii, the origin lies in their triangle exactly when their three angles do not all lie in one semicircle. Three independent uniform angles have this property with probability $1/4$.

Two sub-sigma-algebras ${\mathcal{F}}_1,{\mathcal{F}}_2$ are independent whenfor every $A\in {\mathcal{F}}_1$ and $B\in {\mathcal{F}}_2$.

Events $A_1,\dots ,A_n$ are mutually independent whenfor every finite index set $J$.

If two pi-systems satisfy the probability factorization pairwise, their generated sigma-algebras are independent. For one event in the first family, the events in the second sigma-algebra satisfying the factorization form a Dynkin system; apply Dynkin lemma, then repeat with the roles reversed.

For a sequence of random variables $(X_n)$, the tail sigma-algebra isIts events are unaffected by changing any finite initial segment.

The tail sigma-algebra of a sequence of independent random variables is trivial: every tail event has probability zero or one.

The product measure ${⨂}_n{\mu }_n$ is characterized on measurable rectangles depending on finitely many coordinates by the product of their component measures.

A random variable is a measurable function from a probability space to a measurable space.

Random variables $X_1,\dots ,X_n$ are independent when their generated sigma-algebras are independent sigma-algebras. Equivalently, their joint distribution is the product of their marginal distributions.

A family of independent random variables is independent and identically distributed when every variable has the same probability distribution.

The Poisson arrivals see time averages property says that an external Poisson process independent of a stationary system sees the system-state distribution at its arrival times. Thus the fraction of arrivals that accept a state-dependent admission rule $p(n)$ is ${\sum }_n{\pi }_{n}p(n)$.

An $M/G/1$ queue has Poisson arrivals, independent identically distributed service times, and one server. If service has mean $m$ and $\lambda m<1$, its mean busy period isThis follows by decomposing the busy period into the first service and the busy-period descendants of arrivals during that service.

In a stable regenerative queue,where $L$ is mean population, $\lambda$ is throughput, and $W$ is mean sojourn time. Renewal reward proves it by identifying the area under the population path with the sum of customer sojourn times.

The main modes include almost-sure convergence, convergence in probability, convergence in $L^p$, and convergence in distribution.

Random variables $X_n$ converge in distribution, or converge weakly, to $X$ whenfor every bounded continuous function $f$. For real random variables this is equivalent to convergence of the distribution functions at every continuity point of the limiting distribution function.

A sequence of random variables converges almost surely when the set of outcomes on which pointwise convergence fails has probability zero.

If $(X_n)$ are independent and identically distributed random variables with $\mathbb{E}|X_1|<\infty$, thenalmost surely.

Random vectors $X_n$ converge weakly to $X$ exactly when every linear projection $u\cdot X_n$ converges weakly to $u\cdot X$. The reverse implication follows because these one-dimensional limits give pointwise convergence of the multivariate characteristic functions.

Random variables $X_n$ converge in probability to $X$ whenfor every $\epsilon >0$.

Convergence in probability is equivalent toThe forward implication splits at a fixed error threshold; the reverse implication is Markov's inequality.

From $X_n\to X$ in probability, choose $n_k$ with
$\mathbb{P}(|X_{n_k}-X|>2^{-k})<2^{-k}$. The first Borel--Cantelli lemma then gives $X_{n_k}\to X$ almost surely.

Independent indicators with success probabilities $1/n$ converge to zero in probability, but successes occur infinitely often almost surely by the second Borel--Cantelli lemma.

If $X_n\to X$ in probability and ${\sup }_n\Vert X_n{\Vert }_r<\infty$, then $X_n\to X$ in $L^p$ for every $1\le p<r$. The $L^r$ bound gives uniform integrability of the $p$th powers, while convergence in probability controls their bounded part.

For independent identically distributed $Z_i$ with mean zero and variance one,The numerator obeys the central limit theorem, the denominator converges almost surely to one by the strong law, and Slutsky's theorem combines them.

For independent identically distributed random vectors $Z_i$ with mean $\mu$ and finite covariance matrix $V$,

Binomial distributions with trial count tending to infinity and success probability tending to zero converge to a Poisson law when their product converges.

A simple symmetric random walk starting from $S_0$ haswhere the increments ${\xi }_k$ are independent and satisfy

An integrable adapted process $(M_n)$ is a martingale when $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$.

If every increment ${\xi }_n=M_n-M_{n-1}$ of a discrete martingale takes values in $\{-1,1\}$, thenIterated conditioning shows that the increments are independent symmetric signs, so $M_n-M_0$ is a simple symmetric random walk.

An integrable adapted process $(V_n)$ is a supermartingale when $\mathbb{E}[V_{n+1}\mid {\mathcal{F}}_n]\le V_n$.

If a supermartingale $V$ dominates an adapted reward process $Z$, optional sampling gives $\mathbb{E}{Z}_{\tau }\le V_0$ for every bounded stopping time $\tau$.

A random time $\tau$ is a stopping time when $\{\tau \le n\}\in {\mathcal{F}}_n$ for every $n$.

For a discrete martingale and stopping time $T$,The indicator is measurable at time $n$, so the stopped process is a martingale. If it is uniformly bounded and $T<\infty$ almost surely, bounded convergence gives $\mathbb{E}{M}_T=\mathbb{E}{M}_0$ directly.

For simple symmetric random walk stopped on hitting $-a$ or $b$ and $0<z<1$, choose $w>1$ byThen $z^n{w}^{\pm X_n}$ are martingales. Solving the two boundary equations gives

For bounded stopping times $\sigma \le \tau$, a supermartingale satisfies $\mathbb{E}[V_{\tau }\mid {\mathcal{F}}_{\sigma }]\le V_{\sigma }$; equality holds for a martingale.

In discrete time, a process $A_n$ is predictable or previsible when $A_n$ is ${\mathcal{F}}_{n-1}$-measurable for every $n\ge 1$.

If $M$ is a martingale and $A$ is predictable, then, subject to integrability,is a martingale.

For a square-integrable martingale transform and positive conditional increment variance,

For a stopping time $T$,The indicator is predictable, so a stopped integrable martingale is a martingale.

Let $S_0=0$ be a simple symmetric random walk, let $T_a$ be its first hitting time of the positive integer $a$, and reverse every increment after $T_a$. The resulting path is again a simple symmetric random walk. Consequently,