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For independent standard Gaussian random variables, the partial sums have characteristic functions $\exp (-u^2{v}_n/2)$, where $v_n={\sum }_{k\le n}{a}_k^2$. A finite almost-sure limit implies convergence of these characteristic functions to one continuous at zero; $v_n\to \infty$ would instead give zero at every nonzero $u$. Conversely, bounded $v_n$ makes the partial sums an martingale bounded in L2 space, which converges almost surely and in L2 space.

A chi-bar-squared distribution is a mixture distribution ${\sum }_{j=0}^m{w}_j{\chi }_j^2$, where $w_j\ge 0$ and ${\sum }_j{w}_j=1$, and ${\chi }_0^2$ denotes a point mass at zero. It arises when a Gaussian likelihood is optimized under inequality constraints: projection onto a constraint cone selects different dimensions according to which inequalities are active. For example, if $W,Z$ are independent standard Gaussian random variables, then $Z^2+(-W{)}_{+}^2$ has distribution $\frac{1}{2}{\chi }_1^2+\frac{1}{2}{\chi }_2^2$. On $W\ge 0$ it is $Z^2$; on $W<0$ it is $Z^2+W^2$. Symmetry makes the sign of $W$ independent of $W^2$, proving the stated weights and component laws.

A complex Wishart matrix is the Gram matrix of an $n$-by-$m$ matrix of independent centered complex Gaussian random variables with an indicated variance convention. It is a positive semidefinite matrix. In the standard convention with joint entry density proportional to $e^{-\mathrm{Tr}{X}^{\ast }X}$ and $n\ge m$, its eigenvalue density contains the weight $x^{n-m}{e}^{-x}$ and a squared Vandermonde determinant.

Continuous-time Gaussian white noise is a random distribution rather than an ordinary finite-variance function at every time. Integrating it against deterministic kernels gives Gaussian random variables with covariance $q\int f(s)g(s)ds$. For independent isotropic components, a three-dimensional dot covariance has amplitude $3q$. An ordinary rapidly decaying force covariance approximates this model only above its correlation time.

If a linear functional $Z=\int a(\mathbf{r})V(\mathbf{r})d\mathbf{r}$ of a real Gaussian random field exists in mean square, it is a Gaussian random variable. Completing the square in its density givesFor a complex linear functional $\varphi$, apply this to $Z=2\mathrm{Re}\varphi$; then $\frac{1}{2}\mathrm{Var}Z=\mathbb{E}|\varphi -\mathbb{E}\varphi {|}^2+\mathrm{Re}\mathbb{E}(\varphi -\mathbb{E}\varphi {)}^2$.

For $Z$ with a standard normal distribution and $a\ge 0$, split the Gaussian integral at zero and complete the square to obtain $\mathbb{E}{e}^{a|Z|}=2e^{a^2/2}\Phi (a)\le 2e^{a^2/2}$, where $\Phi$ is the standard normal distribution function. This converts a bound on an absolute Gaussian random variable into a two-sided Gaussian tail bound using Markov inequality.

For a centered Gaussian random variable in a Banach space, its first Gaussian chaos is the closed linear subspace of L2 space generated by its continuous linear observations. Every element is a centered Gaussian random variable, and every finite collection has a multivariate normal distribution: approximate in L2 space and pass to the characteristic function. The first Gaussian chaos supplies the scalar coordinates of its Cameron-Martin space of a Gaussian random variable in a Banach space.

Let $A_i$ be independent random variables with normal distribution $N(\mu ,{\sigma }^2)$, ${\sigma }^2>0$, and let $B$ be an independent exponential random variable of rate $\lambda >0$. At speed $L$, $(B+{\sum }_{i=1}^L{A}_i)/L$ has the good rate functionThe product large-deviation principle gives the joint cost $\lambda b+(z-\mu {)}^2/(2{\sigma }^2)$ for $b\ge 0$, and the contraction principle for large deviations gives its infimal convolution under $b+z=x$. The minimum occurs at $b=(x-\mu -\lambda {\sigma }^2{)}^{+}$. The fixed exponential perturbation changes the far upper tail even though it vanishes in probability after division by $L$; a fixed Gaussian random variable divided by $L$ instead has exponential equivalence with zero at speed $L$.

Let $X_0=0$ and let $X$ be a centered Gaussian process whose increments are independent of the past filtration, with deterministic variance $v(t)$. Its displayed exponential is a positive martingale. Indeed, $X_t-X_s$ is a centered Gaussian random variable with variance $v(t)-v(s)$, so the Gaussian moment-generating function givesThe sign-reversed process has the same variance and the same conclusion. Continuity of paths is unnecessary for this conditional-expectation argument.

Let $X$ be a standard Gaussian random variable and $S_j$ independent symmetric signs, independent of $X$. The stationary sequence $W_j=S_{j}X$ has standard Gaussian marginals and identity covariance, but is not jointly Gaussian. For two coordinates its moment-generating function is $(1+e^{2t^2})/2$, rather than $e^{t^2}$. Consequently a field's Gaussian one-point densities and two-point covariance alone do not justify an exponential moment of a Gaussian linear functional.

For a centered real Gaussian random variable $\varphi$, its characteristic function gives $\mathbb{E}{e}^{i\varphi }=\exp [-\mathrm{Var}(\varphi )/2]$. Applied to a random phase screen, this controls the coherent and diffuse wave fields.

A Gaussian random variable is a random variable whose probability distribution is a normal distribution. For $\sigma >0$, write $X=\mu +\sigma Z$ with a standard normal random variable $Z$; conversely, standardization of a normal random variable recovers $Z=(X-\mu )/\sigma$. For example, $X\sim N(100,15^2)$ has mean $100$, standard deviation $15$, and variance $225$. The outcome $x=115$ corresponds to standard score $z=1$, one standard deviation above the mean.

Some definitions also allow a Gaussian random variable with $\sigma =0$. This means $X=\mu$ almost surely, with variance zero and probability distribution equal to the Dirac measure ${\delta }_{\mu }$. It has no probability density function with respect to Lebesgue measure: $\mathbb{P}(X=\mu )=1$. The positive-scale density formula and division by $\sigma$ apply only when $\sigma >0$. MIT's constructive definition explicitly includes the constant case. ^[ref].

For centered, unit-variance jointly Gaussian random variables with correlation coefficient $\rho$, the displayed identity holds. Realize the pair as projections onto unit vectors of angle $\theta =\arccos \rho$. Rotational symmetry of the two-dimensional Gaussian random vector direction makes sign disagreement have probability $\theta /\pi$. The expectation is therefore $1-2\theta /\pi =(2/\pi )\arcsin \rho$. Perfectly correlated and anticorrelated endpoints follow directly; the sign at zero does not affect the result.

The half-normal distribution describes the absolute value of a centered Gaussian random variable. Its unit-scale version is $H=|Z|$ for a standard normal random variable $Z$. It folds the negative half of the standard normal distribution onto the positive half, retaining the size of a deviation while discarding its sign. If a centered Gaussian random variable models measurement error, its magnitude therefore has a half-normal distribution.Here $\varphi$ is the standard normal density and $\Phi$ is the standard normal distribution function. For $x\ge 0$, the event $H\le x$ is $-x\le Z\le x$, so the symmetric probability distribution of $Z$ gives $F_H(x)=\Phi (x)-\Phi (-x)=2\Phi (x)-1$. The upper-tail probability is therefore $\mathbb{P}(H>x)=2(1-\Phi (x))$. For example, $\mathbb{P}(H\le 1)\approx 0.6826894921$. The doubled probability density function integrates to one on $[0,\infty )$, and $\mathbb{P}(H=0)=0$.

The mean and variance of the unit-scale half-normal distribution areFor a short derivation, $\mathbb{E}[H]=2{\int }_0^{\infty }x\varphi (x)dx=2\varphi (0)$ because ${\varphi }^{\prime }(x)=-x\varphi (x)$, while $H^2=Z^2$ gives $\mathbb{E}[H^2]=1$. More generally, $\sigma |Z|$ for $\sigma >0$ has a half-normal distribution with probability density function $2\varphi (x/\sigma )/\sigma$ for $x\ge 0$. Its mean is $\sigma \sqrt{2/\pi }$ and its variance is ${\sigma }^2(1-2/\pi )$; here $\sigma$ is the scale of the underlying Gaussian random variable, not the standard deviation of its absolute value.

For Brownian motion started at $x$, the time integral is a Gaussian random variable withThe variance follows by integrating the covariance kernel $\mathrm{cov}(B_r,B_s)=\min (r,s)$ twice.

For centered jointly Gaussian random variables, an odd product has zero expectation, and an even product's expectation is the sum over all pairings of the products of the pair covariances. In particular,For a proof, the moment-generating function is $\exp (t^T\Sigma t/2)$. Differentiating once in each variable at zero leaves precisely products of covariance entries indexed by complete pairings. No independence assumption between the coordinates is needed.

An ordinary time integral of random variables taken in the L2 space of the underlying probability space. Under strong measurability and ${\int }_a^b(\mathbb{E}|X_t{|}^2{)}^{1/2}dt<\infty$, it exists as a Bochner integral; for a mean-square continuous process it is also the limit of Riemann sums in that space. If the process is centered and the covariance is integrable, Fubini's theorem givesFor a Gaussian process, the integral is a Gaussian random variable, by approximating it in mean square by finite linear combinations of jointly Gaussian random variables.