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By the union bound,For an $L^2$ random variable, $t^2\mathbb{P}(|X_1|>t)\to 0$. Taking $t=\epsilon \sqrt{n}$ makes the right side tend to zero, proving $M_n/\sqrt{n}\to 0$ in probability.

For a nonnegative random variable $Z$ and $a>0$, the pointwise inequality $Z\ge a{1}_{\{Z\ge a\}}$ gives

The number of stops is $X\sim \mathrm{Bin}(n,1/n)$. For each fixed $k\ge 0$,the mass function of a Poisson random variable of mean one.

The finite sigma-algebra ${\mathcal{F}}_n=\sigma (A_1,\dots ,A_n)$ is partitioned by the nonempty eventsThese are its atoms. Define the random variableand give it any finite value on the union of the null atoms. It is ${\mathcal{F}}_n$-measurable and integrable. Every $A\in {\mathcal{F}}_n$ is a union of atoms, soThis constructs the requested variable directly, without invoking the general existence theorem for conditional expectation.

For an integrable real random variable $X$, its cumulant-generating function is the extended-real convex functionwhere $\psi (\theta )=+\infty$ if the exponential moment diverges. Its Legendre transform of a cumulant-generating function is

For a centered random variable $X$, being sub-Poisson in the right tail with variance parameter ${\sigma }^2$ meansIt is sub-Gamma in the right tail with variance parameter $v$ and scale parameter $c$ when

A centered random variable $Y$ is sub-Gaussian with variance parameter ${\sigma }^2$ whenThe Chernoff bound, applied to $Y$ and $-Y$, yieldsThe tail integral formula for moments and the substitution $u=t^2/(2{\sigma }^2)$ now give

For a nonnegative integer-valued random variable $X$, its probability generating function isThe generating function of a sum of independent random variables is the product of their generating functions.

For a discrete random variable $X$, its probability mass function isIt is nonnegative and sums to one over the possible values of $X$.

For a centered complex random variable $Z=A+iB$, the pseudo-covariance is $P=\mathbb{E}{Z}^2$. Together with $V=\mathbb{E}|Z{|}^2$ it determines the real covariance matrix: $\mathbb{E}{A}^2=(V+\mathrm{Re}P)/2$, $\mathbb{E}{B}^2=(V-\mathrm{Re}P)/2$, and $\mathbb{E}AB=\mathrm{Im}P/2$.

A random matrix is a matrix whose entries are random variables. Its eigenvalues form a random configuration, and its empirical spectral measure records their distribution with equal mass on each eigenvalue. Entry assumptions and normalization determine the spectral regime.

A random measure assigns a measure ${\mu }_{\omega }$ to each outcome so that $\omega \mapsto {\mu }_{\omega }(A)$ is a random variable for every measurable $A$. A finite-variation process gives pathwise signed random measures through Lebesgue–Stieltjes integration.

A random vector is a random variable whose values are vectors or, equivalently, an ordered tuple of scalar random variables on one probability space.

For positive mixture weights $w_j$ summing to one, set $C_j={\sum }_{i\le j}{w}_i$ and choose $J=j$ when $C_{j-1}\le U<C_j$, where $U$ has a uniform distribution. Then $V=(U-C_{J-1})/w_J$ has a uniform distribution independently of $J$: for $0\le v\le 1$, $\mathrm{Pr}(J=j,V\le v)=w_{j}v$. This permits exact finite mixture model sampling with a recycled continuous random variable.

A reparameterization gradient differentiates an expected value by representing the simulated random variable as a differentiable transformation of parameter-independent noise. For $X_{\eta }=g_{\eta }(\epsilon )$, the identity ${\nabla }_{\eta }\mathbb{E}[F_{\eta }(X_{\eta })]=\mathbb{E}[{\nabla }_{\eta }{F}_{\eta }(g_{\eta }(\epsilon ))]$ requires justified differentiation under the integral sign. For a normal distribution, use $X=m+e^{\rho }\epsilon$ with $\epsilon$ drawn from the standard normal distribution.

The second moment of a real random variable is $\mathbb{E}[X^2]$. Its variance is $\mathbb{E}[X^2]-(\mathbb{E}X{)}^2$ when these quantities are finite.

A shape parameter changes the form of a probability distribution without merely translating it or rescaling its random variable.

A centered random variable $X$ is sub-Gamma in the right tail with variance parameter $v$ and scale parameter $c$ whenfor $0\le \lambda <c^{-1}$.

A centered random variable $X$ is sub-Poisson in the right tail with variance parameter ${\sigma }^2$ whenfor every $\lambda \ge 0$.

For $q>0$ and a random variable $X$ with finite absolute $q$th moment, Tonelli theorem gives