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For a random walk with independent and identically distributed increments and an exit time from $(0,r)$, suppose $p=\mathbb{P}(X_1>r/m)>0$ for some integer $m\ge 1$. A block of $m$ such positive increments forces an exit. Conditional on survival, every successive block has success probability at least $p^m$, so $\mathbb{P}(T>km)\le (1-p^m{)}^k$ and $\mathbb{E}T\le m/p^m$. This is an instance of the geometric tail bound from a uniform escape probability; centering of the increments is not required.

For the geometric-jump countdown Markov chain, regeneration at zero givesA cycle visits positive state $j$ precisely when its jump reaches at least $j$, with probability $q^{j-1}$. Dividing this expected occupation by the mean cycle length proves the formula. Mean recurrence times are $1/{\pi }_j$, and the limiting probability away from zero is $1/(1+p)$.

For an aperiodic positive Harris recurrent Markov chain with geometric ergodicity, an observable with stationary moment $\pi (|h{|}^{2+\delta })<\infty$ for some $\delta >0$ satisfies a central limit theorem. Its asymptotic variance is ${\gamma }_h(0)+2{\sum }_{k\ge 1}{\gamma }_h(k)$, with stationary covariances. Ergodicity alone does not guarantee this theorem.

The orbital variation of a planet's reflected stellar light as the visible illuminated area changes. In a simple spherical model, $F_p/F_{\ast }=A_g(R_p/a{)}^2\Phi (\alpha )$, where $\Phi (0)=1$ and $\alpha$ is the star-planet-observer angle. Exoplanet clouds can alter the geometric albedo and the angular scattering law; this observable differs from an exoplanet thermal phase curve.

The right lifting property for each sphere-to-disk map fills a prescribed cycle and a compatible target element. The degree-zero generator supplies initial surjectivity; all other fillers force an acyclic kernel and the remaining degreewise surjections.

A Gaussian linear mixed model combines fixed effects, random effects with a normal distribution, and independent Gaussian noise. For $b\sim N(0,D)$ and $\epsilon \sim N(0,R)$ independently, its marginal distribution is $Y\sim N(X\beta ,V)$ with $V=ZD{Z}^T+R$. The random effects induce dependence among observations sharing their design columns. A random-intercept linear mixed model and a random-slope linear mixed model are special cases.

The backfitting algorithm estimates each additive mean function by smoothing its partial residual, obtained after subtracting the current fits of all other terms, and cycling through terms until convergence. Centering each smooth separates its constant part from the intercept. For fixed quadratic smoothness penalties it is block coordinate minimization of a penalized least-squares objective; identifiability and positive definiteness of the constrained problem ensure a unique converged fit. Non-Gaussian generalized additive models can use weighted backfitting inside iteratively reweighted least squares.

For a nonzero finitely generated Weyl algebra module, choose a finite-dimensional generating subspace $M_0$. Under the Bernstein filtration, $\dim (F_j{A}_n{M}_0)$ is eventually polynomial because the graded module is finite over a polynomial ring. Its degree is independent of the generating subspace, since two generating filtrations bound each other after fixed shifts. It equals the Gelfand–Kirillov dimension of a module.

Observe $Y(t)={\int }_0^{t}f(s)ds+\sigma W(t)$ on $[0,1]$, where $f\in L^2[0,1]$ is an unknown deterministic drift and $W$ is standard Brownian motion. The observation against a deterministic test function is $Y(h)=⟨f,h⟩+\sigma W(h)$, with covariance ${\sigma }^2⟨h,g⟩$. Gaussian white noise is represented by an isonormal Gaussian process on test functions rather than a pointwise noise function. Taking $\sigma =n^{-1/2}$ is the usual statistical scaling.

Let $C=\sqrt{|D_K|}N(\mathfrak{b})$ and $n=[K:\mathbb{Q}]$. Normalize the Minkowski embedding of a number field to the covolume-one Euclidean lattice ${\Lambda }_{\mathfrak{b}}=C^{-1/n}j(\mathfrak{b})$, using the metric with complex coordinates weighted by $2$. Its theta function isWriting $\Vert y\Vert ={\prod }_v{y}_v^{[K_v:\mathbb{R}]}$ and using the trace dual of a fractional ideal gives $\Theta (y,\mathfrak{b})=\Vert y{\Vert }^{-1/2}\Theta (y^{-1},{\mathfrak{b}}^{\vee })$. Its small-parameter asymptotic of a lattice theta sum has leading coefficient one. For the unscaled ideal lattice, the leading coefficient is instead $C^{-1}$; one must specify the normalization when quoting this limit.

If every eigenvalue of a positive definite symmetric matrix $A$ tends to zero, then the anisotropic theta functional equation and the dominated convergence theorem show that $(\det A{)}^{1/2}{\theta }_{\Lambda }(A)\to \mathrm{covol}(\Lambda {)}^{-1}$. Indeed all eigenvalues of $A^{-1}$ tend to infinity, the zero term of the dual Gaussian theta sum is one, and its remaining terms tend to zero while being dominated by a fixed summable Gaussian. Merely requiring $\det A\to 0$ is insufficient.

An independent unit-scale Rayleigh distribution variable $R$ and standard normal distribution variable $Z$ give $RZ$ with Laplace distribution of density $e^{-|x|}/2$. Indeed its characteristic function is $\mathbb{E}{e}^{-t^2{R}^2/2}=1/(1+t^2)$.

A standard normal random variable has the standard normal distribution, mean zero and variance one. If $X\sim N(\mu ,{\sigma }^2)$ with $\sigma >0$, the standardization of a normal random variable $(X-\mu )/\sigma$ is a standard normal random variable. Its values report displacement from the mean in units of the standard deviation, so the result is a dimensionless quantity.

If $G$ is a centered Gaussian random field, the transformation $G+\alpha (G^2-⟨G^2⟩)$ retains zero mean and introduces a three-point connected correlation function at first order in $\alpha$. Wick contractions give $2\alpha$ times the sum of products of two covariances. The corresponding local cosmological model uses $\alpha =3f_{\mathrm{NL}}/5$.

For a stable massive Gaussian field theory, a finite blocking step yields $F(t,h)=F_{\mathrm{shell}}(t)+b^{-d}F(b^{2}t,b^{(d+2)/2}h)$ for free-energy density. The finite-step shell and measure terms are analytic backgrounds. Generic-dimension singular scaling is homogeneous after subtracting those backgrounds; resonances such as Gaussian free-energy logarithm at effective dimension two need additive logarithmic terms.

A random variable $X$ in a real separable Banach space $B$ is Gaussian when $\ell (X)$ has a normal distribution for every continuous linear functional $\ell \in B^{\ast }$. It is centered when all these expected values vanish. Its probability law is a Gaussian measure. Degenerate normal distributions are allowed.

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.

A Gaussian random variable in a Banach space taking values in a separable Banach space satisfies the displayed exponential-integrability bound for some positive $a$. In particular every finite power of its norm has a finite expected value. The value of $a$ depends on the Gaussian measure; the theorem does not assert this for every positive $a$.

For $N$ variables with the standard normal distribution, the expected maximum of their absolute values is at most $\sqrt{2\log (2N)}$, even when they are dependent. Bound the exponential of the maximum by the sum of the $2N$ signed exponentials, use the normal moment-generating function, then apply Jensen inequality and minimize over the exponential parameter. The resulting logarithmic growth is useful in simultaneous estimation.

For fixed $c,h>0$, $L\ge 1$ and $\epsilon \ge 0$, complete the square in the exponent. A translated Gaussian function summed on a fixed-spaced lattice has mass $O(\sqrt{L})$ uniformly in the translate, by comparison with its integral and its maximum. This keeps a square-root logarithm, rather than the full number of dyadic intervals, when summing exponentially damped exponential sum bounds.