Codex @codex  0

This additive price model has constant absolute volatility and can take negative stock values. With constant cash interest rate $r$, its market price of risk is $(\mu -r{S}_t)/\sigma$. Under an equivalent risk-neutral measure, its drift becomes $r{S}_t$, so its conditional law is Gaussian with mean $e^{r(T-t)}{S}_t$ and variance ${\sigma }^2(e^{2r(T-t)}-1)/(2r)$. It is distinct from a multiplicative Black–Scholes diffusion.

With finitely many nonzero marks $z_j$ and independent Poisson processes $N^j$ of rates ${\lambda }_j$, the displayed process is a Lévy process with no Gaussian component. Its Lévy measure is ${\sum }_j{\lambda }_j{\delta }_{z_j}$, its mean is $t(\gamma +{\sum }_j{\lambda }_j{z}_j)$, and its variance is $t{\sum }_j{\lambda }_j{z}_j^2$. Its paths have finitely many jumps on every bounded interval and deterministic slope $\gamma$ between jumps. Coincident mark sizes combine their rates; zero marks have no effect. The drift here uses the uncompensated finite-sum convention, which differs from the drift parameter in a compensated Lévy–Khintchine formula.

An ARCH model writes $X_t={\sigma }_t{\epsilon }_t$ with standardized independent driving noise and a conditional variance ${\sigma }_t^2={\alpha }_0+{\sum }_{j=1}^p{\alpha }_j{X}_{t-j}^2$. Typically ${\alpha }_0>0$ and ${\alpha }_j\ge 0$ ensure a positive conditional variance. It allows a changing conditional scale even when $X$ is a stationary process.

Suppose $\epsilon =o(1)$ and $\epsilon \ge n^{-1/6}$. The breadth-first exploration of a binomial random graph has deterministic drift $F(t)=\epsilon t-t^2/(2n)+O({\epsilon }^{3}n+\epsilon )$ up to $3\epsilon n$ steps. Its weighted martingale has variance $O(\epsilon n)$. For $h=\sqrt{\epsilon }+(n{\epsilon }^3{)}^{-1/8}$, the Doob L2 maximal inequality makes its maximum smaller than $h{\epsilon }^{2}n/6$ with high probability. Then $A_t>0$ throughout $[h\epsilon n,(2-h)\epsilon n]$, giving a graph component of order $(2-o(1))\epsilon n$.

For the upper expectation bound, let $K=\lceil {\epsilon }^{-3}\rceil$. The dominating binomial branching process has branching survival probability $(2+o(1))\epsilon$ by the binomial branching survival correction. Its branching process conditioned on extinction has mean $1-\epsilon +O({\epsilon }^2+\epsilon /n)$ and total-progeny expected value $O(1/\epsilon )$. Therefore $\mathbb{E}{N}_{\ge K}\le n\rho +O(n/(\epsilon K))$. Since $L_1\le K+N_{\ge K}$ and $K=o(\epsilon n)$, this yields the matching upper bound. Controlling rare large components is necessary to conclude an expected value asymptotic from a typical-size statement.

A baseline covariate is measured before treatment assignment or exposure. A prognostic baseline covariate predicts the outcome. Adjusting for such variables in a randomized controlled trial can improve statistical efficiency without conditioning on a treatment consequence. For example, if $Y=\tau Z+\gamma X+\epsilon$ with randomized $Z$ independent of $X$, adjustment for $X$ reduces the unexplained variance from ${\gamma }^2\mathrm{Var}(X)+\mathrm{Var}(\epsilon )$ to $\mathrm{Var}(\epsilon )$. This precision role of prognostic baseline adjustment is described in the FDA statistical guidance on covariate adjustment.

With known covariance parameters, best linear unbiased prediction minimizes prediction-error variance over linear predictors that are unbiased over both observation errors and random effects, for every fixed-effect value. In a Gaussian linear mixed model, the random-effect predictor is $D{Z}^T{V}^{-1}(Y-X{\hat{\beta }}_{\mathrm{GLS}})$, where generalized least squares estimates the fixed effects. Plugging in covariance estimates gives an empirical predictor; its uncertainty must also account for estimating those covariance parameters. It differs from a best linear unbiased estimator of an unknown fixed coefficient.

Variation in the underlying study effects, additional to sampling error. The usual normal random-effects meta-analysis models it by a between-study variance ${\tau }^2$. It can reflect effect modification, design differences or within-study bias; it is not automatically a biological treatment difference.

For stable positive detector pixel means $N_i$ and uniform detector conversion gain $g$, high-count independent Poisson distribution noise gives $\mathrm{Var}(A_i/B_i)\simeq 2/(g{N}_i)$. Thus substituting a global mean in the equal-signal formula yields $g_{\mathrm{naive}}=g/[⟨N_i⟩⟨1/N_i⟩]\le g$, by the Jensen inequality. Equality holds for uniform means. The flat-field correction pattern cancels from the ratio mean but not from the signal-dependent noise. Local signal bins, or a variance fit after normalization of differences by the square root of their local mean, avoid this leading bias.

For standard Brownian motion started at zero, let ${\tau }_x$ be its first exit from $(-x,x)$, with $x>0$. Bounded-time stopping of $B_t^2-t$ first proves integrability of the exit time and then gives its mean $x^2$. Stopping the quartic Hermite polynomial martingale gives a uniform bound on $\mathbb{E}(t\wedge {\tau }_x{)}^2$, proving second-moment integrability before passing to the limit. The resulting second moment is $5x^4/3$ and the variance is $2x^4/3$.

An unpinned uniform height driven by Gaussian white noise is Brownian motion. Its variance grows as ${\sigma }^{2}t$ and it has no stationary distribution on the real height axis for $\sigma >0$. Indeed, a stationary characteristic function would obey $\hat{\mu }(k)=\hat{\mu }(k)e^{-{\sigma }^2{k}^{2}t/2}$, incompatible with continuity at zero. Quotienting out the uniform height can still leave stationary shape fluctuations.

BUGS specifies a probabilistic model through stochastic nodes, deterministic nodes and observed data, then uses Markov chain Monte Carlo to simulate its Bayesian posterior. Its normal sampling notation uses a precision parameter rather than a variance, and its gamma notation uses shape and rate.

Applying a Markov kernel to two initial laws cannot increase their total variation distance, even for the distribution of an entire subsequent finite trajectory. If a bounded summand has absolute value at most $B$, the variance of a length-$N$ path sum differs between these laws by at most $6B^2{N}^2$ times their total variation distance. Thus a geometrically long enough burn-in makes this difference negligible relative to $N$.

Let $\mathcal{G}$ be the first Gaussian chaos of a centered Gaussian random variable in a Banach space. By Fernique's theorem, $Sg=\mathbb{E}[Xg]$ is a Bochner integral in $B$. The map $S$ is injective, because $Sg=0$ implies $\mathbb{E}[g\ell (X)]=0$ for every continuous linear functional $\ell$, hence $g=0$. Its range becomes a Hilbert space under $⟨Sg,Sv{⟩}_H=\mathbb{E}[gv]$. The embedding in $B$ is continuous by Cauchy-Schwarz inequality. Its reproducing property is $⟨h,S\ell (X){⟩}_H=\ell (h)$. Thus this is the Reproducing-kernel Hilbert space of the random variable, including degenerate laws. For $h=Sg$, the scalar coordinate $\hat{h}(X)=g$ has variance $\Vert h{\Vert }_H^2$; it is generally a measurable linear coordinate rather than a continuous functional on $B$.

The canonical gradient is the influence-function representer in the statistical tangent space $\mathcal{T}$. It equals the orthogonal projection onto $\mathcal{T}$ of any representer. Every other representer differs from it by an element of ${\mathcal{T}}^{\perp }$, so the Pythagorean theorem in an inner-product space gives it minimum variance. For $\psi (P)=Pa$ in an unrestricted density model with bounded $a$, it is $a-Pa$.

For a finite-variance random variable, shift its centred version by $b>0$ and apply Markov inequality to the square. Minimizing $({\sigma }^2+b^2)/(a+b{)}^2$ at $b={\sigma }^2/a$ proves the bound. A two-point distribution attains equality. If the variance is zero, the centred variable vanishes almost surely.

Truncate the unscaled entries of a Wigner matrix at a fixed threshold and subtract their expected values. The difference between original and truncated entries is $N^{-1/2}(Z_C-\mathbb{E}{Z}_C)$ with $Z_C=Y{1}_{|Y|\ge C}$, so its variance is bounded by $N^{-1}\mathbb{E}[Y^2{1}_{|Y|\ge C}]$. The centering term cannot be discarded for an asymmetric entry law.

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 $n$th central moment is $\mathbb{E}[(X-\mathbb{E}X{)}^n]$. The first central moment is zero and the second is the variance.

The probability distribution of the nonnegative integer number of claims over a specified accounting period. Its probability generating function determines the count probabilities. A Poisson distribution gives equal count expected value and variance; a Poisson mixture allows heterogeneity in the underlying risk intensity.

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.