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The equal-variance form of Slepian's lemma is the following. If $U,V$ are centered vectors with multivariate normal distributions, $\mathbb{E}{U}_i^2=\mathbb{E}{V}_i^2$ for every $i$, and $\mathbb{E}{U}_i{U}_j\ge \mathbb{E}{V}_i{V}_j$ whenever $i\ne j$, thenfor every threshold vector $(u_i)$. In particular $\mathbb{P}({\max }_i{U}_i\le u)\ge \mathbb{P}({\max }_i{V}_i\le u)$. No nonsingularity of the covariance matrices is required.

Here the pointwise variances $2t^{2H}$ and $2t^{2H^{\prime }}$ differ. To make the comparison rigorous, use the increment form, namely the Sudakov-Fernique inequality: if centered multivariate normal distributions satisfy $\mathbb{E}(U_i-U_j{)}^2\le \mathbb{E}(V_i-V_j{)}^2$ for every pair, then $\mathbb{E}{\max }_i{U}_i\le \mathbb{E}{\max }_i{V}_i$. Unlike the preceding form of Slepian's lemma, this statement has no equal-pointwise-variance hypothesis.

For completeness, the increment form follows by Gaussian integration by parts. Make $U,V$ independent, define $Z_r=\sqrt{1-r}U+\sqrt{r}V$ for $0<r<1$, and letThe $p_i$ are softmax function weights, with $p_i\ge 0$ and ${\sum }_i{p}_i=1$. Direct differentiation gives ${\partial }_i{F}_{\tau }=p_i$ and ${\partial }_{ij}{F}_{\tau }=\tau (p_i{1}_{i=j}-p_i{p}_j)$. The Gaussian integration by parts identity therefore yieldsThe last summands are nonnegative because $D_{ii}+D_{jj}-2D_{ij}=\mathbb{E}(V_i-V_j{)}^2-\mathbb{E}(U_i-U_j{)}^2$. Sinceintegration in $r$ and the limit $\tau \to \infty$ prove the Sudakov-Fernique inequality. All derivatives needed here are bounded for a fixed $\tau$. Singular covariance matrices can be handled by adding small independent centered normal noise with the same variance to both vectors, proving the identity for the nonsingular vectors, and letting the noise variance decrease to zero.

Use the printed normalization of fractional Brownian motion, which omits the conventional factor $1/2$. Its increment variance isFor $s,t\in [0,1]$ and $H>H^{\prime }$, $|t-s{|}^{2H}\le |t-s{|}^{2H^{\prime }}$. Enumerate $T$ by increasing finite subsets $T_n$ whose union is $T$, with $0\in T_n$. The Sudakov-Fernique inequality givesBoth Gaussian processes vanish at zero almost surely. Thus these maxima are nonnegative and increase with $n$; monotone convergence theorem givesTheir continuous modifications agree with the original Gaussian processes simultaneously on the countable set $T$, after discarding a single null event. Continuity and density of the rational parameters make each rational supremum equal to the supremum over $[0,1]$ for these modifications.

The expected suprema are finite. Indeed, the canonical distance of $Y_{H^{\prime }}$ is $d(s,t)=\sqrt{2}|t-s{|}^{H^{\prime }}$. A Euclidean grid supplies a closed-ball metric covering number bound $N(\epsilon )\le 1+\lceil (\sqrt{2}/\epsilon {)}^{1/H^{\prime }}\rceil$ for $0<\epsilon \le \sqrt{2}$. The Dudley entropy integral bound for an anchored centered separable Gaussian process isThe integral is finite since its integrand grows at most like a constant times $\sqrt{1+\log (1/\epsilon )}$ near zero. The same argument applies to $X_H$.

For any real path vanishing at zero,A centered Gaussian process has the same probability law as its negative. Combining this symmetry with the increment comparison gives the stronger boundIn particular the requested comparison isThe common normalization factor $\sqrt{2}$ affects both sides equally. The comparison is valid for every $0<H^{\prime }<H<1$, including the equality cases $s=t$ or $|s-t|=1$ in the increment bounds.

For any finite list $t_1,\dots ,t_k\in [0,1]$, the random vector $(U_{t_1},\dots ,U_{t_k},B_1)$ has a multivariate normal distribution, since it is a linear image of a vector of Brownian motion coordinates. The Brownian covariance kernel and linearity of covariance giveThe independence of uncorrelated jointly normal variables implies that $(U_{t_1},\dots ,U_{t_k})$ is independent of $B_1$. Extending from finite-coordinate cylinder events by independence extended from generating pi-systems provesThus the Brownian bridge is independent of its endpoint. Its covariance kernel is $\mathrm{Cov}(U_s,U_t)=\min (s,t)-st$, confirming that $U$ is the standard Brownian bridge. This is the Brownian bridge independence from its endpoint property.

In factor analysis, the independent multivariate normal distributions of $Z_i$ and ${\xi }_i$ imply that their linear combination is also multivariate normal. Its expected value is zero and its covariance matrix isThereforeThe noise variance is assumed positive. The factor contribution is positive semidefinite and has rank at most $k$, while the noise makes the marginal covariance nonsingular.

Let $Z_{ir}=1_{\{\text{observation }i\text{ belongs to rat }r\}}$. After integrating out the random intercepts,The fixed-effect design matrix has four independent columns, so the given $A$ has dimensions $176\times 172$, with $A^{T}A=I_{172}$ and $A^{T}X=0$. Consequently the error contrasts $U=A^{T}Y$ have a multivariate normal distribution with mean zero and covariance matrix $W=A^{T}VA$.

The restricted likelihood from orthogonal error contrasts is their density. Equivalently, the estimates maximizeThis is restricted maximum likelihood, not maximization over individual rat effects. The unknown $\beta$ disappears because $A^{T}X=0$. Replacing $A$ by $AQ$ for an orthogonal $Q$ rotates $U$ and conjugates $W$, preserving the determinant and quadratic form; hence the objective is independent of the chosen orthonormal basis. The software's integrated REML log-likelihood can differ by an additive constant depending only on $X$, which has no effect on these variance estimates.

A valid covariance matrix must be real symmetric and a positive semidefinite matrix, with $C_{ii}={\sigma }_i^2\ge 0$. Conversely, every such matrix is the covariance of a possibly degenerate multivariate normal distribution. The diagonal correlations are ${\rho }_{ii}=1$; the printed strict inequality can only concern distinct indices. Pairwise bounds do not suffice: the $3$ by $3$ correlation matrix with diagonal $1$ and every off-diagonal entry $-0.9$ has eigenvalue $-0.8$ along $1$ and is invalid. Even strict pairwise inequalities allow singularity: off-diagonal entries $-1/2$ give an eigenvalue zero.

For the inverse requested in the question, assume in addition that $C$ is a positive-definite matrix. Write $x=({\hat{\mu }}_i)$, $\Lambda =C^{-1}$ and $S=1^T\Lambda 1>0$. Up to a constant, the log-likelihood is $-\frac{1}{2}(x-\mu 1{)}^T\Lambda (x-\mu 1)$. Its score function and second derivative areThe correlated Gaussian common-mean estimator is thereforeIts expectation is $\mu$, and direct covariance propagation givesThe Fisher information is $-\mathbb{E}{\ell }^{\prime \prime }(\mu )=S$, so this variance attains the Cramer-Rao bound and the estimator is an efficient estimator.

If $C$ is singular, an ordinary Lebesgue density and $C^{-1}$ are unavailable. When $1$ lies in the range of $C$, the same formulas hold on that range with its Moore-Penrose inverse. Otherwise there is $u\in \ker C$ with $u^{T}1\ne 0$, and $u^{T}x=\mu u^{T}1$ determines $\mu$ exactly with zero variance. The usual nonsingular Cramér–Rao calculation then does not apply.

Forwith symmetric $D$,Substitution into the current from part (d) yieldsIt vanishes for all $\mathbf{x}$ exactly whenThus the stated multivariate normal distribution is a stationary solution whenever the zero-current assumption makes $D$ symmetric positive definite. This is the Stationary Gaussian density for a linear Fokker-Planck equation.

An ${\mathbb{R}}^d$-valued process $(X_t{)}_{t\ge 0}$ is a Brownian motion started at $x$ when:

Stack the observations as $y=(y_1^T,y_2^T{)}^T$ and setLet $K_f(a,b)$ and $K_g(a,b)$ denote matrices obtained by evaluating the two Gaussian process covariance kernels. Independence of the quasar light curve, gravitational microlensing, and Gaussian noise processes giveswhere $E_i=\mathrm{diag}({\sigma }_{i,1}^2,\dots ,{\sigma }_{i,N}^2)$. Thus $y\mid \theta$ is a multivariate normal distribution and its Gaussian-process marginal likelihood isThe off-diagonal blocks are essential: both images contain the same delayed Ornstein-Uhlenbeck process.

Write $a=(2,1{)}^T$, so the data map is $a^{T}u$. The Gaussian likelihood and prior distribution giveCompleting the square, or using Gaussian conjugacy for a normal linear model, identifies a multivariate normal distribution with precision matrix $C_{\delta }^{-1}=I_2+{\delta }^{-2}a{a}^T$. Since $a^{T}a=5$, its posterior mean and covariance matrix areThe parameters areFor example, $(I_2+{\delta }^{-2}a{a}^T)C_{\delta }=I_2$, which checks the square completion.

The orthogonal vectorsare eigenvectors of the covariance matrix. Their respective variances areThey are independent under the posterior distribution, by independence of uncorrelated jointly normal variables. Thus uncertainty is different in the two directions even for positive noise: the data constrain $e_{\parallel }$, whereas $e_{\perp }$ spans the null space of the data map.

As $\delta \to 0$, the posterior distribution converges to the law ofThis is supported on $2u_1+u_2=m$, and its posterior mean is the minimum-norm least-squares solution. The prior distribution still determines the distribution along the unobserved null space; it has not disappeared. This example of a Gaussian posterior in the zero-noise limit distinguishes exact recovery of an observed component from recovery of the whole unknown.

The conditional mean estimator is the posterior meanprovided the first moment is finite. A maximum a posteriori estimate is any maximizer of the posterior density relative to Lebesgue measure:Existence of the posterior density alone does not guarantee a finite posterior mean or existence of a maximizing point.

For the stated multivariate normal distribution and independent standard Gaussian noise, the negative log of the posterior density, up to a constant, isIts precision matrix $A^{T}A+{\Sigma }^{-1}$ is a positive-definite matrix, so Gaussian conjugacy for a normal linear model givesThe two forms agree becauseThe covariance matrix is $(A^{T}A+{\Sigma }^{-1}{)}^{-1}$, and the maximum a posteriori estimate equals the posterior mean for this multivariate normal distribution.

For $d=1$ and $X\sim N(0,{\sigma }^2)$, the characteristic function isFor a multivariate normal distribution with covariance matrix ${\sigma }^2{I}_d$, the coordinates are uncorrelated and hence independent by independence of uncorrelated jointly normal variables. ThereforeThis is the characteristic function of an isotropic Gaussian vector.

Because $S_1$ is Gaussian, the terminal wealth $W_1$ is normal. For constant absolute risk aversion utilitythe moment-generating function of a normal variable givesMaximizing this expected utility is equivalent to maximizingafter dropping a constant. The unique first-order condition isand strict concavity givesIt agrees with part (a) exactly whenFor the usual risk-averse convention $\gamma >0$, the asserted correspondence assumes $w_1>(1+r)w_0$; under that natural target-return condition the choice is unique.

The hat matrix $P$ is an orthogonal projection of rank $p$, so $I-P$ is an orthogonal projection of rank $n-p$. Because $\epsilon /\sigma$ is a standard multivariate normal distribution vector, Cochran's theorem givesThus ${\hat{\sigma }}^2\stackrel{d}{=}({\sigma }^2/n){\chi }_{n-p}^2$. Substituting this into the preceding formula gives the Distribution of the Akaike information criterion in a normal linear model:

Letbe the hat matrix. Thendepends only on the orthogonal projection $HY$, whiledepends only on the orthogonal residual projection $(I-H)Y$.

The random vectors $HY$ and $(I-H)Y$ are jointly multivariate normal, and their cross-covariance isUncorrelated jointly Gaussian vectors are independent. Therefore

A process $(X_t{)}_{t\ge 0}$ is Brownian motion in ${\mathbb{R}}^d$ when $X_0=0$, its paths are almost surely continuous, and for $0\le t_0<\cdots <t_k$ the increments $X_{t_j}-X_{t_{j-1}}$ are independent centered Gaussian vectors with covariance $(t_j-t_{j-1})I_d$.

The estimator is an affine transformation of the multivariate normal distribution. Its mean is $\beta$, and direct covariance calculation gives

Every linear combination of $X_{t_1},\dots ,X_{t_n}$ is a deterministic constant plus one stochastic integral of a deterministic $L^2$ function against $B$. Part a makes every such combination Gaussian. By the linear-combination characterization of a multivariate normal distribution, $(X_{t_1},\dots ,X_{t_n})$ is jointly Gaussian, so $X$ is a Gaussian process.

Write the Cepheid measurement as $\hat{\mu }=\mu +{ϵ}_{\mu }$, where ${ϵ}_{\mu }\sim N(0,{\sigma }_{\mu }^2)$, and define the calibrated absolute magnitude dataThus $q=(q_1,\dots ,q_K{)}^T$ has a multivariate normal distribution with mean $M_{0}1$ and covariance matrix

The Hubble law and the definition of distance modulus giveConsequently, withthe Hubble-flow observations are independent random variables satisfying $x_i\sim N(M_0-\theta ,{\sigma }_{\mathrm{tot}}^2)$. Apart from a factor independent of the parameters, the joint likelihood function isThe expression extends by continuity to a singular limiting covariance such as ${\sigma }_I=0$.