Multivariate normal distributions

Let $V_{0}$ and $V_{1}$ be positive definition matrices. For $0 \leq \theta \leq 1$ let $g_{\theta }$ denote the $N\left(0,(1-\theta ) V_{0}+\theta V_{1}\right)$ density on $\mathbb {R}^{n}$. Show that

by following these steps.

Suppose $X_{n}$ has a multivariate normal distribution, and $X_{n} \leadsto X$. Show that $X$ also has a multivariate normal distribution.

Let $Z_{1}, Z_{2}, \ldots$ be independent random variables, each distributed $N(0,1)$. Show that the distribution of $M_{n}: = \max_{i \leq n} Z_{i}$ concentrates largely in a range of order $(\log n)^{-1 / 2}$ by following these steps. Define $\alpha_{n}: = (2 \log n)^{1 / 2}$ and $L_{n} = \frac{1}{2} \log \log n$.

Let $X$ and $Y$ be random variables for which both $\mathbb {P}\left(X > x\right) \leq \mathbb {P}\left(Z > x\right)$ and $\mathbb {P}\left(X \leq -x\right) \leq \mathbb {P}\left(Z \leq -x\right)$, for all $x \geq 0$. Show that $\mathbb {E}\left[\exp (t X)\right] \leq \mathbb {E}\left[\exp (t Y)\right]$ for all nonnegative $t$.

Let $A$ be a closed subset of $\mathbb {R}^{2}$, with sections $A_{y}: = \left\{ x \in \mathbb {R}:(x, y) \in A\right\}$ having Gaussian measure $g(y): = \gamma_{1} A_{y}$. Let $B$ denote the 1-shift, as in Lemma $\langle 14 \rangle$.

Lemma 14: For a closed set $A \subseteq \mathbb {R}^{2}$ with sections $A_{y}: = \left\{ x \in \mathbb {R}:(x, y) \in A\right\}$, define $h(y)$ by $\Phi (h(y)): = \gamma_{1} A_{y}$ ($\Phi$ the standard normal distribution function), and let $B: = \left\{ (x, y): x \leq h(y)\right\}$ be the 1-shift of $A$ --- the set obtained by sliding and stretching each $x$-section of $A$ into a semi-infinite interval with the same $\gamma_{1}$ measure, so that the $y$-sections of $B$ have the same $\gamma_{1}$ measure as the corresponding sections of $A$. Then $B$ is an improvement over $A$: $\gamma_{2} B \geq \gamma_{2} A$ and $\gamma_{2} B^{r} \leq \gamma_{2} A^{r}$ for every $r \geq 0$ (where $A^{r}$ denotes the $r$-neighborhood of $A$).

Reduce the one-dimensional version of Theorem $\langle 7 \rangle$ to the case where $A$ is a finite union of intervals, by following these steps. Let $\Phi (\alpha ): = \gamma_{1} A$.

Theorem 7: The Gaussian measure of the neighborhood $A^{r}$ is minimized, for a given value of $\gamma_{k} A$, by choosing $A$ as a closed half-space. More generally, $\gamma_{k} A^{r} \geq \Phi (r+a)$, for every Borel subset $A$ of $\mathbb {R}^{k}$ with $\gamma_{k} A \geq \Phi (a)$.

(Slepian's (1962) inequality) Let $X = \left(X_{1}, \ldots , X_{n}\right)$ and $Y = \left(Y_{1}, \ldots , Y_{n}\right)$ both have centered multivariate normal distributions with $\operatorname {Var}\left[X_{j}\right] = \operatorname {Var}\left[Y_{j}\right]$ for each $j$ and $\mathbb {E}\left[X_{j} X_{k}\right] \leq \mathbb {E}\left[Y_{j} Y_{k}\right]$ for all $j \neq k$. Prove that $\mathbb {P}\left(\cup_{j}\left\{ X_{j} > \alpha_{j}\right\} \right) \geq \mathbb {P}\left(\cup_{j}\left\{ Y_{j} > \alpha_{j}\right\} \right)$ for all real numbers $\alpha_{1}, \ldots , \alpha_{n}$.

Let $S$ be a nonnegative random variable with $\mathbb {P}\left(S > M+r\right) \leq C_{0} \exp \left(-\frac{1}{2} r^{2} / \sigma^{2}\right)$ for all $r \geq 0$, for positive constants $C_{0}, M$, and $\sigma^{2}$.