Types of Convergence of Random Vectors

Let $X_1, \dots , X_n$ be a sample from $\operatorname {Exp}\left(\alpha \right), \alpha > 0$. Let $Y = \frac{1}{n}\sum_{i=1}^n X_i = \overline{X}$. Find constants $a(\alpha ), \sigma^2(\alpha )$ such that

Let $\left\{ \xi_{n}\right\}_{n=1}^{\infty },\left\{ \eta_{n}\right\}_{n=1}^{\infty }$ and $\left\{ \zeta_{n}\right\}_{n=1}^{\infty }$ be sequences of random variables. Prove that if $\xi_{n} \stackrel{d}{\rightarrow } \xi ,\left|\xi_{n}-\eta_{n}\right| \leq \zeta_{n}\left|\xi_{n}\right|$ and $\zeta_{n} \stackrel{\mathbb {P}}{\rightarrow } 0$, then $\eta_{n} \stackrel{d}{\rightarrow } \xi$.

Given a set of independent identically distributed random variables $X_{1}, \ldots , X_{n}$ with distribution $\mathcal{N}\left(0, \sigma^{2}\right)$. Consider the statistics $Y=\frac{1}{n} \sum_{i=1}^{n}\left|X_{i}\right|, Z=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2}$ and $T=\sqrt{\frac{2}{\pi }} Z / Y$. Find the limit in distribution of the expression

Let $\left\{ \xi_{n}\right\}_{n=1}^{\infty }$ and $\left\{ \eta_{n}\right\}_{n=1}^{\infty }$ be two sequences of random variables, such that for each $n \geq 1$ the variables $\xi_{n}$ and $\eta_{n}$ are independent. Let $\xi_{n} \stackrel{\mathbb {P}}{\longrightarrow } \xi , \eta_{n} \stackrel{\mathbb {P}}{\longrightarrow } \eta$. Using the method of characteristic functions, prove that $\xi$ and $\eta$ are also independent.

Let $X_{1}, \ldots , X_{n}$ be independent random variables having a Laplace distribution with parameter $\sigma$, i.e. with density

Consider $Y=\frac{1}{n} \sum_{i=1}^{n}\left|X_{i}\right|, Z=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2}$. Using the multivariate central limit theorem, find the limit in distribution of the expression

where $T=Z^{2} /\left(4 Y^{3}\right)$.

It is known that the condition for convergence in distribution of a sequence of random variables $\left\{ \xi_{n}\right\}_{n=1}^{\infty }$ to a random variable $\xi$ can be weakened: $\xi_{n} \stackrel{d}{\rightarrow } \xi$ if for any bounded, uniformly continuous function (in the usual definition it was continuous) $f(x)$ we have $\mathbb {E}\left[f\left(\xi_{n}\right)\right] \rightarrow \mathbb {E}\left[f(\xi )\right]$ as $n \rightarrow \infty$. Using this condition, prove the following generalization of Slutsky's lemma: let $\xi_{n} \stackrel{d}{\rightarrow } \xi$ and $\eta_{n} \stackrel{d}{\rightarrow } C= \operatorname {const}$. Then for any continuous function $\varphi : \mathbb {R}^{2} \rightarrow \mathbb {R}$ we have

Prove Theorem 1.4: let $\xi_n \xrightarrow [n \to \infty ]{d}\xi$ be random vectors of dimension $m$, and let $h(x): \mathbb {R}^{m} \to \mathbb {R}$ be a function differentiable at the point $a \in \mathbb {R}^{m}$. Let $b_n \to 0, b_n \neq 0$. Then

where $\nabla h|_{a}$ is the gradient of the function $h$ taken at the point $a$.

Prove the following generalization of Theorem 1.4 in the one-dimensional case. Let the function $f(x)$ have a derivative at the point $x=$ 0. Then if for the sequences of random variables $\left\{ \xi_{n}\right\}_{n=1}^{\infty },\left\{ \eta_{n}\right\}_{n=1}^{\infty }$ we have $\xi_{n} \eta_{n} \stackrel{d}{\rightarrow } \eta$ and $\eta_{n} \stackrel{\text{ P }}{\rightarrow } 0$, then