1
vote
1answer
19 views

Cauchy sequence of random variable

I know (from calculus) the Cauchy convergence theorem for sequence of real numbers...how to show it with random variables? I have $S_n=\sum_{i=1}^n X_i;$ $X_i$ being iid centered r.v., $S_n$ is a ...
2
votes
0answers
76 views

Is my proof correct? are the arguments right?

my assumptions: (i) $\lim_{t \to \infty}F_{t}(x)=F(x) \ \forall\ x\ \in\ C(F)$(set of continuity points of F) with $F_{t}(x)$ family of distribution functions and $F$ distribution function (ii) ...
1
vote
1answer
46 views

Weak convergence of distribution family

I know the convergence in distribution and the weak convergence. but I have two questions: First one: does weak convergence implies pointwise convergence or it is the same? And second one: I have ...
1
vote
2answers
32 views

An $L^1$ convergence problem

Is the following true? If $X_n$ converges almost surely to a non-negative random variable $X$ having finite expectation, and if $E(X_n)$ converges to $E(X)$, then $E|X_n - X|$ converges to $0$? ...
1
vote
1answer
40 views

Space of Gaussian Functions is Closed in $L^2$

Let $\Omega, \mu$ be a probability space. A measurable function $f: \Omega \rightarrow \mathbb{R}$ is called Gaussian if $$\mu (f^{-1}(A))=\frac{1}{\sigma \sqrt{2\pi }}\int_Ae^{-x^2/2\sigma ^2} dx$$ ...
1
vote
2answers
32 views

Need help understanding the difference between a.s. convergence and convergence in probability.

I have problem understanding the difference when I look at the alternative definition of a.s. convergence. I know how it is defined originally, but it is the alternative definition which makes it ...
2
votes
1answer
22 views

Convergence in total variation

There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ...
-2
votes
0answers
22 views

Almost surely convergence with stationary random vectors

I dont seem to be able to incorporate the stationarity condition into any of limit theorems I know. I cannot see how the Birkhoff almost everywhere ergodic theorem could be used as I cannot see how ...
1
vote
1answer
49 views

Extreme Value Theory - Show: Normal to Gumbel

The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory. How can we show that? We have $$P(\max X_i \leq x) = ...
0
votes
1answer
39 views

Let $Y=\sup_n|X_n|$, Show that $Y<\infty$ a.s.

Suppose $\lim_{n\to\infty}X_n=X$ a.s. and $|X|<\infty$ a.s. Let $Y=\sup_n|X_n|$, Show that $Y<\infty$ a.s. If $\lim_{n\to\infty}X_n=X$ a.s. then $S_1:=\{w:\lim_{n\to\infty}X_n=X\}$ has ...
0
votes
1answer
19 views

Convergence in probability of a random variable

I need to prove that $(X_n^2 -X)^2\to 0$ in probability $\Rightarrow X_n^2\to X$ in probability. I tried solving it with the triangle inequality, but it didn't get me anywhere. Is there another ...
1
vote
0answers
36 views

Central limit theorem does not converge to random variable

Recently, we investigated whether the expression in the central limit theorem converges to a random variable pointwise almost sure? The answer was negative due to $P ( \text{limsup} ...
4
votes
1answer
39 views

How to show convergence in distribution

Let $([0,1],B,\lambda)$ (B Borel Sigma-algebra) and $\lambda$ the Lebesgue measure. I want to show that this sequence converges in distribution. $$X_n(\omega)= \left(\begin{matrix} 1, & ...
2
votes
2answers
58 views

Properties of a sequence of iid rv's

I cannot do part a), and Im fairly sure that b) and c) will follow from it. If possible could I please have a solution to part a) and hints if you feel necessary to parts b) and c).
2
votes
1answer
21 views

Modes of Convergence of a particular random variable

Let $X_n \sim U([-1/n,1/n])$ be uniform random variables on $[-1/n,1/n]$ for $n \in \mathbb{N}$. Do the $X_n$ converge, and if yes in what sense? I think it converges pointwise as for any $x \in ...
0
votes
0answers
14 views

Searching for theorems that prove almost sure convergence from convergence in probability

As we can see that almost sure convergence implies convergence in probability, and the converse is not necessarily true. But now I would like to prove a particular sequence of random variable ...
2
votes
1answer
32 views

Convergence almost surely and B-C lemmas

Showing the expectation is straightforward. I am not sure how to use the Borel-Cantelli lemmas to show the almost surely part.
3
votes
0answers
27 views

Asymptotic Bounds for the distribution of $f_n(X_n)$.

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of $\mathbb{R}^{k}$-valued random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ converging almost surely to $X$. ...
2
votes
1answer
19 views

Convergence in distribution problem

I want to prove that, in $(\mathbb{R},B(\mathbb{R}))$, we have that $\frac{1}{n}\sum_{i=1}^{n}\delta_{\frac{i}{n}}$ converges to $U_{[0,1]}$. We need to prove, by definition, that $\lim_{n \to ...
0
votes
1answer
22 views

Convergence in Probability and in Quadratic Mean for a sequence of random variables

I have been trying to determine whether a sequence of random variables, $X_1,X_2,\ldots,X_n$, such that $$P\left(X_n= \frac{1}{n}\right)=1-\frac{1}{n^2}\\ \text{and}\\ ...
2
votes
2answers
54 views

Almost sure convergence of a sequence of random variables

Once again I've encountered a problem, which might not be difficult: I'm given a sequence of random variables $ (X_n) $, each with density function $g_n(x) = nx^{n-1} \textbf{1}_{(0,1]} $. I am to ...
1
vote
1answer
41 views

Convergence in probability of iid normal random variables

Let $X_1, X_2,\ldots$ be a sequence of iid normal random variables with zero mean and unit variance. I read the following as a trivial example: (1) $X_n \to X_1$ in law, (2) $X_n \not\to X_1$ in ...
1
vote
1answer
28 views

Types of Convergence (Random Variables)

Suppose that for every $n\ge 1$, the law of $X_n$ is given by $P[X_n=n^2]=\beta_n$ and $P[X_n=0]=1-\beta_n$, determine if $(X_n)_{n\ge 1}$ converges in probability, in $L^1$ or almost sure to zero, ...
1
vote
0answers
32 views

How do I prove the special case of the central limit theorem?

Let $(X_n)$ be an i.i.d. sequence such that $\mathbb P(X_1=1)=\frac{1}{2}+\varepsilon$ and $\mathbb P(X_1=-1)=\frac{1}{2}-\varepsilon$ for some $\varepsilon\in(0, \frac{1}{2})$. I'd like to show that ...
3
votes
1answer
187 views

Convergence of marginal distribtution

Here I have a question which looks a little bit weird: $(q_n)_n$ is sequence of probability density functions of the couple $(x,y) \in \mathbb R^2$, $p_n$ is the marginal density of $q_n$, i.e. ...
1
vote
1answer
23 views

Convergence of random variable

I've been facing the following problem: Let $(X_k, Y_k)_k$ be a sequence of $2$-dimensional, independent random variables, each with uniform distribution over $ B(0,k) $ Verify if the following ...
0
votes
1answer
30 views

tightness of sequence of degenerate probabilities

If $\delta_x$ denotes for $x\in \mathscr{R} $, the degenerate distribution at $x$, prove that the sequence $\delta_{x_n}$ of probabilities on $(\mathscr{R,B})$ is tight iff $x_n$ is bounded. This is ...
0
votes
1answer
23 views

Existence of expected value with complex power

Suppose that $X$ be a random variable taking values on $(0,+\infty)$ with density function $f(x)$ and we have $\mathbb{E}(X^2)<\infty$. Can we conclude that $$\mathbb{E}(X^{-2-it}) ...
2
votes
0answers
32 views

problem on almost sure convergence

Let {$X_i$} be iid with finite second moment. Let $Y_n = \frac {2} {n(n+1)} \sum_{i=0}^n i*X_i $, n$\ge$1 Show that $Y_n \to E(X_1) $ I tried to define $Z_i = \frac {2} {(n+1)} i*X_i $ Then $Y_n = ...
1
vote
1answer
44 views

convergence in probability of function of random variables

Suppose that $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d random variables. If we have $E(|X_1|^k) <\infty$ for some $k>0$ and $f(x)$ is a bounded continuous function on $\mathbb{R}$. Is the ...
0
votes
0answers
36 views

Markov Chain Geometric Convergence

Perhaps due to my non-mathematician background (lack of Measure Theory knowledge), I have some difficulties about the Markov Chain Theory. Given an ergodic (irreducible and aperiodic) Markov Chain ...
0
votes
1answer
57 views

Convergence in probability implies convergence in mean under one additional condition

Prove that if random variables $X_n$ are dominated by an integrable random variable then $E[X_n] \to E[X]$ follows if $X_n$ converges to $X$ in probability. Hint: Use the following theorem : A ...
0
votes
0answers
20 views

Expectation of O_p(1) process

Suppose $\{X_n \}$ is bounded in probability, i.e. $Prob(|X_n| > M_\epsilon) = \epsilon$ for all $n > N_\epsilon$, $M_\epsilon < \infty$. Is there any condition(s) to guarantee that ...
2
votes
2answers
154 views

Convergence in probability to a non-measurable limit

Let $(\Omega, \mathcal{F}, P)$ be a probability space. Denote the Borel field on $\mathbb{R}$ by $\mathcal{B}$. Let $\mu: \Omega \rightarrow [0,\infty)$ be a not-necessarily-measurable function and, ...
0
votes
1answer
39 views

Convergence in probability of sample variance

$X_n$ s are a sequence off iid random variables with E($X_n$) = $\mu$, V($X_n$)= $\sigma$$^2$ and $\bar X = \sum$ $\frac{X_i}{n}$. Then show that $\frac1n$ $\sum (X_i - \bar X )^2\to\sigma^2$ in ...
3
votes
0answers
50 views

A basic problem on weak convergence

Let $f_n$ be $2^n$ times the indicator of the set of $x$ in the unit interval for which the digit from $n+1$ to $2n$ is zero in the dyadic expansion of $x$ (lets call it $A_n$). I have to show that ...
3
votes
2answers
61 views

A problem on random series

Suppose $X_1, X_2,\dots, $ be an independent sequence of random variables and $E[X_n] = 0 \forall n$ and $\sum_{n=1}^{\infty} \operatorname{Var}(X_n) < \infty$. I need to prove that ...
4
votes
0answers
218 views

Weak and Probability convergences

I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ ...
1
vote
1answer
34 views

Understanding a proof about convergence in probability

Let $X, X_1, X_2, \ldots : \Omega \to (0,\infty)$ be random variables such that $X_n \xrightarrow{\mathbb{P}} X$. I'd like to show from the definition that $\frac{1}{X_n} \xrightarrow{\mathbb{P}} ...
1
vote
1answer
35 views

A basic problem on random series/ law of large numbers

Consider the following two statements : i) Suppose that $X_1, X_2, \dots$ are independent and identically distributed and $E[X_1^-] < \infty, E[X_1^+] = \infty$. Then $n^{-1} \sum_{k=1}^{n}X_k ...
1
vote
0answers
33 views

Application of the Weak Law of Large Numbers.

I have in my problem that $X_1,\ldots,X_n$ is a random sample from a distribution with probability density $f(x; \theta)=\theta x^{\theta-1}, 0<x<1$. Furthermore, $-\log X_i ...
0
votes
1answer
31 views

The proportion of $\omega$s in $A$ converges almost surely to $P(A)$

Let $A$ be an event in $(\Omega,\mathcal{F},P)$. We generate independent inquiries from $\Omega$ in accordance to $P$. Show that the proportion of $\omega$s in $A$ converges almost surely to ...
2
votes
1answer
69 views

Convergence of a sequence of reciprocals

Let $X_n \geq 0$ be sequence of nonnegative random variables converging a.s. and in $L^2$ to a positive constant $c > 0$: $0 \leq X_n \xrightarrow{a.s.,L^2} c >0$ What can we say about the ...
2
votes
1answer
37 views

On the importance of independence in the central limit theorem

While discussing the central limit theorem, my text makes the following remark: The independence of the random variables $X_1, X_2, \ldots$ is essential. To see this, take $X_1 = X_2 = \ldots = X$ ...
1
vote
2answers
61 views

Is $Z_n→Z$ in distribution and if $z_n→0$ then $z_nZ_n→0$ in distribution

I'm trying to prove the statement made by Did in the comments: Is $Z_n→Z$ in distribution and if $z_n→0$ then $z_nZ_n→0$ in distribution. So we need to prove that $$\forall t>0: ...
1
vote
1answer
58 views

How to make use of the hint for proving $\text{CLT} \implies \text{WLLN}$?

I've seen an exercise where one is asked to prove that the central limit theorem implies the weak law of large numbers. The author gave the following hint: "First prove that convergence in ...
2
votes
1answer
48 views

Three questions about ucp convergence

We say that a sequence of processes $X^n$ converges to a process $X$ uniformly on compacts in probability if for all $\epsilon >0, t>0$ $$P[\sup_{s\le t}|X^n_s-X_s|>\epsilon]\to 0 $$ for ...
1
vote
1answer
26 views

Why is this set an event?

As a part of a setup for another problem, my text remarks that it can be used without a proof that if $X_1, X_2, \ldots$ are random variables then $$C:=\{ \omega\in\Omega \ | \ \sum X_n(\omega) \ ...
1
vote
1answer
39 views

How does one show that the series converges almost surely?

Let $X_1, X_2, \ldots:\Omega\to \mathbb R$ be random variables. Define $C:=\{ \omega \ | \ \sum X_n(\omega) \text{ converges} \}$. There is such $q\in(0,1)$ that for all $n\in \mathbb N: P\{ |X_n| ...
0
votes
1answer
46 views

A basic doubt on probabilistic and almost sure convergence

Suppose we have a sequence of random variable $X_n$ such that $X_n$ converges in probability to $X$. Now, consider the set $\{\omega: X_n(\omega) -> X(\omega)\}$. Assume that the convergence is ...