1
vote
1answer
70 views

A sequence of nonconstant i.i.d. random variables converges with probability zero

Proove: $X_{n} iid, X_{n}$ not constant a.s. $\iff P(X_{n}$ $converges)=0$ My idea for "$\Rightarrow$": $X_{n}$ not constant a.s. $\iff \forall$ c $\in \mathbb{R}$, $\varepsilon$ > 0: ...
0
votes
1answer
58 views

What is the mean and variance of $Y$, where $Y$ is sum of iid's

Here's my work for part a. I could use clarification on part b and d. Is part d the same as part a ($E[A_n] = E[Y]$) ? a) $$E[Y_n] = E[\frac{X_n}{2^n}]$$ ($X$'s are iid so...) $$= \frac{E[X]}{2^n} ...
0
votes
0answers
63 views

Law of large numbers weak vs strong

Does someone have an example where the strong law of large numbers do not hold, but the weak law do hold ? If you think there is no such example, please explain why there are 2 laws of large numbers ...
0
votes
1answer
62 views

If a sequence of random variables converges to both $X$ and $Y$ almost surely, then $X$ and $Y$ have the same distribution

Show that if .${X_n}\mathop \to \limits^{a.s} {\rm{ }}X$. and ${X_n}\mathop \to \limits^{a.s} {\rm{ }}Y$ ,then X and Y have the same distribution. Proof Let $A = \{ \omega \in \Omega :X(\omega ) ...
0
votes
1answer
23 views

Convergence of expecations implies convergence of positive and negative parts?

If we have $E|X_n| \rightarrow E|X|$ does that imply \begin{equation} \lim_{n\rightarrow\infty} E X_n^\pm = X^\pm \end{equation} How about if we only have $EX_n \rightarrow EX$? Is this true in ...
0
votes
1answer
24 views

(Multidimensional) Standard Brownian Motion: Convergence

Relating to this question, I have a further one, and hope, someone can help me. I know that $$\left(X_j - X_{j-1}\right)_{j=1}^t \xrightarrow{d} \left(Y_j\right)_{j=1}^t.$$ Further, we know that ...
0
votes
1answer
23 views

convergence to standard brownian motion

Could you help me with the following: I have that $$T(x):=\frac{X(nx)-E[X(nx)]}{\sqrt{n}} \xrightarrow{d} N(0, \frac{x^k}{k})$$ for each fixed $x>0$, where we also have that $\frac{X(nx)}{t}$ is ...
1
vote
1answer
35 views

Bolzano–Weierstrass theorem for random variables?

I am wondering if there is something similar to the Bolzano–Weierstrass theorem for random sequences. Namely, let $\{x_n\}$ be a bounded random sequence. Is it true that, under some reasonable ...
2
votes
2answers
46 views

Sandwiching Limsups & liminfs of expectations

Why is it that if we sandwich a liminf of an expectation between two equal quantities we get that the limit exists? Can we somehow deduce the limsup from that and conclude that it's the same or am I ...
3
votes
1answer
35 views

If $\sum_{n \geq 1}X_n$ converges a.s. then $\forall a > 0: \sum P(|X_n|>a) < \infty$

I'd like to show that for $(X_n)_{n\geq 1}$ a sequence of real-valued and independent random variables, If $\sum_{n \geq 1}X_n$ converges a.s., then $\forall a > 0: \sum P(|X_n|>a) < ...
0
votes
1answer
27 views

Show convergence in mean

Suppose $X(t)$ is a positive i.i.d. random process in discrete time with finite mean $a$. Let $Y(t)=\min \{a, X(t)\}$. Question is: (1)Can we claim that ...
2
votes
0answers
40 views

Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
2
votes
1answer
35 views

Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$

Let $X_1, X_2, ...$ be a sequence of real-valued random variables. Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$ Attempt: Suppose ...
2
votes
1answer
41 views

Convergence in distribution/Distribution of X

For each $n = 1, 2, ....$, suppose that $X_n$ is a discrete random variable with range $\{1/n, 2/n, ..., 1\}$ and $\hspace{15mm}\mathrm{Pr}(X_n = j/n) = \frac{2j}{n(n+1)}$, $j = 1,...,n$. Does ...
1
vote
1answer
123 views

Conditions for convergence of moments

Let ${X_n}$ be a sequence of r.v. such that $X_n\xrightarrow [d]{}X$, with $E(X)$ finite, and with $E(|X_n|^{1+\delta})\leq K<\infty$ for all $n$. We know that: a) For $\delta>0$, we have ...
1
vote
1answer
26 views

weak convergence of probability measures and unbounded functions with bounded expectation

Assume that $\mu^n$ are probability measures on $R$ that convergence weakly(-*) to $\mu$, i.e for all $f \in C_b (R)$ (bounded and continuous), we have that $\int f(x) \mu^n(dx) \rightarrow \int f(x) ...
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
30 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
1answer
65 views

Show that convergence in the mean implies convergence of the means [closed]

Question: Let $X_n$, n = 1,... denote a sequence of real-valued random variables; $X_n$ is said to converge in mean if $\hspace{20mm}$$$\lim_{n\to\infty} E[|X_n-X|] = 0$$ Show that if $X_n$ ...
1
vote
0answers
39 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} ...
2
votes
1answer
34 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.
-1
votes
0answers
19 views

Independence and limits [duplicate]

Suppose $X_n$, $Y_n$ are independent real valued random variables for every natural n. And that the limit as n tends to infinity almost surely exists finitely, say X, Y respectively. Is it necessary ...
1
vote
1answer
29 views

Convergence of random harmonic series

The problem is to show that the random harmonic series $X_n:=\sum_{n=1}^{\infty}\frac{\nu_n}{n}$ with $P[\nu_n = 1] = P[\nu_n = -1] = \frac{1}{2}$ converges. It is obvious that the harmonic series ...
3
votes
0answers
28 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 $X$. ...
0
votes
1answer
28 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
67 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
28 views

Borell Cantelli Application

If i got that $\mathbb{P}(\underbrace{|X_{n}|>n^{\frac{1}{2}+\epsilon}}_{=:A_{n}})\leq \exp\left(-\frac{n^{2\epsilon}}{8}\right)$ with $\epsilon \in (0, 0.5)$. I know that ...
0
votes
1answer
25 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}) ...
0
votes
0answers
22 views

weak convergence of a sequence of cdfs

Suppose $F_n \to F $ weakly, $ x \in c(F)$ and $ x_n $ is a real sequence converging to x. Prove that $F_n(x_n) \to F(x) $. Here $F_n$, $F$ are cdfs and $c(F)$-set of continuity points of $F4. I ...
0
votes
1answer
48 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 ...
1
vote
1answer
28 views

Rate of convergence of mean in a central limit theorem setting

I recently asked a question here that was the following: If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = P(Z_i=+1) = \frac 12,$ then we have by the Central Limit Theorem that ...
1
vote
0answers
18 views

What is the minimum standard deviation for a normal PDF such that one tail is always larger than that of a second normal PDF (different means)?

Say I have two weighted normal distributions, $$ f_1(x) = \frac{a}{2 \sigma_1} e^{-\frac{(x-\mu_1)^2}{2\sigma_1^2}} $$ and $$ f_2(x) = \frac{1-a}{2 \sigma_2} e^{-\frac{(x-\mu_2)^2}{2\sigma_2^2}} $$ ...
1
vote
0answers
21 views

Convergence of sequences of random variables

Let $X_1, X_2, ...$ and $Y_1, Y_2, ...$ be two sequences of nonnegative random variables. Assume that each $n$ random variable $Y_n$ is uniform in the interval $[0, X_n]$. Show that if ...
0
votes
1answer
52 views

Two different sequences of random variables each converge in distribution; does their sum?

My question is about basic probability. We have two sequences of random variables, $ \{ X_n \}$ and $\{ Y_n \}$, such that each converge in distribution - i.e. there exist random variables $X$ and ...
1
vote
0answers
26 views

Convergence in distribution and moments

Let us assume that we are given real random variables $X_n$ that converge in distribution to $X$. Moreover, it is known that $\sup_n \mathbb{E}[g(X_n)] < \infty$, where $g$ is a measurable function ...
0
votes
0answers
13 views

Differentiability of random processes.

I know the appropriate criterions for mean-square differentiability of random processes. These criterions are connected with covariance function of a process. Are there any criterions for ...
0
votes
0answers
39 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 ...
1
vote
1answer
57 views

Almost Sure convergence.

Given $(X_n, n\in\Bbb N)$ and $(Y_n, n\in\Bbb N)$ sequences of random Variables. For all $n\in\Bbb N$ it is : $X_n=Y_n$ almost sure. Now the question: Is then $P(X_n=Y_n \forall n\in\Bbb N)=1$? ...
0
votes
0answers
22 views

Sufficient conditions for Uniform Law of Large Numbers

I would need a Uniform Law of Large numbers for $f_T(\theta)$ over $\Theta$ when $f$ is the indicator function and, thus, not continuous over $\Theta$. Do you know about any sufficient conditions?
3
votes
0answers
32 views

Convergence of a matrix product

Let $A=o_{a.s.}(1)$; $A:k\times k$ matrix and $Vu=O_p(1)$; $V:k\times k$; $u: k\times 1$. Specifically, $Vu$ converges in distribution to $\mathcal N(0,I_k)$. Can we show that $VAu=o_p(1)$ or ...
0
votes
1answer
31 views

stats - limiting distribution of $X_i$

Suppose $P(X_{n}=i) = \frac{n+i}{3n+3},$ for $i = 0, 1, 2$. Find the limiting distribution of $X_{n}$. Is the cdf $F_{x_{n}}$ like: " if x = 0, $F_{x_{n}} = \frac{n}{3n+3}$; if x = 1, $F_{x_{n}} = ...
0
votes
2answers
30 views

stats - limiting distribution

Let $0 < p < 1$, and let $X_{n}$ have p.d.f. $f_{n}(x) = ( 1 – p ) ( n + 1 ) ( 1 – x )^{n} + p n x^{n – 1}$, for 0 < x < 1, zero elsewhere. Find the limiting distribution of $X_{n}$. ...
0
votes
1answer
49 views

stats - determine limiting distribution

Let $Y_{1} < Y_{2} < ... < Y_{n}$ be the order statistics of a random sample from a distribution with pdf $f(x) = e^{-x} , 0 < x < \infty$, zero elsewhere. Determine the limiting ...
2
votes
1answer
56 views

Convergence in Probability of a Sequence of Exponential Random Variables

If $X$ is an exponential random variable with $\lambda = 3$ and $Y_n = \frac{X^n}{n}$, I am trying to prove whether or not $Y_n$ converges in probability. My original approach was the following: ...
0
votes
0answers
28 views

How to generalize the convergence together lemma

How can we generalize the convergence together lemma that says: If $X_n \to X_{\infty}$ in distribution and $Y_n \to c$ ; $c$ is a constant in distribution/probability . Then $ X_n+Y_n \to ...
0
votes
1answer
34 views

Convergence in Probability for a sequence

Given sample space $\Omega=[0,1]$ and P( ) the uniform probability measure define random variable $X_1,X_2,.....$ by $X_{2n}=\begin{cases} e^{2n} & \text{if $\omega\ \epsilon\ [0,\frac{1}{2n}]$} ...
0
votes
1answer
27 views

$P(\sum_{k=1}^n Y_k <(1-\varepsilon)\log n)=0$ if $Y_n=\min_{1\le k\le n}X_k$ where $X_n$, $n\ge 1$ are i.i.d. Unif$(0,1)$

$P(\sum_{k=1}^n Y_k <(1-\varepsilon)\log n)=0$ if $Y_n=\min_{1\le k\le n}X_k$ where $X_n$, $n\ge 1$ are i.i.d. Unif$(0,1)$. The original question was to show $\sum_{k=1}^n Y_k/\log n$ goes to 1 in ...
2
votes
1answer
70 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 ...
1
vote
0answers
17 views

Is there a relation between Bounded convergence and convergence in distribution?

Is there a relation between Bounded convergence theorem and convergence in distribution ? more specifically, If we have g $\geq$ 0 continuous. and $X_n \to X_{\infty} $ in distribution, Can we ...
1
vote
0answers
65 views

Expectation of the inverse of an almost surely convergent sequence

Let $A_t$ be a bounded positive-semidefinite random matrix sequence, which converges a.s. to a positive definite matrix $A$. Let $B > 0$ be a fixed positive definite matrix. Consider the random ...