1
vote
1answer
117 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
18 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
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. ...
1
vote
1answer
58 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
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} ...
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.
0
votes
0answers
18 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
27 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
26 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
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
24 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
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}) ...
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 ...
1
vote
1answer
42 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
22 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
16 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
18 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
43 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
24 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
11 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
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 ...
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
21 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
29 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
48 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
50 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
25 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
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 ...
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
63 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 ...
4
votes
1answer
51 views

Convergence of Random Variables in mean

If $$E[|X_n-X|^r]\rightarrow0$$ prove that $$E|X_n^r|\rightarrow E|X^r| $$ for every $r\ge 1$ This is the very notation used. I believe it should be: $$E[|X_n|^r]\rightarrow E[|X|]^r $$ Attempt I ...
0
votes
0answers
24 views

Rate of convergence almost surely

I would like to see the rate (ordre) of convergene almost surely of a scheme they said from that $ \forall \beta \in ]0,1/2[,\qquad > Dt^{\beta}.\sup_{t\in[0,T]}|\widehat{X_{t}} -X_{t}| ...
0
votes
0answers
42 views

Properties convergence of random variables

I want to understand the concepts of convergence of random variables better and therefore I wanted to find out how certain concepts that are straightforward for canonical convergence behave in this ...
1
vote
0answers
25 views

Convergence rate of an estimator

Say we are interested in estimating some unknown real scalar parameter $\alpha$ using data. Suppose the estimator $\widehat \alpha_N$ of $\alpha$ using the data is consistent. I want to know what it ...
3
votes
1answer
52 views

Convergence almost everywhere implies convergence in measure, the proof thereof

Let $(E, \mathcal{E}, \mu)$ be a measure space, and $(f_n)_{n\in\mathbb{N}}$ and $f$ be measurable functions $(E, \mathcal{E}, \mu)\longrightarrow (\mathbb{R}, \mathcal{B})$. The first part of Theorem ...
0
votes
0answers
53 views

convergence in law of Cauchy random variables.

$(X_n)_{n\geq 1}$ is a sequence of random variables independent and identically distributed with Cauchy distribution of parameter $\alpha$ , $\alpha > 0$. Let $$ Y_n= \dfrac{\max_{1 \leq k \leq ...
1
vote
1answer
46 views

How to show that a sequence of random variables doesn't converge in probability?

Say, we have the sequence of random variables defined on $\Omega=[0,1]$ with uniform distribution: $$X_n(\omega) := \begin{cases} \omega, & \text{if $n$ is odd} \\ 1-\omega, & \text{if $n$ ...
0
votes
0answers
43 views

SLLN for nearly-identical observations

When observations $X_i\sim\mu$ are IID the usual SLLN along with separability of the underlying metric space yield that the empirical measure $\hat\mu_n=\frac1n\sum_{i=1}^n\delta_{X_i}$ converges ...
3
votes
2answers
48 views

Almost sure convergence + convergence in distribution implies joint convergence in distribution?

I'm wondering, if I have two sequences of random variables $(X_n)$ and $(Y_n)$, defined on the same probability space, such that $X_n\stackrel{a.s.}{\rightarrow}X$ and $Y_n\stackrel{d}{\rightarrow}Y$, ...
0
votes
0answers
33 views

Why is Expectation Maximization algorithm guaranteed to converge to minimum, even local?

I have read a couple of explanations of EM algorithm (e.g. from Bishop's Pattern Recognition and Machine Learning and from Roger and Gerolami First Course on Machine Learning). The derivation of EM is ...
1
vote
1answer
37 views

Almost everywhere convergence of a function of rv

Let f(x) a continuous function on $\mathbb {\bar{R}^+}$(extended positive real line, $x\in(0,\infty]$). Take $y\in \mathbb R^+$. We can say that $\lim_{y\to 0}\frac{x}{y}=\infty $ almost everywhere ...
2
votes
1answer
47 views

Uniform convergence in integrated survival function implies uniform convergence of distribution functions?

For a probability distribution function $F$ supported on a bounded interval $[a,b]$, the integrated survival function (ISF) is defined as $$\Psi_F(t)=\mathbb E_F\max\{X-t,0\}=\int_t^b(1-F(x))d x.$$ ...
0
votes
1answer
136 views

Almost sure convergence of a product of random variables

Let $X_1, X_2, ...$ be a sequence of random variables such that: $X_i = \begin{cases} 1 &\mbox{with probability } p = 1-1/n^2 \\ 2 & \mbox{with probability } p = 1/n^2 \end{cases} $ and let ...
4
votes
2answers
68 views

Convergence of random increment of two bins

If bins A and B are initialized to 1, and then I continually increment one of them by 1 with a probability proportional to their values, would A/B ever converge? To demonstrate ...