0
votes
0answers
8 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
24 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
29 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
17 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?
1
vote
0answers
27 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
24 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
20 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
30 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 ...
1
vote
0answers
46 views

Almost sure convergence of a sum of independent exponential random variables?

I'm in difficult with this exercise... I hope someone can help me. Let $X_1,X_2,...$ be independent random variables, $X_n\sim \exp(\lambda_n)$, where $$0 < \lambda_n\rightarrow \lambda , \lambda ...
2
votes
1answer
19 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
20 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
30 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
25 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
65 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
16 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
60 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
46 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
19 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
32 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
18 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
40 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
34 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
39 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
39 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
32 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
30 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
31 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
37 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
86 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 ...
1
vote
1answer
38 views

questions on mean-square convergence for a AR(P) example

In the following example related to AR(P) process, I have two questions I marked these two questions with two different colors. Question 1) (I marked with yellow), why that sum is mean-square ...
1
vote
1answer
75 views

Almost sure weak convergence of empirical measure

Do empirical measures converge weakly to the measure almost surely? In particular suppose $\mu$ is a Borel probability measure on $\mathbb R^d$ and that $X_1,X_2,\dots$ are IID drawn from $\mu$. Let ...
1
vote
0answers
47 views

Convergence in distribution for $\frac{Y}{\sqrt{\lambda}}$

Given a sequence of independent r.v's $\{X_n\}_{n\geq 1}$ such that $P(X_n=x)=\frac{1}{2}$ if $x=-1$ and/or $x=1$ Let $N\in Po(\lambda)$ be independent of $\{X_n\}_{n\geq 1}$ and we set that ...
2
votes
1answer
64 views

If $P(X=k)=1/2^k$ for every $k\ge1$ and $h_k(x)$ converges to identity function $h(x)=x$, does $E(X|h_k(X))$ converge almost surely?

$P(X=k)=1/2^k$,k=1,2,3... $h_k(x)$ converges to identity function,i.e $h(x)=x$. The question is that whether $E(X|h_k(X))$ converges almost surely. If true, how to prove it? If not, please give an ...
0
votes
0answers
32 views

Question around Central Limit theorem for sum of binomial with limit of sum as sum of poisson

Let P be Poison random variable of intensity T,i.e. P ~ $P_{o}(T) $, and $Y_{k} ~ B(1,p) $, $k \geq 1$ be an iid sequence of random variables. Define $N_{t} = \sum_1^t P_{i} $ and $X_{s} = ...
1
vote
0answers
21 views

Sufficient onditions for convergence in distribution and a.s.

Given $X{}_{j}$ for $j=1,2,3,\ldots$ are i.i.d. continuous random variables with density $f\left(x\right)$, where $f\left(x\right)=0$ for $-\infty<x<1.$ Let $Y{}_{n}=\sqrt[n]{X_{1}\cdot ...
2
votes
1answer
49 views

does convergence of $X_{n}$ imply convergence of $VarX_{n}$?

Can we conclude that $VarX_{n}\rightarrow VarX$ if $X_{n}$ converges to $X$ almost surely and $VarX_{n}<\infty$ and $VarX<\infty$? What about if $X_{n}$ converges in mean square?
4
votes
2answers
50 views

Limiting case of Binomial(n,p)/n?

Let the random variable $X$ have distribution $X \sim \text{Binomial}(n,p)$. Let $Y = X/n$. What is the limiting distribution of $Y$, as $n \to \infty$? Does it have a simple distribution? Of ...
0
votes
0answers
14 views

Probability Convergence of Distributions

Suppose that $X_n$ and $Y_n$ are random variables over possibly different probability spaces but $d(X_n) = d(Y_n)$, that is, they induce equal Borel probability measures on $\mathbb{R}$. If $X_n\to 0 ...
1
vote
0answers
29 views

Convergence in mean with logarithm

The problem is to prove that $E[\log|X_1-\bar{X}_n|]$ converges towards $E[\log|X_1|]$ for i.i.d. continuous random variables $X_1,\ldots,X_n$ with $E[X_i]=0$ and $Var[X_i]=1$, for example for Laplace ...
1
vote
1answer
257 views

Conditions for convergence of moments given uniform convergence of distribution functions

Setup: Let $S_n = n^{-1} \sum_{i=1}^n X_n$ denote a sample mean and let $S_n^*$ denote a stationary bootstrap re-sample of $S_n$. Let $F_n(x)$ denote the cumulative distribution function of $\sqrt{n} ...
0
votes
0answers
45 views

Does a a.s. convergence of $X_n\to X$ and a $E|X_n|^{2+\epsilon}\to E|X|^{2+\epsilon}$ combined, imply a $L^2$ convergence?

Does a a.s. convergence of $X_n\to X$ and $E|X_n|^{2+\epsilon}\to E|X|^{2+\epsilon}$ convergence combined, imply a $L^2$ convergence $X_n\to X$ (namely, $E|X_n- X|^2 \to 0$)? Or more practically: ...
0
votes
0answers
26 views

Convergence in probability of excheangeable random variables

I would like to prove the following result: Let $X_1,X_2,\ldots$ be a sequence of i.i.d. random variables (non necessarily Gaussian) and let $Y_{n,k}=\log|X_k-\bar{X}_n|$ if $X_k\ne\bar{X}_n$ and 0 ...
1
vote
1answer
40 views

Prove $\lim_{n\to \infty} P(n^{1/2} \cdot (\bar{x} - 1 ) < t) = P(Z < t) $ ($X$ exponential)

$x_{i}$, $i = 1, 2, 3, \dots$ i.i.d. from exponential ($\lambda = 1$) $Z$-standard normal r.v. $$ \lim_{n\to \infty} P(n^{1/2} \cdot (\bar{x} - 1 ) < t) = P(Z < t) $$ It seems like I could ...
1
vote
1answer
37 views

Does the following function converge to zero almost certainly?

Let $X_n(w) = nI_{(n-1,n)}(w)$ be a function on the real numbers with the Borel sigma algebra. Consider Lebesgue measure, denoted by $\mu$. Can this be a random variable? Does this converge to 0 ...
2
votes
0answers
63 views

Understanding the difference between uniform convergence , convergence everywhere and almost sure convergence

I'm trying to get the intuition behind convergences . I understand convergence in distribution and convergence in mean-square error. However I'm vague about the difference between - almost sure ...
1
vote
1answer
24 views

Extending the one series theorem to $k$-dependent sequences

The problem goes: For some $k \ge 1$, let $\{X_i\}$ be a sequence of $k$-dependent random variables. That is, for each $n\ge 1$, $\sigma(X_1,\ldots,X_n)$ and $\sigma(X_{n+k+1},\ldots)$ are ...
3
votes
1answer
86 views

$X_n \rightarrow X$ in probability. Does $X_n \Rightarrow X$?

The question is the title. But please read this post to see my thoughts - I am interested in your feedback on this argument. Suppose $X_n \rightarrow X$ in probability. Does $X_n \Rightarrow X$? I ...
0
votes
0answers
34 views

On convergence in distribution

Let $\delta$ and $\theta$ be two estimators of the scaler parameters $\delta_0$ and $\theta_0$, respectively. Suppose that conditions are satisfied such that as the sample size $n \to \infty$, then ...
3
votes
1answer
77 views

Almost sure convergence proof

Cud someone please explain the proof of $ P(X_n \to X)=1 $ iff $$ \lim_{n \to \infty}P(\sup_{m \ge n} |X_m -X|>\epsilon) \to 0 $$. Im not able to understand the meaning of the various sets they ...