1
vote
0answers
29 views

Prove that a.s.$\lim\limits_{t\to\infty}\frac{N_t}{t}=\frac{1}{\mu}$

Consider a diligent janitor who replaces a light bulb the instant it burns out. Suppose that the first bulb is put in at time zero and let $X_i$ be the lifetime of the i-th bulb. Suppose ...
-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 ...
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 ...
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.
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
25 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 ...
1
vote
0answers
33 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 ...
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 ...
3
votes
2answers
53 views

How to show a random variable diverges from its mean?

I am trying to prove the following statement which seem not to be very complicated but I cannot find a straightforward way to prove it (can it even be wrong?): Suppose $X_n$ are a sequence of random ...
1
vote
0answers
19 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 ...
1
vote
1answer
19 views

Confusion about random variables and convergence in probabilty and distribution

I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution: In my syllabus ...
2
votes
1answer
52 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
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 ...
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 ...
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 ...
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 $n$ ...
1
vote
0answers
20 views

Can convergence in distribution say anything about mean-square convergence rate?

Suppose I have a sequence $\{x_n\}$ that I already know converges in the mean-square-sense ($\lim_n E |x_n|^2\to 0$). Suppose further I know that the sequence $\{x_n\}$ converges in distribution to ...
1
vote
1answer
43 views

A small question about convergence in distribution

If we have two sequence of random variables $X_n$ and $ Y_n$ such that $P(|X_n - Y_n|) > \epsilon \to 0$ for any $ \epsilon > 0$. If $X_n$ converges in distribution to some distribution (for ...
1
vote
1answer
34 views

Uniform $L^1$ convergence of a double sequence of random variables.

Let $X_n^m$ be a double sequence of integrable random variables such that $X_n^m \rightarrow_n X^m$ in probability, where $X^m$ is another sequence of integrable random variables. Assume that such ...
2
votes
1answer
197 views

convergence of sequence of random variables and cauchy sequences

Let $(X_n)$ be a sequence of real random variables on $(\Omega,\mathcal A,\mathbb P)$. Then 1. and 2. are equivalent: There exists a random variable $X$, s.t. $X_n\to X$ $P$-almost sure for $n\to ...
0
votes
1answer
138 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 ...
-1
votes
1answer
84 views

Convergence of the empirical distribution function

Let $\alpha\in\mathbb R^d$, with $\alpha\neq 0$. Take a sequence of iid random variables $X^{1},\dots,X^{n}$ with values in $\mathbb R^d$ and denote $R$ the cdf of $\alpha X^1$ (where the product is a ...
1
vote
1answer
44 views

Let $Y_{n}$ be a bounded sequence with mean $\mu$ and variance $\sigma_{n}^{2}$, I want to show the follow [closed]

Let $Y_{n}$ be a bounded sequence of independet random variables with mean $\mu$ and variance $\sigma_{n}^{2}$ , Suppose $Y_{n} \xrightarrow{a.e}Y$ and $\sigma_{n}^{2} \to0. $ then show that $Y=\mu ...
0
votes
2answers
148 views

Sum of two independent random variables converges in distribution [closed]

Show that if $X_n$ and $Y_n$ are independent random variables for $1 \le n \le \infty$, $X_n \Rightarrow X_{\infty}$, and $Y_n \Rightarrow Y_{\infty}$, then $X_n + Y_n \Rightarrow X_{\infty} + ...
2
votes
1answer
104 views

To show $ \liminf \ X_n/\log(n) = 0 $ almost surely

For i.i.d s $ \{X_n,\ n\geq 1\} $ that are $ exp(1) $ random variables, I need to prove that $$ \liminf \ \frac{X_n}{\log(n)} = 0\ \ \ \ \text{almost surely}$$ I have found that $\mathbb{P}(X_n ...
1
vote
1answer
105 views

Convergence in probability of product and division of two random variables

How can I prove the following: Let $X_i$ and $Y_i$, $i = 1, \ldots, n$, $X$ and $Y$ be random variables defined on the probability space $(\Omega, \mathcal F, \mathbb P)$ and assume that $X_n$ ...
3
votes
0answers
131 views

Show that $(X_{n},Y) \to^{\mathcal{D}} (X,Y)$ AND if $X=h(Y)$ where $h$ is a Borel function that $X_{n}\to^{P} X$

Let $X_{n}$, $X$, and $Y$ be real-valued r.v.'s all defined on the same space $(\Omega, \mathcal{A},\mathbb P)$. Assume that $\lim_{n \to \infty}\mathbb E\{f(X_{n})g(Y)\}=\mathbb E\{f(X)g(Y)\}$ ...
0
votes
1answer
27 views

The weak law of great numbers from the central limit theorem.

I couldn't derive the weak law of large numbers from the central limit theorem for iid random variables with $0 < \operatorname{Var}(X) < \infty$. The central limit theorem gives $$\frac{\sum ...
1
vote
1answer
50 views

Solution to Billinglsey (1995) problem 20.22

Let $Y_1\leq Y_2\leq ...$ be random variables s.t. $\mathrm{plim} Y_n = Y$. Show that $Y_n \to Y$ with probability 1. Some hints? My strategy would be to prove that $\sum P(\lvert Y_n -Y \rvert > ...
3
votes
1answer
90 views

Three series of Kolmogorov

Let $X_n\geqslant 0$ be a sequence of independent random variables. The following are equivalent: $i) \sum_{n=1}^{\infty}{ X_n} <\infty$ a.s $ii)$ $\sum_{n=1}^{\infty}{ \mathbb P(X_n>1)} ...
1
vote
1answer
80 views

Convergence in probability of the means of a uniformly integrable sequence

Suppose $\{X_n\}$ is a uniformly integrable sequence of independent random variables with zero mean. Prove that $1/n \sum\limits_{i=1}^n X_i \rightarrow0 $ in probability. I tried to ...
3
votes
1answer
224 views

Weak convergence in the Skorohod space $D([0, T])$ $\forall T$ implies Weak convergence in $D([0, \infty))$?

Assume that $Z_n$ are random variables taking value in the Skorohod space $D([0, \infty),Y)$ (endowed with its usual Skorohod topology) of right-continuous functions $[0, \infty) \to Y$, where $Y$ is ...
1
vote
1answer
54 views

Convergence in Probabiliy

$X_n$ converges to $0$ in probability and a sequence of constants $|c_n|$ diverges to infinity. Can someone please help me prove that $X_n - c_n$ does not converge to $0$. (I am totally blank as to ...
1
vote
1answer
49 views

Convergence in Probability of a sum of RVs

Let $X_1, X_2, ..., X_n, ...$ be independent random variables. Assume that for each $n$, the random variable $X_n$ is distributed uniformly on $[0,n]$. Find a sequence $a_n$ such that $(X_1^2 + ... + ...
2
votes
1answer
230 views

Convergence in probability of the sum of random variables

Let $\{\xi_t\}$ be i.i.d random variables (may not have expectation in general) and $\{z_t^{n}\}$ - i.i.d. Bernoulli random variables ($t\in \mathbb{Z}_+$) such that $$ z_t^{n} = \begin{cases} 0 \ ...
1
vote
1answer
32 views

Convergence in probability of the sum of scheme of series

Could you please help with this one. It looks like smth simple but I can't figure it out. Let $\{x_{in}\}, \ i=1,\dots, n, \ n=1,\dots,\infty$ be the scheme of series of random variables. For each ...
2
votes
3answers
213 views

Proof of Levy's zero-one law

Let $(\Omega, \mathcal{F},\mathbb P)$ be a probability space and let $X$ be a random variable in $L^1$. Let $(\mathcal{F}_k)_k$ be any filtration, and define $\mathcal{F}_{\infty}$ to be the minimal ...
4
votes
1answer
59 views

What does this hint mean and how is it useful to solve the problem?

I am doing a problem on convergence of random variable. There was a hint given, but I am still struggling to understand the hint. Here is the problem: Let $Y_n$ be uniformly distributed on ...
0
votes
0answers
77 views

Integration by part for Lebesgue integral

I tried to prove one theorem in convergence of random variables and found myself in a little bit of trouble when doing integration by part. The reason being it involves a Lebesgue integral which I am ...
0
votes
1answer
77 views

Finding the value of distribution function of a converging random variable

There is this example in a note that I think this is supposed to be a simple problem, but I still find it not as straightforward. Consider a sequence of random variables $X_n\equiv1/n,X\equiv0$. Then ...
0
votes
1answer
46 views

Convergence of random variable to a negative constant

Let $X_n$ be the sequence of R.Vs and $X_n\overset{P}{\rightarrow}A$ (or $X_n\rightarrow A$ almost surely) where $A<0$ I want to prove that $Pr[X_n < 0] \rightarrow 1$ (or $X_n < A$ almost ...
1
vote
1answer
81 views

$X_n \sim \text{Exponential}(\lambda_n)$, independent, $\sum 1/\lambda_n = \infty$, then, $\sum X_n=\infty$ a.s.

Let $\{X_n\}$ be a sequence of independent Exponential random variables with mean $$ E(X_n)=\frac{1}{\lambda_n}, $$ where $$ 0 < \lambda_n < \infty. $$ If $$ \sum \frac{1}{\lambda_n} = \infty, ...
1
vote
0answers
60 views

convergence of discrete random variables with finite entropy

Let $Z$ be the set of discrete random variables on some probability space. Define the quantity $d(X_1,X_2)=h(X_1 \mid X_2)+h(X_2 \mid X_1)$ between two random variables $X_1, X_2 \in Z$. For $X \in Z$ ...
1
vote
2answers
69 views

Is it true that $\lim_{n\to\infty}E[X_n] = 0$ if $X_n\to 0$ in probability?

Is there any counter example that: Let $X_1, X_2,\dots$ be a sequence of random variables that converge to $0$ in probability. That is, for any $c > 0$ $$\lim_{n\to\infty} P(|X_n - 0| > c) = ...
-1
votes
1answer
41 views

Problem about Convergence in Probability (3) [closed]

Let $X_1,X_2,\dots$ be a sequence of random variables that converge to $0$ in probability. That is, for any $\varepsilon > 0$, $\lim\limits_{n\rightarrow +\infty} Pr(|X_n-0|>\varepsilon) = 0$ ...
-1
votes
2answers
39 views

Problem about convergence in Probability (2) [duplicate]

Let $X_1,X_2,\dots$ be a sequence of random variables with $$ \lim_{n\rightarrow+\infty}E\left[\left|X_n\right|\right]=0 $$ Is it true or false that the sequence $X_n$ must converge to $0$ in ...
0
votes
1answer
116 views

Convergence to a constant in probability but not almost surely

Please give a example that a sequence of random variables that converge to a constant $c$ in probability but fail to converge to $c$ with probability $1$. Thanks very much.
1
vote
2answers
68 views

Must the sequence $X_n$ converge to $0$ in probability?

Let $X_1, X_2,\dots$ be a sequence of random variables with $\lim_{n\to +\infty} E[|X_n|] = 0$. Is it correct or wrong that the sequence $X_n$ must converge to $0$ in probability?
0
votes
2answers
114 views

Convergence of sum of random variables

Let $X_n$, $n\geq 0$, be i.i.d. random variables such that: $\mathbb E(X_1)=0$, and $0<\mathbb E(|X_1|^2)<\infty$. Given that $\alpha >\frac{1}{2}$, I need to show that ...