Tagged Questions
1
vote
0answers
21 views
Convergence in distribution and convergence of expectation.
Let $\{X_n\}_{n\geq1}$ and $\{Y_n\}_{n\geq1}$ be two sequences of uniformly integrable iid random variables with distributions $F_n(x)$ and $G_n(x)$, respectively. If
$$|F_n(x)-G_n(x)|\leq ...
2
votes
0answers
32 views
measure theory and convergence
1) Let $\Omega=[0,1]$, $F = B([0,1])$, $P$ be Lebesgue measure on $[0,1]$ ($P([a,b])=b-a$).
Set
$$A_n^i:=\left[{\frac{i-1}{n},\frac{i}{n}}\right]$$
and
$$X_n^i(\omega):=\chi_{A_n^i}(\omega)$$
...
1
vote
1answer
30 views
A problem on almost sure convergence
Consider a sequence of random variables defined on the standard unit interval probability space :
$ X_n = 2^n \text{when} \frac{1}{2^n} \leq \omega \leq \frac{1}{2^{n-1}}$
...
2
votes
1answer
82 views
Borel-Cantelli Lemma
I have some difficulties understanding the following:
Let $(X_n)$ be a sequence of independent random variables s.t.
$P[X_n=1]=1−P[X_n=0]=\frac{1}{n}$
After using the Borell Cantelli lemma, I ...
0
votes
0answers
33 views
Does $f(X_n)\to f(X)$ in probability imply $X_n\to X$ in probability?
Does $f(X_n)$ converge in probability to $f(X)$ imply $X_n$ converge in probability to $X$?
0
votes
1answer
17 views
Dose X converges in probability to Y converges in probability to a constant z implies X converges in probability to z
Suppose we have $\frac{1}{n}\sum_j^n X_{ij}$ converges in probability to $Y_i$ and $\frac{1}{n}\sum_y^n Y_{j}$ converges in probability to a constant $z$, where $Y_i$ is not the expectation value of ...
3
votes
1answer
43 views
Convergence of a random variable
I'm working on a problem and I appreciate if you can guide me how to proceed.
Assume that $\sqrt{n}\big(X_n - \mu\big)\rightarrow^d N(0,1) \text{ as } n\rightarrow \infty$.
By the symbol ...
1
vote
1answer
29 views
Continuity of conditional expectation in $L_p$
I'm looking at a probability space $(\Omega,\mathcal{F},P)$. Let $1\leq p<\infty$, and let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. I'm then asked to show that, for $X\in L_p(P)$, ...
1
vote
2answers
43 views
Is it true that $\lim_{n\to\infty}E[X_n] = 0$ if $X_n\to 0$ in probability?
Is there any counter example that:
Let $X_1, X_2,\dots$ be a sequence of random variables that converge to $0$ in probability. That is, for any $c > 0$
$$\lim_{n\to\infty} P(|X_n - 0| > c) = ...
-2
votes
1answer
28 views
Problem about Convergence in Probability (3)
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
32 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 ...
4
votes
1answer
64 views
Is Cesaro convergence still weaker in measure?
I've encountered a question I couldn't answer, and I would appreciate any help:
Is it true that $f_n \xrightarrow{m}0$ $\Rightarrow$ $ \frac{1}{n} \sum_{k=1}^{n}f_k \xrightarrow{m}0$?
Where ...
1
vote
2answers
45 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
0answers
31 views
a sequence of random variables that converge to a constant c in probability but fail to converge to c with probability 1?
Any example that a sequence of random variables that converge to a
constant c in probability but fail to converge to c with probability 1?
1
vote
2answers
50 views
If $a_n \rightarrow 1$, does $X_n \overset{p}{\rightarrow} X$ implies $a_n X_n \overset{p}{\rightarrow} X$?
The title pretty much sums it up.
I'm trying to prove that $S_n^2 \overset{p}{\rightarrow} \sigma^2$. So far, I have:
$S_n^2 = \frac{n}{n-1} \frac{1}{n} \sum_{i=1}^n X_i^2 - \frac{n}{n-1} ...
1
vote
1answer
50 views
Question related to central limit theorem
Let $X_1, X_2, X_3, \ldots, X_n$ be a sequence of $i.i.d$ random variables from Gamma$(1,1)$.
Let $S_n=\sum_{i=1}^nX_i$. Then show that for all $a>0$, $\lim_{n\to\infty}{P[S_n<a]=0}$.
Please ...
3
votes
1answer
117 views
A Coupled Random Walk on the xy-Plane
Consider a point on the $xy$-plane whose position is updated in iterations. In each iteration, the point undergoes, with equal probability, either an $A$- or a $B$-update, defined as follows:
...
4
votes
0answers
158 views
I need help about some compactness arguments
I need help to find some compact sets for my engineering problem. Through this page I learned quite much about it however since I have neither read a book yet nor have an experience I am not able to ...
1
vote
1answer
32 views
$X_n \overset{a.s.}{\longrightarrow} X$ and $X_n \overset{L^1}{\longrightarrow} Y$ implies $X = Y$ a.s.?
If I have a sequence of random variables $\{X_n\}_{n \geq 0}$ such that
$$X_n \overset{a.s.}{\longrightarrow} X \quad\textrm{and}\quad X_n \overset{L^1}{\longrightarrow} Y$$
then is it always true ...
3
votes
0answers
40 views
How to show $nP\{|X|>n\}\to 0$ as $n\to\infty$, but $X$ is not integrable.
How can I construct a random variable $X$ such that: $nP\{|X|>n\}\to 0$ as $n\to\infty$, but $X$ is not integrable.
3
votes
2answers
53 views
Let $\{X_n\}$ be i.i.d integrable r.v.s, show that $\frac{1}{n}\max_{1\leq j\leq n}|X_j|\to 0 \quad \mbox{a.e.}$
This problem is an exercise in Probability theory,independence,interchangeable, martingale(Chow), exercise 4.1.10.
Let $\{X_n,n\geq 1\}$ be independent identical distributed integrable random ...
0
votes
1answer
40 views
Bounded away from 0 almost surely
Suppose that for a succession of random variables $X_n \to 1$ almost surely.
Does it hold that $P[X_n\geq 1/2]=1$ for $n$ large enough?
0
votes
1answer
51 views
Conditions that imply Lindeberg's condition
Suppose that $Z_{1},Z_{2},\ldots$
are i.i.d. random variables with mean 0 and variance 1, and define $X_{nk}=\sigma_{nk}\cdot Z_{k}$
If
$$
\frac{\underset{1\leq k\leq ...
0
votes
1answer
47 views
Probability Central Limit Application
Let $X_{1},X_{2},\ldots$
be independent and suppose that $P\left(X_{j}=\sqrt{j}\right)=P\left(X_{j}=-\sqrt{j}\right)=\frac{1}{2}$
for all $j\in\mathbb{N}
.$ We want to study the asymptotic ...
2
votes
1answer
74 views
Asymptotic Distribution by Central Limit Theorem
Let $X_{1},X_{2},\ldots$ be i.i.d. exponential random variables with mean $1$ and variance $1$. Let $$Y_j=\sqrt{j}\left(X_j-1\right)$$
for all $j\in\mathbb{N}$. I want to find the asymptotic ...
1
vote
1answer
62 views
Weak Convergence to Exponential Random Variable
Assume that $X_1$, $X_2$,... are independent random variables uniformly distributed on $[0,1]$. Let $Y^{(n)}=n\inf\{X_i,1\leq i\leq n\}$. I am asked to prove that it converges weakly to an exponential ...
0
votes
1answer
47 views
Prove that $(X_1 X_2\cdots X_n)^{1/n} \to c$ as $n\to\infty$ where $c$ is a constant
This is a assignment question, a part of my homework. So I need hints to start towards the solution. I was thinking that under the given conditions of the problem the random variables $\log X_1$, ...
0
votes
1answer
59 views
Mean hitting times and monotone convergence theorem
Take the random walk on $\mathbb{N}= \{0,1,2, \ldots\}$ which starts at $x$ and jumps to the right with probability $p$ and to the left with probability $1-p=q$. Let $T_p$ denote the first hitting ...
2
votes
2answers
50 views
Convergence of an Ergodic process
I'm having trouble working through the math of the problem below. I believe the problem as described is an ergodic process. I've written a simple simulation of the problem, that converges to 66.6...6% ...
6
votes
2answers
80 views
Closeness of probability measures
Consider a set of probability measures $\{P_n\}$. Suppose $P_n$ converges to $P^*$ weakly and
$$
\int \xi^2 P_n(d\xi)< \infty.
$$
Can we claim
$$
\int \xi^2 P^*(d\xi)<\infty
$$
and
$$
\lim_{n\to ...
2
votes
1answer
84 views
Do we need extra assumption for convergence in probability
I am trying to convince myself that if $X_i$ are iid random variables with $\mathbb{E}(X_i)=\mu$ and $Var(X_i)=\sigma^2<\infty$ then $\frac{1}{n}\sum_{i=1}^n X_i^2$ converges in probability to ...
1
vote
1answer
57 views
Convergence of a non-increasing sequence of random variables to a constant.
I have a non-increasing sequence of random variables $\{Y_n\}$ which is bounded below by a constant $c$ $\forall \omega \in \Omega$. i.e $\forall \omega \in \Omega$, $Y_n \geq c$, $\forall n$. Is it ...
3
votes
3answers
85 views
Two equivalent definitions of a.s. convergence of random variables.
Could someone prove that $\mathbb{P}[\omega:\lim_{n\to\infty}X_n(\omega) = X(\omega)] = 1$ iff $\lim_{n\to\infty}\mathbb{P}[\omega:\sup_{k>n}|X_k(\omega) - X(\omega)|>\epsilon] = 0$ ? Here ...
1
vote
3answers
63 views
Difference between $\lim_{n\to\infty}P(|X_n-X|<\epsilon)=1$ and $\lim_{n\to\infty}P(|X_n-X|=0)=1$
What is difference between $\lim_{n\to\infty}P(|X_n-X|<\epsilon)=1$ and $\lim_{n\to\infty}P(|X_n-X|=0)=1$? The former is the definition of convergence in probability. I'd like to get some intuitive ...
2
votes
1answer
62 views
Proving $\lim_{n\to\infty}P\,(X_n=X) = 1$ when the sequence of random variables $X_n$ converges to $X$ a.s.
This is a homework problem. The problem is proving that $\lim_{n\to\infty}P\,(X_n=X) = 1$ when the sequence of random variables $X_n$ converges to $X$ almost surely.
I think the problem is ...
0
votes
1answer
97 views
An example of a sequence of random variables in Wikipedia, that converges in probability but not almost surely
This Wikipedia page says that the sequence of random variables $X_n$ that assume $1$ with probability $1/n$ and assume $0$ with probability $1 - 1/n$ converges to $0$ in probability but not almost ...
2
votes
1answer
125 views
Convergence with probability one and convergence in law
Suppose $\sqrt{n}(X_n - \mu)\overset{d}{\longrightarrow}N\left(0,\sigma^2\right)$. Prove that $X_n \overset{p}{\longrightarrow}\mu$ is true.
I see that it's not true in general and I can ...
2
votes
1answer
122 views
Limit inferior, limit superior and Borel Cantelli lemmas
I am having trouble trying to understand the topic of my question. For reference please use Virtual Laboratory of Probability and Statistics.
Let's start with limit superior:
$$\limsup_{n \to \infty} ...
2
votes
1answer
60 views
Monotone convergence
Consider $X$ as non-decreasing non-negative function. Consider $\mu$ and $\nu$ as two probability measures on $(\mathbb{R},\mathcal{B})$ for which we know $\mu([t, \infty)) \geqslant \nu([t, \infty)) ...
1
vote
2answers
82 views
Do these random variables converge to 0?
Suppose a sequence $U_n$ of random variables satisfies the following conditions: For each $n$,
$$P(U_n = 1) = 1/n$$
and
$$P(U_n = 0) = 1 - 1/n.$$
Can someone tell me whether this converges almost ...
2
votes
1answer
116 views
A problem about almost sure convergence.
Suppose $X_1, X_2, ...$ are independent random variables with $P(X_n=\sqrt n)=1/\sqrt n $ and $P(X_n=0)=1-1/\sqrt n$. Let $S_n=X_1+X_2+\cdots+X_n$ for all $n$. Show that $S_n/n \rightarrow 1 \space ...
5
votes
3answers
87 views
Example for a simple game of chance where the average value converges fast to the expected value
I want to introduce the concept of expected value (for a discrete random variable, finite case) in a high school class. My first idea was to start with a simple game of chance and let the students ...
3
votes
1answer
109 views
Question on convergence almost surely
I'm working on the problem below. I've proved one side of it, but I need help on the other side.
Consider $X_1, X_2, \ldots$ as independent random variable where:
$\Pr(X_n = k) = (1-p_n)p_n^k$ for $k ...
0
votes
0answers
36 views
Proof of convergence in distribution of a discrete random variable [duplicate]
Possible Duplicate:
Proof of convergence in distribution of a discrete random variable
I'm working on a question on "convergence in distribution" and I appreciate if you could guide me on ...
1
vote
1answer
62 views
A problem about almost sure converges of a random variable with infinite variance.
Let $X_1,X_2,...$ be independent random variables such that for all positive integers $k$, we have $P\left( X_{k}=k^{2}\right) =\dfrac {1} {k^{2}}$, $P\left( X_{k}=2\right)=\dfrac{1}{2}$, and $P\left( ...
1
vote
1answer
58 views
A question about convergence in probability.
Suppose $X_2,X_3,\ldots$ are independent random variables.
Assume that $X_k$ has the exponential distribution with parameter $\lambda_k=\dbinom{k}{2} $ for all $k$, which means $ E[X_k] = ...
4
votes
1answer
231 views
Convergence in distribution (weak convergence)
Let $X_n$ and $X$ be random variables taking values in the metric space $(S,d)$.
The sequence $(X_n)_n$ is convergent to $X$ in distribution (or weakly) if
$E[f(X_n)] \to E[f(X)]$ for all $f:S\to R$ ...
2
votes
2answers
140 views
Non-decreasing sequence of random variable convergence in probability implies it also converges almost surely.
The problem stated as follow:
Suppose $X_1 \leq X_2 \leq \cdots$ and $X_n \xrightarrow[]{p} X$. Show that $X_n \to X$ a.s.
I'm think about may be use the continuity of probability measure, but I ...
2
votes
1answer
112 views
Independence of two limits
Let $(X_n)$ and $(Y_n)$ be two sequence of random variables. $(X_n)$ and $(Y_n)$ are independent to each other. If $(X_n)$ and $(Y_n)$ have limits in distribution. $(X_n)$ tends to $X$, and $(Y_n)$ ...
2
votes
1answer
144 views
Convergence in mean squared?
I'm working the question below:
$X_1, X_2, \dots $ are uncorrelated random variables with $E(X_i) = \mu_i $ and $\operatorname{var}(X_i)/i \rightarrow 0$ as $i \rightarrow \infty$. Now, let $S_n = X_1 ...
