Questions tagged [probability-limit-theorems]
For question about limit theorems of probability theory, like the law of large numbers, central limit theorem or the law of iterated logarithm.
1,126
questions
4
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1) Is there other way to provide a proof WITHOUT using the theorem about order of series? 2) Changing the hypothesis how can I solve this?
QUESTION: Let $X_1, X_2, \cdots$ be independents random variables such that $P(X_n=-n^{\theta})=P(X_n=n^{\theta})=1/2$. If $\theta > -1/2$ prove that the Lyapunov condition works and this sequence ...
0
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0answers
45 views
Show that $\frac{S_{n}}{n}$ converges in distribution.
Let $(X_n:n \geq 0)$ be a simple random walk on $\mathbb{Z}$ such that $X_0 = 0$ and $X_n = \sum_{i=1}^n s_j$ where $s_{j}$'s are iid Rademacher variables. Define $S_n = \min(n, \inf\{m \geq n/2:X_m=0\...
0
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1answer
28 views
Durrett's probability Theorem 3.2.11 Proof
I am trying to understand the proof of Thm 3.2.11 in Durrett's Probability Theory and Examples (5th edition), however, I cannot understand (i) -> (ii)
I can understand that the (baby) Skorokhod's ...
-1
votes
0answers
25 views
Prove $\sum X_n$ converges with probability $1$.
I got a problem with this question. Let ${X_n}$ be independent random variables, if $P(|X_n|\le 1) = 1$ and $E(X_n)=0$. Show that if $\sum X_n$ converges in probability, then it converges with ...
1
vote
1answer
68 views
Limiting distribution of $Y_n=\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n X_i^3}$ where $X_i$'s are i.i.d $N(0,1)$
Suppose $X_1,X_2,\ldots,X_n$ are i.i.d standard normal variables. I am looking for the limiting distribution of $$Y_n=\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n X_i^3}$$
Since $E(X_1^3)=0$, I don't think I ...
4
votes
1answer
51 views
How to apply Borel-Cantelli Lemma to find if something converges almost surely to 0.
Suppose that $Z_1,Z_2,...$ are random variables with $Z_n∼Exp(1)$. (We do not assume that these random variables are independent.) Show that $Z_n/\big(\ln^2(n)\big)$ converges to $0$ almost surely.
...
1
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1answer
34 views
Statistic converging to a consistent statistic is convergent--why is this wrong?
I was doing a problem in statistics and I was trying to show that a certain estimator is consistent. Basically, we have an estimator $\hat{\theta}$ which is known to be consistent for some parameter $\...
0
votes
1answer
36 views
Calculating probability limit
I tried taking conditional probability on $\epsilon,$ to change the question in a form where we are taking plim of $\mu^2$ plus some noise. However, I'm having difficulties showing the noise part ...
0
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0answers
33 views
Probability convergence / Convergencia [closed]
I have to prove the almost sure convergence. I really do not know how to start this, i have been struggling with this all the day and i am not going anywhere. The $X_1, X_2,...$ are random variables ...
0
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0answers
14 views
A non- trivial algorithm to estimate the log-exp expected value?
While working on a project, I am facing the an expression in the form
$$\frac1N\log\mathbb{E}_{\textbf{X}}e^{\frac12\textbf{X}^TA\textbf{X}}$$
for $\textbf{X}$ uniformly distributed over $\{\pm1\}^N$. ...
2
votes
1answer
33 views
Finding the limit of $\frac{S_n}{\sigma\sqrt{n}}-\frac{S_{2n}}{\sigma\sqrt{2n}}$ using the central limit theorem
Background Knowledge:
Central Limit Theorem:
Assume that $X_1,\dots,X_n$ are i.i.d random variables with mean $\mu$ and variance $\sigma^2$. >Then, $\lim_{n\to\infty} \sqrt{n}(\frac{\bar{X}_n-\mu}{...
2
votes
1answer
53 views
Problem in convergence in probability involving Poisson distribution.
Let $X_1,X_2,\dots$ be identically distributed and independent random variables. If distribution of $X_1$ is Poisson($\lambda$).
Let $\bar{X_n} =\frac{\sum_{i=1}^{n}X_n}{n}$ and $Y_n=(1-\frac{1}{n})^{...
0
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0answers
59 views
nature of the spread parameter in the Levy distribution
The reflected Levy distribution is given by
$$
p(x):=
\left\{
\begin{array}{lcl}
\sqrt{\frac{\sigma}{2\pi}}e^{-\frac{\sigma}{2(\mu-x)}}\cdot (\mu-x)^{-\frac{3}{2}} &,& \text{if}\; x\leq\mu\\
0 ...
-1
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1answer
28 views
suppose P is a probability function defined in the real line R, prove these two points
$P([0,\infty) = \lim_{n\rightarrow\infty}P([0, n])$
and
$\lim_{n\rightarrow\infty} P([n,\infty])=0$
I have been trying to construct sequences $F_n = E_n/(\bigcup_{i=1}^{n-1}E_i)$ but I can't seem to ...
0
votes
0answers
49 views
Convergence in probability of $\frac{1}{n \ln (n)} \sum_{k=1}^{n} X_{k}$ to $\frac{1}{\ln (2)}$ for i.i.d RVs with $ P\left(X_{1}=2^{k}\right)=2^{-k}$
Let the random RVs $X_{1}, X_{2}, X_{3}, \ldots$ be independent and identically distributed such that $\mathbb{P}\left(X_{1}=2^{k}\right)=2^{-k}$ for $k=1,2,3, \ldots$. I want to prove that
$$
\frac{1}...
3
votes
2answers
37 views
Calculating Upper Bounds for Probability when Extrapolating
This question is from the MIT 6.042J 2005 Final exam.
The problem asks us to find an upper bound on the probability that a program runtime will be $\geq60$ seconds given that the expected runtime is ...
3
votes
1answer
150 views
Aproximation of the Normal Distribution by the Normal Density Function
In Feller's introduction to probability the next lemma is stated:
"As $x\rightarrow \infty$
$\tag1 \frac{1-R(x)}{x^{-1}n(x)} \rightarrow 1$
Where $R(x)$ is the normal distribution and $n(x)$ is ...
0
votes
0answers
25 views
How to determine the expected distribution of the population?
I have a sample with a size n = 30. I know the mean and standard deviation of the sample. I also know the distribution of the sample which is normal. With this data, is it possible to determine the ...
2
votes
1answer
70 views
A probability question, need help.
Suppose we have $N$ people, $N\in\mathbb{Z}_+$. These people arrive at a place at a random time $t_1, t_2, \dots, t_N$, where $t_k\in(0,1)$, following certain distribution (it is unknown in my case, ...
4
votes
2answers
66 views
Uniform integrability of $\sqrt{n} Y_n$ for $Y_n \sim \mathcal{N}(0, \frac{\sigma^2}{n})$
Let $Y_n \sim \mathcal{N}(0, \frac{\sigma^2}{n})$.
Then the set $\{\sqrt{n} Y_n\}_{n \ge 1}$ is uniformly integrable since
$$
\sqrt{n} Y_n \sim \sqrt{n}\mathcal{N}\bigg(0, \frac{\sigma^2}{n}\bigg) = \...
0
votes
1answer
28 views
Why we need symmetry when we apply Chebyshev's inequality?
I have a question about the given solution of this problem.
Problem: A farmer uses $1000$ gallons of water on average per day. The variance of the water rate does not exceed $40000$ gallons. What's ...
1
vote
1answer
35 views
Obtain $\lim _{n \rightarrow \infty} \mathbb{E}\left\{\left|1-X_{n}\right|\right\},$ if it exists for $X_{1},X_{2},X_{3}.. RVs$ and under conditions
$X_{1}, X_{2}, \ldots .$ are nonnegative $\mathrm{RVs}$, such that $ \boldsymbol{X}_{n} \stackrel{\text { a.s. }}{\rightarrow} \mathbf{0}$ (a.s stands for almost sure convergence) and $\lim\limits _{n ...
0
votes
2answers
47 views
Understanding convergence in probability, why do we write $P(|X_n-X|>\epsilon$) instead of $P(|X_n-X|<\epsilon$)?
The definition states :
a sequence ${X_n}$ of random variables converges in probability towards the random variable $X$ if for all $\epsilon \gt 0$ : $\lim_{n\to \infty}Pr(|X_n-X|\gt \epsilon)=0$\
Why ...
0
votes
1answer
27 views
Limsup and in probability convergence [closed]
Suppose I have two sequences $X_{n}$ and $Y_{n}$. We know that $$\limsup_{n\rightarrow\infty}X_{n}=\infty\quad a.s.\text{ and }|X_{n}-Y_{n}|\overset{p}{\rightarrow}0\text{ as }n\rightarrow\infty.$$ ...
1
vote
0answers
46 views
Let $X$ be a random variable and $P(X<u)\leq \Phi (u)\leq P(X\leq u)$ for all $u$. Is X a continuous random variable?
Let $X$ be a random variable and $P(X<u)\leq \Phi (u)\leq P(X\leq u)$ for all $u$. Everything is defined on the real line and $\Phi(\cdot)$ is standard Normal.
Is X a continuous random variable and ...
0
votes
0answers
51 views
Proof of consistency of $M$-estimator
I am trying to prove proposition 2.1 below. As you can see, the proof provided by Huber is not really a proof at all. All the necessary notations and background can be found in my other question on ...
5
votes
0answers
38 views
Convergence rate law of iterated logarithm for a Brownian motion
The law of iterated logarithm has the following implication for a standard Brownian motion $(W_t, t\geq 0)$,
$$
\mathbb{P}\left(\limsup_{t\downarrow 0}\frac{W_t}{\sqrt{2t\ln\left(\ln\left(\frac{1}{t}\...
5
votes
0answers
113 views
Conditions for monotonic convergence in the CLT?
(Revised to assume $\text{support}(X_i)=\mathbb{N}$ in response to counterexamples noted in comments.)
Let $S_n=X_1+...+X_n$, where the $X_i$ are i.i.d. with $\text{support}(X_i)=\mathbb{N}$ and $0<...
0
votes
0answers
19 views
Probability density function of a weighted infinite sum
Hi this is my first question here, I hope someone could help me.
I am struggling with the computation of the probability density function (pdf) of a random variable U, which is the infinite sum of ...
2
votes
1answer
133 views
Question about continuity of stopping time probability in proof of Dirichlet problem in Stein and Shakarchi
In Chapter 6 in the book Functional analysis by Stein and Shakarchi the following theorem (Dirichlet problem) is proved:
$\mathcal{R}$ denotes a bounded open set and a point $y$ is called regular if
$...
1
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0answers
21 views
High dimensional asymptotics of a random projection
Let $X_i$ be iid zero mean, unit variance random vectors taking values in $\mathbb R^{d(n)}$ where ${d(n)}$ is a sequence (dependent on a positive integer $n$) that grows such that ${d(n)}/n \...
1
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0answers
23 views
convergence in densities imply convergence in law
(review question for self-study)
If I have a sequence of densities $f_n$ and a limit density $g$, that is, $f_n \to g$ pointwise, does it follow that the random variables $X_n$ with pdfs $f_n$ ...
0
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1answer
26 views
Limit distribution of gaussian norm and Paley-Zygmund inequality (solution verification)
Let $X_i \sim \mathcal N(0, I_{d(n)})$ be iid normal random vectors taking values in $\mathbb R^{d(n)}$ where ${d(n)}$ is a sequence (dependent on a positive integer $n$) that grows such that ${d(n)}/...
3
votes
1answer
60 views
Bounded random variables $X,Y$ satisfying $\mathbb{E}(X^mY^n) = \mathbb{E}(X^m)\mathbb{E}(Y^n)$ for every $m, n\in\mathbb{N}$ are independent
Suppose $X, Y$ are bounded random variables and we have that for every $m, n$ positive integers, $\mathbb{E}[X^mY^n] = \mathbb{E}[X^m]\mathbb{E}[Y^n]$. Then show that $X, Y$ are independent.
I have ...
3
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2answers
117 views
Slowpoke and Doubles probability — does it converge as the numbers of laps increases?
I posted this on puzzling stack exchange three months ago and it was immediately closed as off-topic, for being a "fairly straightforward probability calculation".
I have the solution to the ...
0
votes
2answers
50 views
Almost sure convergence in parameters preserves convergence in distribution
Let $X_n$, $n\in\mathbb{N}$ denote a sequence of real-valued random variables that converges in distribution to the standard normal distribution. In addition, each $X_n(c)$ is a function of a real-...
2
votes
1answer
35 views
Calculating the plim of a posterior belief about a Bernoulli r.v. formed from observing an infinite sequence of other Bernoulli r.v.'s
Suppose $Y$ is a Bernoulli random variable s.t. Prob$\{Y=1\} = \alpha\in(0,1)$. Further suppose I observe a sequence $\{X_t\}_{t=1}^\infty$ of i.i.d. Bernoulli random variables s.t. $\text{Prob} \{X_t=...
6
votes
1answer
115 views
Does the law of large numbers hold for covering numbers?
I am self-studying empirical process theory.
I have encountered the covering number $N(\delta,\mathcal{G},P)$, as well as the empirical version $N(\delta,\mathcal{G},P_n)$.
It seems intuitive to ...
0
votes
0answers
13 views
Excess Kurtosis and Convergence Rate to Central Limit
For $X_i$ i.i.d., does the kurtosis of the sum $\frac{1}{n}\sum_{i=1}^{n}$ tell us something about the convergence rate to the central limit? The basic central limit theorem tells us that if $X$ has ...
7
votes
2answers
395 views
If $X_n \to X $ in probability then $f(X_n) \to f(X)$ in probability for a Borel function $f$
$(X_n)_n$ is a sequence of identically distributed random variables, $f:\mathbb{R} \to \mathbb{R}$ a Borel function.
Prove that if $X_n$ converges in probability to $X,$ then $f(X_n)$ converges in ...
0
votes
2answers
55 views
Derivative of conditional expectation of integral of stochastic process
Let $T>0$. Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathbb{F}=(\mathcal{F}_u)_{u\in[0;T]}$ be a filtration such that $\mathcal{F}_T=\mathcal{F}$. Let $(\alpha_u)_{u\...
1
vote
0answers
35 views
Is there any method to solve this probability and find an closed form expression?
Suppose I have Shannon's rate equation $$R = \sum_{i=1}^{N}\log\left(1+\frac{P\gamma_i}{\sigma^2}\right).$$ Here $N$ is the number of OFDMA subcarriers, and $\gamma_i$ are channel coefficients that ...
0
votes
1answer
25 views
Proof of Weak Law of Large Numbers with non-zero covariance
Let $ X_{1}, \cdots, X_{n} $ be a sequence of dependent random variables having the same finite mean $ \mu=\mathrm{E}\left(X_{1}\right), $ the same finite variance $ \sigma^{2}=\operatorname{Var}\left(...
0
votes
1answer
23 views
poisson process limit, SLLN
The number of cars that arrive at a parking lot follow a Poisson process with $\lambda > 0$.
Be $T_1,T_2 ...$ The time of arrive between each car, so that $T_1 + T_2 + T_2 ... + T_n$ is the ...
0
votes
1answer
41 views
Normalized sum converges in distribution to $N(0,1)$
Let $X_i$ be a sequence of i.i.d. random variables with $E(X_i)=0$ and $Var(X_i)=\sigma^2>0$. Prove that the distributions of $(\sum_{i=1}^{n} X_i)/ (\sqrt(\sum_{i=1}^{n}X_i^2))$ converge weakly ...
2
votes
1answer
70 views
a.s convergence of a series
$(X_n)_n$ is a sequence of independent random variable, such that $E[X_n]=0.$ Let $Y \in L^2$ such that for all $n \in \mathbb{N}^*,Y-\sum_{k=1}^nX_k$ is independent of $(X_1,...,X_n).$
Prove that for ...
1
vote
1answer
48 views
Law of large number integral asymptotic results
Let $(X_n)$ be a i.i.d. random variables, following a uniform distribution. Using the weak law of large numbers,
$$\lim_{n +\infty} \int_0^1 ... \int_0^1 \frac{\phi(x_1) + ... + \phi(x_n)}{\psi(x_1) + ...
1
vote
0answers
33 views
Convergence of a weighted discrete measure
Preamble
In a recent problem that I have found is about the limit of weighted delta functions in two dimensions.
Background
Let $\Omega \subset \mathbb{R}^2$ be bounded with sufficently smooth ...
2
votes
1answer
82 views
Prove that MGF defined on an open set is infinitely differentiable
I am doing an exercise in the book "Applied Stochastic Analysis" by E-Li-Vanden-Eijnden, and I meet this problem (on Page 26):
(Exercise 1.19) Prove that if the moment generating function $...
6
votes
1answer
150 views
Why do my Stack Exchange reps follow a power law?
I noticed a pattern while looking at my network profile the other day, and I'm wondering if it's a fluke, or if there is something deep to it.
My reps for my top five Stack Exchange communities ...
