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.

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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 ...
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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\...
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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 ...
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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 ...
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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 ...
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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. ...
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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 $\...
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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 ...
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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 ...
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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 $\{\pm1\}^N$. ...
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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}{...
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1answer
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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})^{...
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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 ...
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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 ...
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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}...
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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 ...
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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 ...
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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 ...
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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, ...
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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) = \...
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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 ...
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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 ...
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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 ...
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1answer
27 views

Limsup and in probability convergence [closed]

Suppose I have two sequences $X_{n}$ and $Y_{n}$. We know that $$\limsup_{n\rightarrow\infty}X_{n}=\infty\quad a.s.\text{ and }|X_{n}-Y_{n}|\overset{p}{\rightarrow}0\text{ as }n\rightarrow\infty.$$ ...
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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 ...
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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 ...
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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}\...
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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<...
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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 ...
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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 $...
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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 \...
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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 $f_n$ ...
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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
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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 ...
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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 ...
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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-...
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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=...
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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 ...
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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 ...
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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 ...
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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\...
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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 ...
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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(...
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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 ...
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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 ...
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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 ...
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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) + ...
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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 ...
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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 $...
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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 ...

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