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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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Approximation of mean of a rational function of random variables

Let $\xi_i$ with $i\in\{1,\dots,n\}$ be iid random variables and let $Q(x,y)$ be a rational function. I need to compute one $x$ that satisfies $$\frac{1}{n}\sum_{i=1}^n Q(x,\xi_i)=0.$$ This is a ...
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22 views

Projection estimator in hilbert space [on hold]

Let $ f \in \mathbb{L}^2[0,1] $ $ Y_i = f(i/n) + \epsilon_i $ with $ \epsilon$ independent and centered With basis $ \phi_1 (x) = 1 , \: \: \phi_{2k} (x) = \sqrt{2}cos(2\pi kx) \: \: \phi_{2k+1} (...
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1answer
21 views

Swap the maximum and limiting process

Suppose a sequence $(X_n)_{n \geq 0}$ of random variables converge in distribution to a random variable $X$, where $X_{n}>0$ and $X\sim $exp$(1)$. Suppose further that $(Y_{n})_{n \geq 0}$ converge ...
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1answer
16 views

Weak convergence of measures and boundedness

Let $(X,d)$ be a Polish space. Suppose that $(\mu_n)$ is a sequence of probability measures on $X$ such that $\mu_n\to \mu$ weakly, where $\mu$ is a probability measure on $X$. Is it true that there ...
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1answer
27 views

The limiting distribution of the product of two random variables

Suppose $X_{n}\to X$ in distribution and $Y_{n}\to 0$ in probability, then I want to show that $X_{n}Y_{n}\to 0$ in probability. My idea is that choose $\delta>0$ and we have $\mathbb {P}(|X_{n}...
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1answer
64 views

How do we need to apply the martingale convergence theorem here?

Let $(E,\mathcal E,\mu)$ be a measure space $E_0\in\mathcal E$ with $\mu(E_0)\in(0,\infty)$ $n\in\mathbb N$ $B_1,\ldots,B_n\subseteq\left.\mathcal E\right|_{E_0}:=\left\{B\cap E_0:B\in\mathcal E\...
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7 views

The rate function in large deviations

Why do we use rate functions in the definition of Large Deviations. That is why do we require the function to be lower-semicontinuous?
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Is argmin attained equivalent to local convexity?

I have the following optimization problem: $$ \theta^* \in \arg \min_\theta f(\theta) $$ $$ \widehat{\theta^*} \in \arg \min_\theta \widehat{f}(\theta) $$ With $f(\theta) = \mathbb{E}(g(X; \theta))$ ...
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0answers
31 views

U statistic asymptotic distribution

Can someone provide me hints as to where to attack part (c) from? I tried using results from U-statistics, but having trouble for connecting distribution of $$L_n(\hat{\theta}_n) - E[1\{Y_i > Y_j\} ...
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convergence of submartingale

We know that if $(X_n)_n$ is a submartingale such that $\sup_nE[X^+_n]<+\infty$ then $(X_n)_n$ converges a.s to an integrable random variable. I am searching for a proof which doesn't use Doob's ...
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2answers
47 views

weak convergence: Show that $\mathbf{P}(X_n = q) \to 0$ as $n \to \infty$ when the cdf $F$ of the limit $X$ is continuous at $q$

The question might seem easy to some of you, but I'm still pretty new to probabilty theory (and also not very deep in mathematics) and appreciate any help. Here is the full statement I'm trying to ...
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1answer
87 views

General Convergence of Exponential of Function to Dirac Delta

Let $f:\mathcal{X}\rightarrow \mathcal{F}$ be a function with a unique, positive maximum at $x^*=\arg\sup_{x}f(x)$ where $\mathcal{X}$ and $\mathcal{F}$ are both bounded. Let $f$ be locally smooth ...
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Question about series convergence motivated by selecting subsets of size 3

TLDR; I would like to prove that there is an $L \in (0,1)$ such that: $$\sum_{j=0}^{k/2} \frac{3^{k-j} \binom{n/3}{j}\binom{n/3-j}{k-2j}}{\binom{n}{k}} \to L \qquad (k = \lceil n^{2/3} \rceil, n \to \...
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35 views

Central limit theorem and integrability

If $(Y_n)_n$ is a sequence of independent random variables and identically distributed, and if $\frac{\sum_{k=1}^nY_k}{\sqrt{n}}$ converges in distribution to a random variable Y, does this mean that $...
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49 views

Example for Lévy's continuity theorem

I am searching a sequence of RV $(X_n)$ for which we prove a convergence in distribution to a random variable $X$, using the fact that the characteristic functions $(\varphi_n)_n$ converges pointwise ...
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2answers
85 views

Variant of the Strong Law of Large Numbers

Let $X_1,X_2,\ldots$ be a i.i.d. sequence of random variables with uniform distribution on $[0,1]$, with $X_n: \Omega \to \mathbf{R}$ for each $n$. Question. Is it true that $$ \mathrm{Pr}\left(\...
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1answer
26 views

Rigorous construction of the pointwise limit of a sequence of random variables

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space and let $$X_1,X_2,X_3,... \: \Omega \rightarrow \mathbb{R} $$ be a sequence of random variables. Moreover, let there be an event $A \...
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0answers
29 views

weak convergence and CDF

we know that if $(X_n)_n$ is a sequence of real random variables, then it converges in distribution to a random variable $X$ if and only if $\lim_nF_{X_n}(x)=F(x)$ at every point $x \in \mathbb{R}$ ...
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27 views

stopping time almost surely finite

Let $(X_n)_n$ be a sequence of independent random variables and identically distributed such that $P_{X_1}=p\delta_1+q\delta_{-1}+r\delta_0$ where $0 \leq p,q,r<1$ and $p+q+r=1.$ Let $\alpha, \...
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1answer
59 views

Probability of $\limsup_n\left\{|\frac{\max_{1\leq k \leq n}X_k}{\ln(n)}-1| >\epsilon\right\}$

Let $(X_n)_n$ be a sequence of independent random variables and identically distributed, following the exponential distribution with parameter 1. Let $0<\epsilon<1.$ I want to compute $P\left(\...
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28 views

If $(X_i,ξ_i)$ are mutually independent, then $\text E[|d^{-1/2}\sum_{i=1}^dg'(X_i)ξ_i|^{2q}]$ is bounded by a constant only depending on $q$

Let $d\in\mathbb N$, $X$ be a $\mathbb R^d$-valued random variable on a probability space $(\Omega,\mathcal A,\operatorname P)$ with density $$p(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\in\mathbb R^d$...
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21 views

Non existence of limit of Gibbs distribution

We consider an N-particle system given by the gradient dynamics $dX(t)= -N\nabla H_N (X(t)) dt + \sigma d\beta(t)$ in $(\mathbb{R}^d)^N$, where $\sigma$ is a positive constant. We assume that for $x=...
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1answer
63 views

If $(W_i)_{i\in\mathbb N}$ obeys the strong law of large numbers, what can we say about $\liminf_{d\to\infty}\frac1{d^{2\alpha}}\sum_{i=1}^dW_i$?

Let $d\in\mathbb N$ and $W_1,\ldots,W_d$ be mutually independent, identically distributed and square-integrable real-valued random variables on a probability space $(\Omega,\mathcal A,\operatorname P)$...
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2answers
59 views

Given two biased coins, find probability that one gets $k$ heads before the other

Assume two coins, where the probabilities of flipping heads for the first is $a$ and similarly the heads probability for the second is $b$. I start flipping the two coins in parallel and want to ...
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1answer
21 views

weak convergence and Cumulative distribution function

Let $X$ be a real random variable and $(X_n)_n$ be a sequence of real random variable, such that : $$\exists \alpha>0; \lim_n\int_{\mathbb{R}}|F_{X_n}(x)-F_X(x)|^{\alpha}dx=0$$ I need to prove ...
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1answer
63 views

$X_1, X_2$, … independent variables with support in [0,1] , mean $\mu\in(0,1)$, $Y_n=X_1X_2…X_n,\;\;\;n\geq1$. Study convergence of $Y_n$ [closed]

$X_1, X_2$, ... independent variables with support in the interval [0,1] and equal mean $\mu\in(0,1)$, but not necessarily identically distributed. Study the convergence in quadratic mead and ...
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How do I find probabilities for certain values of a piecewise probability mass function?

Consider the following probability mass function $f$: $ f(x)= \begin{cases} 0&\text{}\, x\lt 0\\ \frac{x}{4}&\text{}\, 0\leq x\lt 1\\ \frac{1}{2}+\frac{x-1}{4}&\text{}\, 1\leq x\lt 2\\ ...
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39 views

Conclude convergence in probability from uniform convergence on a set of limiting probability 1

Let $\kappa_d$ be a Markov kernel on $(\mathbb R^d,\mathcal B(\mathbb R^d))$ for $d\in\mathbb N$ $f_d:\mathbb R^d\times\mathbb R^d\to\mathbb R$ be Borel measurable for $d\in\mathbb N$ $B_d\in\mathcal ...
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1answer
31 views

Coin flipping limit

Let $S_{n}\sim\text{Bin}\left(n,p\left(n\right)\right)$ where $p\left(n\right)$ is the unique solution to the equation $\delta\left(p\left(n\right),n\right)=0$ with $\delta$ being continuous and ...
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1answer
24 views

Replacing continuous variables in a limit with a sequence

I have a question regarding the nuts and bolts involved in the proof of the limit of CDFs. The statement is that Proposition: Let $X$ be a random variable with CDF $F_X(.)$. Then $F_X(.)$ posses the ...
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1answer
103 views

Strong law of large numbers for a scaled sequence of normally distributed random variables

Let $f\in C^3(\mathbb R)$ be positive $g:=\ln f$ $d\in\mathbb N$, $$p_d(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\in\mathbb R^d$$ and $\lambda^d$ denote the Lebesgue measure on $\mathcal B(\mathbb R^...
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55 views

Show that the limit inferior of the sum of third moments of mutually independent Gaussians tends to negative infinity

Let $d\in\mathbb N$, $\sigma:=\ell d^{-\alpha}$ for some $\ell>0$ and $\alpha>0$, $Y$ be a Gaussian $\mathbb R^d$-valued random variable with mean $0$ and covariance matrix $\sigma^2I_d$$^1$, $Z$...
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0answers
23 views

Variance of limiting distribution equal to limit of variance

I have a possibly basic question, which I am not sure on whether or not it is true. Suppose we have a sequence of identically distributed, but not necessarily independent random variables $X_n$ on a ...
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1answer
22 views

Converge in probability with ratio involves both sample average and n.

Suppose $x_1,...x_m$'s are i.i.d. chi-square random variables with 1 degree-of-freedom; $y_1,... y_n$ are i.i.d. chi-square random variables with 1 degree-of-freedom and $\frac{m}{n + m} \rightarrow a ...
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0answers
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An “Edgeworth Series-esque” approximation of ratio distribution using Monte Carlo methods. What is this method called?

I am hoping someone can provide me with the name of the following technique that appears to estimate the density of the ratio of independent random variables (although it could work for other ...
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2answers
83 views

About the limit $\lim_{n \to +\infty} \frac{1}{n^2} \sum_{1 \le a,b \le n} \frac{1}{ \mathrm{gcd} (a,b)} $

This is not homework. My question is: Prove or disprove: $$\lim_{n \to +\infty} \frac{1}{n^2} \sum_{a,b=1}^n \frac{1}{ \mathrm{gcd} (a,b)} = \frac{\zeta(3)}{\zeta(2)}$$ This would represent the ...
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1answer
36 views

Simple problem with convergence in distribution.

Let ${\displaystyle X_{n}\ {\xrightarrow {\overset {}{d}}}\ a\quad}$, where a is constant. Is that true that $P(X_n<a) \rightarrow 0$ ? My intuition tell's me that this is true so i tried to ...
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269 views

If $X_i∼fλ$, $Z∼\mathcal N(0,I_d)$ and $Y=X+\ell d^{-α}Z$ with $α<1/2$, then $\liminf_{d→∞}\text E\left[1∧\prod_{i=1}^d\frac{f(Y_i)}{f(X_i)}\right]=0$

Let $f\in C^3(\mathbb R)$ be positive $g:=\ln f$ $d\in\mathbb N$, $$p_d(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\in\mathbb R^d$$ and $\lambda^d$ denote the Lebesgue measure on $\mathcal B(\mathbb R^...
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Multiple Random Variables convergence

I have this random sample $(X_{1}; Y_{1});(X_{2}; Y_{2});...$ with $E(X) = \mu_{x}$ e $E(Y ) = \mu_{y}$ finite and positive, $Var(X) = \sigma_{x}^{2}$ and $V ar(Y ) =\sigma_{y}^{2}$ finite and ...
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1answer
48 views

Convergence of a Bernoulli sequence of random variables

I am facing a big chalenge to formalize the answer. Thinking all day. Any help? Hint? Consider $X_{1}$,$X_{2}$,$X_{3}$,$X_{4}$,$X_{5}$,... i.i.d Bernoulli(i.e $P(X_{i}=1)=p),P(X_{i}=0)=1-p)$: i) ...
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35 views

Showing that sequence converges in distribution

Consider the following sequence of random variable: $X_{1}$,$X_{2}$,$X_{3}$,$X_{4}$,$X_{5}$, ... i.i.d with $E(X_{i}=\mu)$ and n even. Define: $P_{n}=\frac{2}{n}\sum\limits_{i=1}X_{2i}$ and $I_{n}=\...
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1answer
26 views

These sequences converges in distribution and in probability

I have this exercise. Let $U$ and $V$ two independents random variables with Normal Distribution(0,1). Let, $X_{1}=U$,$X_{2}=V$,$X_{3}=U$,$X_{4}=V$,$X_{5}=U$, ... I) The sequence $X_{1},X_{2},X_{3},...
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1answer
46 views

Why this sequence of random variable converges in probability but does not converge in distribution

How can I, formally, explain why this sequence of random variable below converges in probability but does not converge in distribution? Let $X_{1},X_{2},X_{3},...$ random variables i.i.d such that: $...
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1answer
55 views

Why this sequence converges to $0$ almost surely

I have this sequence of random variables: $$X_{1}(w) = \mathbb{1}_{(1/2,1]}(w), X_{2}(w) = \mathbb{1}_{(0,1/2]}(w), X_{3}(w) = \mathbb{1}_{(3/4,1]}(w), X_{4}(w) = \mathbb{1}_{(1/2,3/4]}(w) ...$$ I ...
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2answers
50 views

Why this sequence of R.V converge in distribution, but doesnt in probability

Why this sequence of R.V converge in distribution, but doesnt in probability? Probability space $([0,1],B,m)$. ($B$ consists of all Borel sets of $[0,1]$, $m$ is the Lebesgue measure.) Let $X_{2n}(ω)...
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1answer
76 views

Tail event example

In Durrett's Probability (4th edition), an example of a tail event (an event in the tail sigma-field $\bigcap_n \sigma(X_n, X_{n+1}, \dots)$) is the following: given independent random variables $X_1, ...
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2answers
71 views

Tilted sum of independent random variables

Let $(X_i)_i$ be a sequence of centered i.i.d. random variables with finite variance. Is it true that $$\frac{\sum_{i=1}^{\lfloor n^{0.6} \rfloor}X_i}{\sqrt{n}}\stackrel{\mbox{a.s.}}{\longrightarrow} ...
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1answer
35 views

Polya's theorem implies Glivenko-Cantelli for continuous distributions

Let $X_n$ denote a sequence of random variables. Here is Polya's theorem: Suppose that $X_n \rightsquigarrow X$ (convergence in law) for a random vector $X$ with a continuous distribution function. ...
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2answers
46 views

Why this sequence converges to $0$ and not to $1$ in probability?

I didn't understand why this sequence bellow converges to $0$ in probability? Shouldn't this sequence converge to $1$ in probability? This is the definition of convergence in probability:
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1answer
25 views

Tail weight of product distributions

Are there any general results relating the tail weight of two (or more) probability distributions to the tail weight of their product distribution (in particular, on the assumption that the ...