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.
0
votes
0answers
7 views
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 ...
0
votes
0answers
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} (...
0
votes
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 ...
1
vote
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 ...
0
votes
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}...
0
votes
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\...
0
votes
0answers
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?
3
votes
0answers
23 views
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 $f(\theta) = \mathbb{E}(g(X; \theta))$ ...
1
vote
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 $$L_n(\hat{\theta}_n) - E[1\{Y_i > Y_j\} ...
1
vote
0answers
20 views
convergence of submartingale
We know that if $(X_n)_n$ is a submartingale such that $\sup_nE[X^+_n]<+\infty$ then $(X_n)_n$ converges a.s to an integrable random variable.
I am searching for a proof which doesn't use Doob's ...
1
vote
2answers
47 views
weak convergence: Show that $\mathbf{P}(X_n = q) \to 0$ as $n \to \infty$ when the cdf $F$ of the limit $X$ is continuous at $q$
The question might seem easy to some of you, but I'm still pretty new to probabilty theory (and also not very deep in mathematics) and appreciate any help. Here is the full statement I'm trying to ...
1
vote
1answer
87 views
General Convergence of Exponential of Function to Dirac Delta
Let $f:\mathcal{X}\rightarrow \mathcal{F}$ be a function with a unique, positive maximum at $x^*=\arg\sup_{x}f(x)$ where $\mathcal{X}$ and $\mathcal{F}$ are both bounded. Let $f$ be locally smooth ...
2
votes
0answers
27 views
Question about series convergence motivated by selecting subsets of size 3
TLDR; I would like to prove that there is an $L \in (0,1)$ such that:
$$\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 \...
1
vote
0answers
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 $...
1
vote
0answers
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 ...
5
votes
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
$$
\mathrm{Pr}\left(\...
0
votes
1answer
26 views
Rigorous construction of the pointwise limit of a sequence of random variables
Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space and let
$$X_1,X_2,X_3,... \: \Omega \rightarrow \mathbb{R} $$
be a sequence of random variables. Moreover, let there be an event $A \...
1
vote
0answers
29 views
weak convergence and CDF
we know that if $(X_n)_n$ is a sequence of real random variables, then it converges in distribution to a random variable $X$ if and only if $\lim_nF_{X_n}(x)=F(x)$ at every point $x \in \mathbb{R}$ ...
1
vote
0answers
27 views
stopping time almost surely finite
Let $(X_n)_n$ be a sequence of independent random variables and identically distributed such that $P_{X_1}=p\delta_1+q\delta_{-1}+r\delta_0$ where $0
\leq p,q,r<1$ and $p+q+r=1.$ Let $\alpha, \...
2
votes
1answer
59 views
Probability of $\limsup_n\left\{|\frac{\max_{1\leq k \leq n}X_k}{\ln(n)}-1| >\epsilon\right\}$
Let $(X_n)_n$ be a sequence of independent random variables and identically distributed, following the exponential distribution with parameter 1.
Let $0<\epsilon<1.$ I want to compute $P\left(\...
1
vote
0answers
28 views
If $(X_i,ξ_i)$ are mutually independent, then $\text E[|d^{-1/2}\sum_{i=1}^dg'(X_i)ξ_i|^{2q}]$ is bounded by a constant only depending on $q$
Let $d\in\mathbb N$, $X$ be a $\mathbb R^d$-valued random variable on a probability space $(\Omega,\mathcal A,\operatorname P)$ with density $$p(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\in\mathbb R^d$...
0
votes
0answers
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=...
2
votes
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 $(\Omega,\mathcal A,\operatorname P)$...
1
vote
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 ...
0
votes
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 ...
1
vote
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 ...
2
votes
0answers
24 views
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\\
...
1
vote
0answers
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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
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^...
0
votes
0answers
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$$^1$, $Z$...
1
vote
0answers
23 views
Variance of limiting distribution equal to limit of variance
I have a possibly basic question, which I am not sure on whether or not it is true.
Suppose we have a sequence of identically distributed, but not necessarily independent random variables $X_n$ on a ...
0
votes
1answer
22 views
Converge in probability with ratio involves both sample average and n.
Suppose $x_1,...x_m$'s are i.i.d. chi-square random variables with 1 degree-of-freedom; $y_1,... y_n$ are i.i.d. chi-square random variables with 1 degree-of-freedom and $\frac{m}{n + m} \rightarrow a ...
1
vote
0answers
15 views
An “Edgeworth Series-esque” approximation of ratio distribution using Monte Carlo methods. What is this method called?
I am hoping someone can provide me with the name of the following technique that appears to estimate the density of the ratio of independent random variables (although it could work for other ...
4
votes
2answers
83 views
About the limit $\lim_{n \to +\infty} \frac{1}{n^2} \sum_{1 \le a,b \le n} \frac{1}{ \mathrm{gcd} (a,b)} $
This is not homework.
My question is:
Prove or disprove: $$\lim_{n \to +\infty} \frac{1}{n^2} \sum_{a,b=1}^n \frac{1}{ \mathrm{gcd} (a,b)} = \frac{\zeta(3)}{\zeta(2)}$$
This would represent the ...
0
votes
1answer
36 views
Simple problem with convergence in distribution.
Let ${\displaystyle X_{n}\ {\xrightarrow {\overset {}{d}}}\ a\quad}$, where a is constant.
Is that true that $P(X_n<a) \rightarrow 0$ ?
My intuition tell's me that this is true so i tried to ...
11
votes
0answers
269 views
If $X_i∼fλ$, $Z∼\mathcal N(0,I_d)$ and $Y=X+\ell d^{-α}Z$ with $α<1/2$, then $\liminf_{d→∞}\text E\left[1∧\prod_{i=1}^d\frac{f(Y_i)}{f(X_i)}\right]=0$
Let
$f\in C^3(\mathbb R)$ be positive
$g:=\ln f$
$d\in\mathbb N$, $$p_d(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\in\mathbb R^d$$ and $\lambda^d$ denote the Lebesgue measure on $\mathcal B(\mathbb R^...
1
vote
0answers
14 views
Multiple Random Variables convergence
I have this random sample $(X_{1}; Y_{1});(X_{2}; Y_{2});...$ with $E(X) = \mu_{x}$ e $E(Y ) = \mu_{y}$ finite and positive, $Var(X) = \sigma_{x}^{2}$ and $V ar(Y ) =\sigma_{y}^{2}$ finite and ...
2
votes
1answer
48 views
Convergence of a Bernoulli sequence of random variables
I am facing a big chalenge to formalize the answer. Thinking all day.
Any help? Hint?
Consider $X_{1}$,$X_{2}$,$X_{3}$,$X_{4}$,$X_{5}$,... i.i.d Bernoulli(i.e $P(X_{i}=1)=p),P(X_{i}=0)=1-p)$:
i) ...
2
votes
0answers
35 views
Showing that sequence converges in distribution
Consider the following sequence of random variable: $X_{1}$,$X_{2}$,$X_{3}$,$X_{4}$,$X_{5}$, ... i.i.d with $E(X_{i}=\mu)$ and n even.
Define: $P_{n}=\frac{2}{n}\sum\limits_{i=1}X_{2i}$ and $I_{n}=\...
1
vote
1answer
26 views
These sequences converges in distribution and in probability
I have this exercise.
Let $U$ and $V$ two independents random variables with Normal Distribution(0,1).
Let, $X_{1}=U$,$X_{2}=V$,$X_{3}=U$,$X_{4}=V$,$X_{5}=U$, ...
I) The sequence $X_{1},X_{2},X_{3},...
3
votes
1answer
46 views
Why this sequence of random variable converges in probability but does not converge in distribution
How can I, formally, explain why this sequence of random variable below converges in probability but does not converge in distribution?
Let $X_{1},X_{2},X_{3},...$ random variables i.i.d such that: $...
2
votes
1answer
55 views
Why this sequence converges to $0$ almost surely
I have this sequence of random variables:
$$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) ...$$
I ...
2
votes
2answers
50 views
Why this sequence of R.V converge in distribution, but doesnt in probability
Why this sequence of R.V converge in distribution, but doesnt in probability?
Probability space $([0,1],B,m)$. ($B$ consists of all Borel sets of $[0,1]$, $m$ is the Lebesgue measure.)
Let $X_{2n}(ω)...
1
vote
1answer
76 views
Tail event example
In Durrett's Probability (4th edition), an example of a tail event (an event in the tail sigma-field $\bigcap_n \sigma(X_n, X_{n+1}, \dots)$) is the following: given independent random variables $X_1, ...
4
votes
2answers
71 views
Tilted sum of independent random variables
Let $(X_i)_i$ be a sequence of centered i.i.d. random variables with finite variance. Is it true that
$$\frac{\sum_{i=1}^{\lfloor n^{0.6} \rfloor}X_i}{\sqrt{n}}\stackrel{\mbox{a.s.}}{\longrightarrow} ...
0
votes
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. ...
1
vote
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:
0
votes
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 ...
