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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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20 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
66 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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Probability convergence of $aX_{n} \rightarrow aX$

I'm trying to prove that $aX_{n} \rightarrow aX$ (where $a\in \mathbb{R}$ and $X_1, X_2,...$ are a sequence of random variables such that $X_{n} \rightarrow X$) and am struggling to understand if my ...
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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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0answers
13 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 ...
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1answer
27 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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0answers
33 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
25 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
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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
54 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
37 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
40 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
66 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
23 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
45 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
14 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 ...
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0answers
32 views

Central Limit Theorem- Lapunow, Linderberg

I have a task: $(X_n)_{n>=1}$ are independent. $P(X_n=0)=1/n$ and $P(X_n=2n)=1-1/n$. Check the weak convergence $\frac{X_1+X_2+X_3+....+X_n}{n}-n$. I tried use the Lapunow theorem or Linderberg ...
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0answers
29 views

Asymptotics of expected value of maximum of normal variables

Let $X_n = \max (|Z_1|, \dots, |Z_n|) $ where $(Z_i)$ are i.i.d. standard normally distributed random variables. I'd like to show $\mathbb{E} [X_n] \sim \sqrt{2 \log n}$. I've shown already that $$\...
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1answer
30 views

Convergence of the sum of iid scaled by $n^\alpha$

I am interested in the convergence of the sequence $\mathbb{P}(|X_1+...+X_n|/n^\alpha<z)$ where $z>0$, $\{X_n\}_n$ is an i.i.d. sequence with mean zero and finite variance. I can easily prove ...
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1answer
23 views

Lindeberg CLT condition on Discrete Uniform independent sequence of random variables

There is a worked exercise in my book, however there is a line that I am not sure sure about. I understand all of the work before and after this line to finish the proof. Here is what we are given: ...
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1answer
28 views

Asymptotic independence on the tail field is equivalent to a probability measure's being trivial

Let $(\Omega, \mathcal B, P)$ be a probability space and $\mathcal A = \bigcap_n \mathcal{A}_n$, where $\mathcal{B} \supset \mathcal{A}_1 \supset ... \supset \mathcal{A}_n \supset ...$ is a decreasing ...
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1answer
46 views

$\lim_{n \to \infty} E[e^{-X*\frac{t}{n}}]^n \approx \lim_{n \to \infty} E(1-\frac{E[X] t}{n})^n = e^{-E[X]*t}$

X is a random variable. I don't understand this passage. Please someone can explain it to me each equality at a time? Thank you! $\lim_{n \to \infty} E[e^{-X\frac{t}{n}}]^n \approx \lim_{n \to \...
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1answer
32 views

Estimators and Confidence intervals

I was curious as to what the relationship between probabilistic values, estimators and confidence intervals are. I was wondering, if I have an estimator of some parameter $\lambda$, and a probability ...
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0answers
34 views

How to prove that the sequence of distributions converges to the Dirac measure?

I'm working on parameter estimation with Bayesian inference. I've already implemented MCMC-algorithm (Metropolis-Hasting) and for test purpose, I simulated a case when I assume a true parameter and ...
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1answer
58 views

independence implies integrability?!

Let $(X_n)_n$ a sequence of independent random variables and identically distributed. Let $$Y_n=\frac{1}{n}\sum_{k=1}^nX_k \ \ \ \ and \ \ \ W_n=\frac{1}{n-1}\sum_{k=1}^n(X_k-Y_n)^2$$ If $Y_n$ and $...
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0answers
31 views

necessary and sufficient condition on $(u_n)$ for a.s convergence of $\sum_n u_nX_n$

I want to know if there is a necessary and sufficient condition on the sequence $(u_n)_n$ such that $\sum_n u_nX_n$ converges a.s. wehere $(X_n)_n$ are independent. I know that this series converges ...
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0answers
38 views

Expectation of exponential random variables

Can someone help me understand the parts I have highlighted in the proof? I have not understood how $|2c...|$ is upper bounded by $1$ and how one computes the expectation of an exponential random ...
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1answer
23 views

Inverse Gaussian Distribution and the Central Limit Theorem

Let the random variables $Y_1,\ldots,Y_n$ be independent and identically distributed (i.i.d.) (standard) Inverse Gaussian random variables with parameters $\mu$ and $\lambda$. Then, let the random ...
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1answer
35 views

Limiting Distribution on order statistics

Here is the question: Find the limiting distribution of the quartile ratio defined as $\frac{X_{(3n/4):n}}{X_{(n/4):n}}$ (the third quartile divided by the first quartile)for the exponential ...
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1answer
120 views

independent variables implies integrability and same distribution

I want to solve this difficult problem, so I need your help, I want to prove that if $X$ and $Y$ are independent real variables such that $X+Y$ and $X-Y$ are independent, then $X^2$ and $Y^2$ are ...
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0answers
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Sum of independent random variable - possibility of introducing concept similar to integration

Let's consider a PDF family - $f(x;i)$, where $i$ ($i$ can vary continuously with 0 to 1) is the parameter and wrt parameters RVs are independent. Now I define a RV like this $X_s=\sum_{i=0}^{i=1} {\...
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0answers
31 views

Probability of getting between 490 and 500 sixes throwing a die 3000 times.

If I throw a die $3000$ times, what is the approximate probability that the number of sixes lies between $490$ and $500$? I am trying to use the central limit theorem or Chebychev inequality to solve ...
0
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1answer
25 views

Almost sure convergence of quadratic form x'Ax

Let $x_n$ be an $n\times 1$ vector of random variables, and $A_n=(a_{ij,n})$ be an $n\times n$ constant matrix. Suppose that $n^{-1}x_n'x_n$ converges almost surely to some limit as $n\rightarrow \...
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0answers
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Feller's Proof of the Basic Limit Theorem for Recurrent Events

I'm having trouble understanding the last passage in Feller's proof of the Theorem above, which can be found in section XIII.11 of "An Introduction to Probability Theory and its Applications, vol.1". ...
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1answer
35 views

Why does the strong law of large numbers require random variables with the same variance?

The strong law of large numbers states that for an infinite sequence of i.i.d random variables $X_1, X_2, ...$, and $\bar{X_n} = \frac{1}{n}(X_1 + \cdots + X_n)$, $$\bar{X_n} \rightarrow \mu \text{ ...
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2answers
24 views

Convergence in probability towards a constant

Prove that if a sequence $X_1,X_2..$ of random variables satisfies the following conditions: $$1.lim_{n\to\infty}\mathbb{E}(X_n)=a,\space a\in\mathbb{R}$$ $$2.lim_{n\to\infty}\mathbb{V}(X_n)=0$$ Then ...
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1answer
27 views

Higher Dimensional Random Walks

I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in an arbitrary direction (uniformely on the unit sphere $S^1$), the expected distance ...
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1answer
27 views

Proving Brownian motions have bounded quadratic variation

I am looking into arguments which yield the result $(dB_t)^2=dt$ and authors first start by letting P be a partition $P=\{t_0, t_1, ..., t_n\}$ of the interval $ [0,T]$ where $t_i=\frac{i}{n}T$ and ...
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4answers
311 views

What is the probability that the sum of digits of a random $k$-digit number is $n$?

Let $X_1, X_2, \dots, X_k$ each be random digits. That is, they are independent random variables each uniformly distributed over the finite set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Let $S = X_1 + X_2 + ...
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0answers
120 views

Where does the bias come from in LogLog and HyperLoglog?

In the original LogLog paper from P. Flajolet, the cardinality estimation function is presented as $$E := \alpha_m m2^{1 \over m} {^{\sum_{j} M^{(j)}}}$$ Where $\alpha$ is needed to correct the ...
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1answer
44 views

Empirical distribution of sorted Gaussian numbers

I wrote a small program that does the following : Pick $N$ independent standard Gaussian numbers (expected value : 0, standard deviation : 1). Call that list $L=\{y_1, \ldots, y_N\}$. Sort that list ...
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1answer
195 views

$X_1,..,X_n$ i.i.d random variable and discrete, local limit theorem

Let $X_1,..., X_n$ be i.i.d discrete random variables which take their value in $\mathbb{Z}$ with a non trivial and finite support. Let $S_n = X_1+...+ X_n$ Prove the existence of $ 0 < C_1 &...
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0answers
29 views

CLT for decaying random variables

Suppose $X_1,X_2,...$ are bounded random variables with compact support, and $\frac{X_1+...+X_n}{\sqrt{n}}\overset{d}{\longrightarrow}N(0,1)$. Is there neccessarily a central limit theorem for the ...
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1answer
51 views

How to prove that the following estimators are biased and consistent?

Given a random variable $X$ following a geometric distribution with parameter $p.$ Then one estimator that can be obtained by considering the second moment $E[X^{2}]=\frac{2-p}{p^2}$, which is $$\hat{...
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2answers
36 views

$\lim_{x\rightarrow+\infty} x^{\alpha+\beta}P(|X|>x)=0\Leftrightarrow\lim_{x\rightarrow+\infty}x^{\alpha}E||X|^{\beta}\cdot1_{\{|X|>x\}}|=0$

I have met with a probability problem which I have no idea to deal with. It says: Let $\alpha>0$, $\beta\geq0$, prove: $\lim_{x\rightarrow+\infty} x^{\alpha+\beta}\mathbb{P}(|X|>x)=0\...
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0answers
12 views

Convergence of row superposition of null array to Poisson point process

Background: The limit of superposition of infinite number of independent point processes is a Poisson process under certain conditions. The conditions for the limit process to exist and be Poisson is ...
3
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3answers
151 views

Limiting a sequence of moment generating functions

I was trying to solve the following problem: Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables with the probability mass function $P\{X_n = \pm1 \} = \frac{1}{2}$, $n \in \...
1
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2answers
41 views

Probability limits of random variable sums

I have $X_1, X_2, X_3, \cdots$ which are independent random variables with the same non-zero mean ($\mu\ne0$) and same variance $\sigma^2$. I would like to compute $$\lim_{n\to\infty} P[\frac{1}n\sum^...
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1answer
14 views

Does this converge in distribution in this case?

Let $X_i$ be i.i.d. sequence of random variables with finite second moment. Then, by the Central limit theorem, $\frac{\sqrt{n}(\bar{X}-E(X))}{\sigma_X} \to N(0,1)$ in distribution. Let $\alpha_n$ ...