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Questions tagged [law-of-large-numbers]

For questions about the law of large numbers, a classical limit theorem in probability about the asymptotic behavior (in almost sure or in probability) of the average of random variables. To be used with the tags (tag:probability-theory) and (tag: limit-theorems).

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Let $X_n$ be a uniform distribution on $(-1,1)$. Let$ Y_n$ ~ Cauchy(0,1). Everything independent.

Let $X_n$ be a uniform distribution on $(-1,1)$. Let$ Y_n$ ~ Cauchy(0,1). Everything independent. Let $Z_n$ = $X_n$ + $Y_n$ I want to study the law convergence of the sample mean of $Z_n$. That is: ...
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31 views

How do I find the numbers of large population? Statistics

I have a vector $$[10000, 1000, 800, 700, 500, 100, 12, 12, 12, 11, 8 , 7,6,4,3,1,0]$$ And I want to find out how many large numbers there are in my vector, which I call my population. In this case,...
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55 views

Strong law of large numbers for Poisson rvs with different parameter

Let $X_n$ be independent Poisson random variables with $E[X_i] = \mu_i$, and let $Y_n = X_1+...+X_n$. I want to show that if $\sum_n \mu_n = \infty $ then $Y_n/E[Y_n] \rightarrow 1$ almost surly. What ...
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50 views

Strong Law of Large Numbers imply Weak Law [closed]

If the Strong Law of Large Numbers imply the Weak Law, why do we have a Weak Law of Large Numbers?
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31 views

Covariance and Law of Large numbers

Say I am taking the average value of the product of two dependent random variables $X$ and $Y$ sampled an infinite amont of times. That is I am computing $\lim_{n \rightarrow \infty} E \left[ \sum_{i=...
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30 views

Problem on Weak Law of Large Numbers

Question- $X_n$ can take only two values $n^a$ and $-n^a$ with equal probabilities. Show that we can apply weak law of large numbers to the sequence of independent random vatiables ${X_n}$ if $a<\...
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Suppose $X_n$ are iid with a symmetric distribution. Then $\Sigma_n \frac{X_n}{n}<\infty ~\mathrm{a.s. iff }~\mathbb{E}|X_1|<\infty$

It seems to be solved by using Kolmogorov strong law of large numbers. Why $X_n$ have symmetric distribution?
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What is the difference between ergodicity and the law of large numbers?

I want to begin by saying that I know absolutely no measure theory. To my knowledge, roughly speaking a stochastic process is ergodic if its time average converges to the expectation (space average) ...
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Weak law of large numbers for reciprocal of normal

In two different journal articles: The First Negative Moment of Skew-t and Generalized Student's t-Distributions in the Principal Value Sense and The first negative moment in the sense of the ...
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2answers
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LLN when $E X = \infty$

Does there exist a random variable $X$ with $\mathbb{E}X = \infty$ and some constants $a_n \to \infty$ such that if $X_1, X_2, \ldots$ are iid $\sim X$, then $$\lim_{n \to \infty} \frac{X_1 + X_2 + \...
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$X_i = i^{\delta}Z_i$, find the range of $\delta$ for which the law of large numbers and the central limit theorem are valid

$Z_1,Z_2,...$ are i.i.d., their expected value is zero, their variance $\sigma^2$, and $E[|Z_i^2|] = m_3 < \infty$. $X_i = i^{\delta}Z_i$, find the range of $\delta$ for which the law of large ...
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1answer
32 views

Question about book solution to estimate $E[e^{XY}]$ when $X$ and $Y$ are independent exponential RVs with $\lambda = 1$

Let $X$ and $Y$ be independent exponential random variables with mean 1. (a) Explain how we could use simulation to estimate $E[e^{XY}]$. (b) Show how to improve the estimation approach in part (a) by ...
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1answer
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Strong law of Large numbers (SLLN4)

I'm trying to prove SLLN(4): Let $\{X_n : n\geq1\}$ be a sequence of $L^1-$integrable independent random variables on a probability space and $S_n = \sum_{j=1}^nX_{j}$ for every $n\geq1$. Let $\phi :...
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Using strong law of large numbers to construct a measure

How can I apply the strong law of large numbers to construct a measure $\{\mu_p\}$ on $([0,1],\mathcal{B}([0,1]))$ such that ${\mu_p}$'s are singular to $\lambda_{Leb}$ and the distribution function ...
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A family of continuous distribution functions with a singular law to the Lebesgue measure

Given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, let $\{X_n: n \ge 1\}$ be a sequence of i.i.d random variables with the common distribution $$\mathbb{P}(X_1 = 1) = p \text{ and } \...
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Does the law of large numbers hold for a large number of different trials?

The Law of Large numbers states that when a large number of repeated trials have been completed, the average of the obtained results will be close to the expected value. However, consider a large ...
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1answer
28 views

Application of Central Limit Theorem to Sales

Consider the following problem and solution. (I am stuck at the modified problem.) Problem There are exactly two phone shops, $A$ and $B$, serving a town of 1000 people. Both shops sell an iPhoneX ...
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1answer
24 views

Application of Strong Law of Large Numbers

I have troubles with the following problem. Can you help me, please? Suppose $\lambda_1 = 2$ and $\lambda_2 = \frac{1}{3}$ and $\lambda_1$, $\lambda_2$ are chosen independently with probability $1/2$ ...
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Law of iterated logarithm for stationary sequences

Let $(X_i)_{i\in\mathbb Z}$ be a sequence of random variables which is stationary and ergodic, but not necessarily i.i.d. Does there exist a real function $f$ such that the following holds? $$ \...
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Variants of weak and strong LLN

Given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, let $\{X_n: n\ge 1\}$ be sequence of square integrable random variables, i.e., $X_n \in L^2(\Omega, \mathcal{F}, \mathbb{P})$ for each $...
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1answer
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Example of a sequence of random variables which are pairwise uncorrelated and identically distributed, but LLN does not hold

Let $X_n$ be a sequence of identically distributed random variables such that $X_n$'s are pairwise uncorrelated and $\mathbb{E}(|X_1|)<\infty$. Then, is it necessary that $\frac{X_1+...+X_n}{n} \...
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Weak LLN holds but strong LLN fails

Show there exist independent random variables $\{X_n\}$ with $X_n\in\{-n, n, 0\}$, $\mathbb{E}(X_n)=0$, and $Y_n=\tfrac{1}{n}\sum_{k=1}^nX_k$ for all $n$, $\mathbb{P}(|Y_n|\ge\epsilon)\to 0$ for all $\...
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Pove that the summation of iid sequence satisfies $\frac{S_n}{n\log n}\rightarrow c\quad \text{in probability}$

Suppose that $X_1,X_2,\cdots,X_n$ are iid sequence with pdf $\frac{2}{\pi (1+x^2)}\cdot 1_{(0,+\infty)}(x)$. Denote $S_n$ as $S_n:=X_1+X_2+\cdots+X_n$. Prove that there exits $c>0$ such that $$\...
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123 views

Law of large numbers and theoretical probability

I didn't exactly know how to phrase the title of this question so a little more information.. I was conducting a small experiment with a class of secondary-school students to demonstrate the law of ...
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1answer
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Expectation of the mean of the sum of random variables [closed]

If $X_i$'s are independent and identified random variables, each with mean $\mu$ and variance $\sigma^2$. Let's say $S_m = \frac{1}{m} \sum_{i=1}^m X_i,~~ m = 1,2,\ldots,M.$ What are the values of $\...
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66 views

With $X \sim Unif(0,1)$ what is the limit of $\frac{n}{x_1^{-1} + \cdots + x_n^{-1}}$

I am confused as to how I can tackle this question: With $X \sim Unif(0,1)$ what is the limit of $\frac{n}{x_1^{-1} + \cdots + x_n^{-1}}$. My assumption is that is $0$. but I would like to show that ...
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If $\lim_{n \rightarrow \infty} \frac{S_n^4}{n^4} = 0$ then $\lim_{n \rightarrow \infty} \frac{S_n}{n} = 0$

If $\lim_{n \rightarrow \infty} \frac{S_n^4}{n^4} = 0$ then $\lim_{n \rightarrow \infty} \frac{S_n}{n} = 0$ where $S_n$ is the sum of $n$ iid RVs with mean zero. My question I'm having trouble ...
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Central Limit Theorem for geometric mean

Suppose that $X_1,X_2,...$ be i.i.d. variable uniformly distributed on (0,1), and let $\tilde{X_n}$ denote the geometric average of $n$ of these variables, i.e.: $\tilde{X_n}=(X_1X_2\cdots X_n)^{1/n}$....
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Almsot surely convergence of series

(a) Suppose that $X_1,X_2,...$ be independent with $P(X_n=n-1)=\frac{1}{n}$, $P(X_n=-1)=1-\frac{1}{n}$. Show that there are no constants ${\mu_n}$ such that $\frac{s_n}{n}-\mu_n \rightarrow 0$ a.s. (...
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Question about convergence of a series - need help understanding a proof for the strong law of large numbers

My question: In the proof below for the strong LLNs, I don't know how the author goes from the $\color{red}{\text{red box}}$ to the $\color{blue}{\text{blue box}}$ (see screenshot below) Getting to ...
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1answer
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Question about the strong law of large numbers (to build understanding)

I'm learning about the LLN and CLT for the first time and I'm having some trouble. I've read through other posts but I have a quirky (likely dumb) question about denominators... The Strong LLN ...
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A Problem on the Limit of an Integral

While I investigate the property of positive random variables, I encountered the following question, not easy to solve. Let $f:\left[0,\infty\right)\rightarrow\mathbb{R}$ be a continuous positive ...
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A Problem on the Limit of a Sequence

While I investigate the property of positive random variables, I encountered the following question, not easy to solve. Let $\left\{ a_n \right\}_{n=1}^{\infty}$ be a sequence that satisfies $a_n &...
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1answer
42 views

Does variance of sample mean converge to zero?

$n$ random variables $X_1,\ldots,X_n$ are an i.i.d. sample. $\bar X_n$ is the sample mean. $\mu$ is the expectation of distribution. Doesn't guarantee a finite variance. Does this always hold? $$E[(\...
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1answer
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Using strong law of large numbers to prove transience

I'm trying to work my way through a problem which defines $N_t$ as a Poisson process of rate $\lambda$ and $ X_n = N_n − n,\quad\text{for }\; n = 0, 1, 2, \ldots $ I've explained why $X_n$ is a ...
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1answer
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Application of law of large numbers on $\frac{X_1^k+X_2^k+…+X_n^k}{n} \overset{p}{\to}E(X_1^k)$

I read through an example in which the author states that the following application of law of large numbers (without proof or explanation): "Let $X_1,X_2,...$ be i.i.d. random variables such that $E|...
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Under finite variance assumption, does weak law of large numbers not imply the strong law?

In one of the proofs of the weak law of large numbers we assume finite variance and use Chebyshev's inequality. But if variance is finite, expectation (lower moment) is finite. And if expectation is ...
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1answer
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Law of large numbers and a product of random variables

A gambler makes a long sequence of bets against a rich friend. The gambler has initial capital C. On each round, a coin is tossed; if the coin comes up tails, he loses 30% of his current capital, but ...
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Question about Strong Law of Large Numbers in Breiman

I was reading through Breiman's Probability on Strong Large Numbers and got stuck on a part of its proof. It goes: Let $\Omega$={0,1}$^\mathbb N$ and $S_n$=$\sum_{k=1}^n\omega_k$. If $C$={$\omega\...
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1answer
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Probability in permutation of balls

There are $n$ balls, of which $r$ balls are red and $(n-r)$ balls are blue. If we select $d$ balls at random (without replacement), what is the probability we select $rd/n$ red balls? In the ...
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Limitation of law of large numbers

The book "Statistical Learning Theory" by Vladimir Vapnik has a part which I cannot understand properly. "According to the classical law of large numbers, the frequency of any event converges to the ...
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Find the limit of $\sum\limits_{r=\lfloor an \rfloor}^{\lfloor bn \rfloor} {n \choose r } p^r (1-p)^{n-r}$ using the central limit theorem

Let $p \in(0,1)$. What is the distribution of the sum of $n$ independent Bernoulli random variables with parameter $p$? Let $0 \leq a < b \leq 1$. Use approprtiate limit theorems to determine how ...
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Intuitive Explanation to a random variable concept

This is from the Wikipedia page on Stationary Processes: Let Y be any scalar random variable, and define a time-series { Xt }, by Xt = Y for all t. Then { Xt } is a stationary time series, for ...
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A sequence for which the running average converges to something other than the expected value

$X_1,X_2,\ldots$ is a sequence of independent random variables and $$P\{X_n = n^2 - 1\} = 1 - P\{X_n = -1\} = n^{-2}$$ Clearly, $E[X_n] = 0$. However, $\frac{1}{n}S_n \to -1$ almost surely where $...
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3answers
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Why doesn't the Law of Large Numbers hold here?

This is from an old homework assignment of mine, which I've since turned in. Say you have an independent sequence of R.V.s such that $\mathbb{P}(X_n= 2^n) = \frac{1}{2^n} = 1 - \mathbb{P}(X_n = 0)$....
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1answer
42 views

Convergence of a sequence of r.v. in probability and almost surely

Let $(X_k)_{k=1}^{\infty}$ be a sequence of independent r.v. where $\mathbb{E}X_k=0$ and $\mathbb{E}X_k^2=k$. Consider $Y_n=\frac{1}{n^2}\sum_{k=1}^{n}X_k$. Show $Y_n$ converges to 0 in probability ...
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1answer
55 views

Rate of convergence of sum of random variables

I'm working on the following problem and I'm a little stuck Suppose $X_1,X_2,\dots$ are iid. (a) If $E|X_1|^\alpha$ is finite for some $\alpha>0$, show that $\max_{1\le k\le n} |X_k|/n^{1/...
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1answer
44 views

Expectation: Is this statment true or is there a counter example [closed]

Let $(X_i)_{i\in (1,...,N)}$ be a sequence of i.i.d random variables with $0<\mathbb{E}(X_i^2)<0$. Is $$\sup_i|X_i|<\infty$$ a consequence of the strong law of large numbers, or is there a ...
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0answers
39 views

Rate of convergence for sum of random variables [duplicate]

I'm working on the following problem and I'm stuck. My plan was to use SLLN, but this gives me an expression for $\sum_{i=1}^n \xi_{i}^p/n$ and it's not clear how to convert it to having ...
3
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1answer
155 views

Law of large numbers for sequence of running minima of i.i.d. Uniform (0,1) random variables

Let $(X_i)$ be i.i.d. Unif$(0,1)$. Define $M_n=\min\{X_1,\ldots,X_n\}$ and $T_n=\sum\limits_{k=1}^n M_k$ for every $n\ge1$. Show that $$\dfrac{T_n}{E(T_n)}\stackrel{P}{\to} 1$$ The problem I am ...