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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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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
45 views

Intuitive explanation of law of large numbers

If $Y_n=\frac{X_1+X_2+...X_n}{n}$ and $X_i's$ are i.i.d and $E(X_i)=0$, when we want to show that $Y_n$ converges to zero almost surely we need to prove that $$\mathbb{P}\{w:Y_n(w)\rightarrow0\}=1.$$ ...
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1answer
42 views

Probability Limit of a Variable

$n$-sample size. For $i \in \{1,\dots,n\}$, $x_i(n)$ is a single draw from a distribution $f(x)$ on some bounded set. Associated with each $x_i(n)$ is a value $a_i(n)$, where $a_i(n)$ are such that $\...
2
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1answer
59 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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1answer
44 views

For missing data problem, show that $\frac{\frac1n{\sum_{i=1}^nD_iY_i}}{\frac1n{\sum_{i=1}^nD_i}}\overset{p}\to E(Y)$.

Consider a missing data $\{(Y_i,D_i):1\le i\le n\}$. If $D_i=1$, $Y_i$ is observed; if $D_i=0$, $Y_i$ is missing. Assume that $Y\bot D$. Denot $p=E(D)$, Show that $$\frac{{\sum_{i=1}^nD_iY_i}}{{\sum_{...
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1answer
50 views

Suppose $\frac1{\sqrt{n}}\sum_{i=1}^nY_i\overset{d}\to N(0,V).$ What is the distribution of$\frac1{\sqrt{n}}\sum_{i=1}^nG(Y_i)$?

Suppose $$\frac1{\sqrt{n}}\sum_{i=1}^nY_i\overset{d}\to N(0,V).$$ Let $G(x)=\int_{-\infty}^xk(u)du$ be a kernel distribution function. Can we obtain the asymptotic distribution of $\frac1{\sqrt{n}}\...
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1answer
89 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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1answer
29 views

Distance between the noise and the corrupted signal

How can one formalize the fact that the law of $X+Z$ where $X \in \mathbb{R}^d$ is any vector-valued random variable and $Z\sim \mathcal{N}(0, \sigma^2 \mathbf{I}_d)$ closely resembles the law of $Z$ ...
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2answers
46 views

Sum of random variables tilted by Bernoullies

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of real valued i.i.d. random variables for which the following convergence in probability holds $$\frac{1}{n}\sum_{i=1}^nX_i\stackrel{n\to\infty}{\...
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0answers
45 views

Poke holes in this proof of the SLLN

I have a proof (sketch) of the Strong Law of Large Numbers, at least the "sufficiency" half of it, that seems a little too easy. This is the version where you only assume i.i.d. random variables, and ...
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1answer
35 views

How to compute $\lim_{n \to \infty}P(C_n>C_0)$?

The unit price of a certain commodity evolves randomly from day to day with a general downward drift but with an occasional upward jump when some unforeseen event excites the markets. Long term ...
2
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1answer
31 views

Convergence (distribution)

$X_1, X_2, X_3....$ are independent random variables. $P(X_n=0)=P(X_n=2)=1/4, P(X_n=-1)=1/2$. Find the limit of: $\frac{4\sqrt{n}(X_1+X_2+...+X_n)-7n}{n+(X_1+X_2+....+X_n)^2}$. I computed: $EX_n=...
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1answer
141 views

Prove that $\frac{1}{n}D_{n}\to \frac{\pi}{4}$ a.s.

Let $X^{n}:=(X_{1}^{n},X_{2}^{n})$ and $(X^{n})_{n}$ be IID random variables where $X^{n}$~$\mathcal{U}(K)$ on a probability space $(\mathbb R^{2}, \mathcal{B}(\mathbb R^{2}), P)$ where $\forall A \in ...
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1answer
49 views

Variant on Gambler's ruin problem?

Suppose I started with $W_1=1$ and decided on a ratio $0<\alpha<1$ such that I invest $\alpha W_n$ of my earning in the next round. I either lose my investment, or win it with probability $p=9/...
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2answers
50 views

Showing weighted average is consistent estimate

Here's the problem statement: Let $X_1$, . . . , $X_n$ be independent random variables with common mean $\mu$ and variances $σ_i^2$ . To estimate $μ$, we use the weighted average $T_n$ = $\sum_{i=1}^n ...
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0answers
21 views

Show that the set of numbers normal to base d has Lebesgue Measure 1 (d=2 and d=3).

I need to prove this for $d = 2, d= 3$. I'm working on $d =2$. The idea is to show that my $x_n$'s are IID so that I can apply the strong law of large numbers. Let $N_2 = \{x \in [0,1] \mid x\text{ ...
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1answer
26 views

Determine the convergence of $\overline{X^2}_{n}$

Let's say I sample $X_{1},X_{2},\dots,X_{n}$ from a random variable X with a distribution. It is not important to know what the distribution is at this point. I am trying to determine whether $\...
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0answers
100 views

Law of Large Numbers contradicts Central Limit Theorem?

My text defines the weak law of large numbers: If $X_1,\ldots,X_n$ are IID, then $\overline{X} \overset{P}{\to} \mu$. And the CLT as: Let $X_1,\ldots,X_n$ be IID with mean $\mu$ and variance $\...
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55 views

Using SLLN to show that the set of numbers normal to base 3 has Lebesgue measure 1

Looking for resources/places to research normal numbers, the normal number theorem, strong law of large numbers, etc. (anything that will help me to solve the problem in the title). I am doing self-...
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1answer
17 views

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: ...
0
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1answer
33 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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1answer
82 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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1answer
76 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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62 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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1answer
46 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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1answer
39 views

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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0answers
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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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0answers
28 views

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
45 views

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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0answers
9 views

$X_i = i^{\delta}Z_i$, 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
40 views

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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0answers
25 views

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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0answers
73 views

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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2answers
25 views

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
30 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
31 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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0answers
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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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0answers
31 views

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
28 views

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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1answer
48 views

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 $\...
3
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2answers
77 views

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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1answer
157 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
31 views

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 $\...
3
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2answers
70 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 ...
1
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0answers
56 views

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 ...
0
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2answers
122 views

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.: $\tilde{X_n}=(X_1X_2\cdots X_n)^{1/n}$....
0
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1answer
31 views

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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0answers
27 views

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
42 views

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 ...