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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(\...
0
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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
61 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
46 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
92 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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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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0answers
535 views

How to use Chebyshev's inequality or the law of large numbers to a probability question?

Let $x$ be a random bit string that takes values $\{1,0\}^n$. Let $r$ be the value of the most significant bit (MSB) of $x$ (and $r$ is a r.v. $1$ or $0$ that are equally likely). Let $g$ be our guess ...
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2answers
48 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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1answer
79 views

Product of Uniform Distribution

I know that there exists some discussions related to my question, however, I couldn't find an explanation for my question. I hope it is not a duplicate. Let $X_n$ be sequence of i.i.d. uniform ...
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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 ...
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
36 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 ...
0
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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/...
2
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1answer
231 views

Simplified Strong Law of Large Number by Using Truncating Function

Consider $X_1,X_2,...$ be i.i.d. random variables with $E|X_i| <\infty$ and let $EX_i := \mu$ and $S_n := \sum_{i=1}^n X_i$. Now, consider the corresponding truncated random variables $Y_k := X_k ...
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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
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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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1answer
134 views

Find $(a_n)$ such that $\frac1{a_n}\sum\limits_{k=1}^nX_k^2\to1$ in probability, for $(X_n)$ independent and $X_n$ uniform on $[0,n]$

Let $X_1, X_2, ..., X_n, ...$ be independent random variables. Assume that for each $n$, the random variable $X_n$ is distributed uniformly on $[0,n]$. Find a sequence $a_n$ such that $(X_1^2 + ... + ...
2
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1answer
72 views

For $(X_n)$ i.i.d. standard normal , find $(a_n)$ such that $\frac1{a_n}\max(X_1, …, X_n)\to1$ in probability

Let $(X_n)$ be independent $N(0,1)$ random variables. Find a numerical sequence $(a_n)$ such that $\frac1{a_n}\max(X_1, ..., X_n)$ converges in probability to $1$ as $n \to \infty$. I'm not quite ...
3
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0answers
102 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: ...
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1answer
122 views

Reference request: Strong Law of Large Numbers for V-statistics

I'm requesting a reference for a Strong Law of Large Numbers theorem for V-statistics (similar to Hoeffding's 1961 paper for U-statistics). That is, I am searching for an almost sure convergence ...
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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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65 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
66 views

Suggestion for probability problem book

I am an undergraduate having a course on probability that currently encompasses the topics: General Theory of Expectation, Modes of Convergence (almost surely/conv in probability), Laws of Large ...
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1answer
40 views
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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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0answers
130 views

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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$X_i = i^{\delta}Z_i$, 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
74 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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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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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
158 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

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
19 views

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? $$ \...
2
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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
29 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} \...
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
49 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 $\...