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).
5
votes
2answers
79 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
$$
\mathrm{Pr}\left(\...
0
votes
1answer
45 views
Intuitive explanation of law of large numbers
If $Y_n=\frac{X_1+X_2+...X_n}{n}$ and $X_i's$ are i.i.d and $E(X_i)=0$, when we want to show that $Y_n$ converges to zero almost surely we need to prove that
$$\mathbb{P}\{w:Y_n(w)\rightarrow0\}=1.$$
...
0
votes
1answer
43 views
Probability Limit of a Variable
$n$-sample size.
For $i \in \{1,\dots,n\}$, $x_i(n)$ is a single draw from a distribution $f(x)$ on some bounded set. Associated with each $x_i(n)$ is a value $a_i(n)$, where $a_i(n)$ are such that $\...
2
votes
1answer
61 views
If $(W_i)_{i\in\mathbb N}$ obeys the strong law of large numbers, what can we say about $\liminf_{d\to\infty}\frac1{d^{2\alpha}}\sum_{i=1}^dW_i$?
Let $d\in\mathbb N$ and $W_1,\ldots,W_d$ be mutually independent, identically distributed and square-integrable real-valued random variables on a probability space $(\Omega,\mathcal A,\operatorname P)$...
0
votes
1answer
46 views
For missing data problem, show that $\frac{\frac1n{\sum_{i=1}^nD_iY_i}}{\frac1n{\sum_{i=1}^nD_i}}\overset{p}\to E(Y)$.
Consider a missing data $\{(Y_i,D_i):1\le i\le n\}$. If $D_i=1$, $Y_i$ is observed; if $D_i=0$, $Y_i$ is missing. Assume that $Y\bot D$. Denot $p=E(D)$, Show that
$$\frac{{\sum_{i=1}^nD_iY_i}}{{\sum_{...
0
votes
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}}\...
0
votes
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^...
0
votes
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 $Z$ ...
2
votes
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}{\...
1
vote
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 ...
0
votes
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
...
2
votes
1answer
32 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=...
2
votes
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 ...
0
votes
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/...
3
votes
2answers
52 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 ...
1
vote
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{ ...
0
votes
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 $\...
3
votes
0answers
104 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 $\...
0
votes
0answers
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-...
0
votes
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
votes
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,...
0
votes
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 ...
0
votes
1answer
77 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?
0
votes
2answers
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=...
0
votes
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<\...
0
votes
1answer
40 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?
2
votes
0answers
133 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) ...
1
vote
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 ...
2
votes
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 + \...
0
votes
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 ...
1
vote
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 ...
1
vote
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 :...
0
votes
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 ...
2
votes
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
$$\mathbb{P}(X_1 = 1) = p \text{ and } \...
0
votes
2answers
25 views
Does the law of large numbers hold for a large number of different trials?
The Law of Large numbers states that when a large number of repeated trials have been completed, the average of the obtained results will be close to the expected value.
However, consider a large ...
1
vote
1answer
30 views
Application of Central Limit Theorem to Sales
Consider the following problem and solution. (I am stuck at the modified problem.)
Problem
There are exactly two phone shops, $A$ and $B$, serving a town of 1000 people. Both shops sell an iPhoneX ...
0
votes
1answer
31 views
Application of Strong Law of Large Numbers
I have troubles with the following problem. Can you help me, please?
Suppose $\lambda_1 = 2$ and $\lambda_2 = \frac{1}{3}$ and $\lambda_1$, $\lambda_2$ are chosen independently with probability $1/2$ ...
1
vote
0answers
19 views
Law of iterated logarithm for stationary sequences
Let $(X_i)_{i\in\mathbb Z}$ be a sequence of random variables which is stationary and ergodic, but not necessarily i.i.d. Does there exist a real function $f$ such that the following holds?
$$ \...
2
votes
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 $...
1
vote
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} \...
0
votes
1answer
50 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
votes
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 $$\...
4
votes
1answer
160 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 ...
0
votes
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
votes
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
vote
0answers
57 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
votes
2answers
124 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
votes
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.
(...
0
votes
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 ...
1
vote
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 ...
