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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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Is the Law of Large Numbers empirically proven?

Does this reflect the real world and what is the empirical evidence behind this? Layman here so please avoid abstract math in your response. The Law of Large Numbers states that the average of the ...
28
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4answers
9k views

Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance

I'm currently stuck on the following problem which involves proving the weak law of large numbers for a sequence of dependent but identically distributed random variables. Here's the full statement: ...
12
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4answers
4k views

What does a probability of $1$ mean?

From a textbook on probability on the Law of Large Numbers: Theorem 3-19 (Law of Large Numbers): Let $X_1,X_2, \ldots , X_n$ be mutually independent random variables (discrete or continuous), each ...
11
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3answers
391 views

Help understanding the weak law of large numbers with respect to statistics

I'm trying to do some self-studying to upgrade my statistics knowledge, and came across this term in a section discussing the weak law of large numbers and Bernoulli's theorem: $$\sum_{k=0}^n k\frac{...
11
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1answer
7k views

The strong and weak laws of large numbers: Why two?

The following questions are entirely based on the corresponding article from Wikipedia. The assumptions of both laws are the same, and the strong law has a more general claim than the one of the weak ...
9
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1answer
718 views

Why weak law of large number still alive?

I know the difference between WLLN and SLLN in terms of a convergence type. Then, as revealed in any statistical textbook saying sufficient conditions to two theorems are the same, I think that we do ...
8
votes
2answers
1k views

Why aren't the strong LLNs and CLT contradicting each other?

Given $n$ i.i.d. random variables $\{X_1, X_2, \dots , X_n\}$, each with mean $M$ and variance $V$, both strong and week LLNs seem to say that the average of the $n$ random variables, $S_n = \frac{X_1 ...
8
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1answer
273 views

Limit of $\frac1{c^n}\iint_{[0,c]^n}\frac{f(x_1) +f(x_2) +\cdots+f(x_n)}{g(x_1) +g(x_2) +\cdots+g(x_n)}\,dx_1dx_2\cdots dx_n$ when $n\to\infty$

A. How do I prove the following sequence converges as $n$ goes to $\infty$ for any $c$, and how do I find the limit? $$ \begin{align} a_1 &=\frac{1}{c} \int _0^c\frac{x_1 }{1+x_1}\;dx_1 \\ \\ ...
8
votes
2answers
1k views

Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables such that $$P(X_n=n+1)=P(X_n=-(n+1))=\frac{1}{2(n+1)\log(n+1)}$$ $$P(X_n=0)=1-\frac{1}{(n+1)\log(n+1)}$$ Prove that $X_n$ ...
8
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1answer
1k views

Uniform Law of large numbers

Could you please help with the proof of the proposition: Let $G(\cdot)$ be bounded, continuous, strictly increasing function on $\mathbb{R}.$ Let ${\xi_t},\, t\in\mathbb{Z}_+$ be i.i.d random ...
6
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1answer
1k views

Strong law of large numbers for Poisson process

My question regards the strong law of large numbers as stated, e.g., in Ethier and Kurtz (1986, p. 456 Eq. (2.5)), as follows: If $Y$ is a unit Poisson process, then for each $u_0>0$, \begin{...
6
votes
1answer
93 views

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 ...
6
votes
1answer
162 views

S.N. Bernstein Law of Large Numbers

while reading a paper named "Network Embedding as Matrix Factorization: UnifyingDeepWalk, LINE, PTE, and node2vec" (http://keg.cs.tsinghua.edu.cn/jietang/publications/WSDM18-Qiu-et-al-NetMF-network-...
6
votes
1answer
232 views

Find the limit of $\int_0^1\cdots\int_0^1 \sin(\sqrt[n]{x_1\cdots x_n}) \, dx_1\cdots dx_n$ using a probabilistic approach

I am looking for the solution of this: $$ \lim_{n \rightarrow \infty} \int_0^1\cdots\int_0^1 \sin(\sqrt[n]{x_1\cdots x_2}) \, dx_1\cdots dx_n $$ I know that $X_1,\ldots,X_n \sim \mathcal{U}[0,1]$....
6
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3answers
797 views

Almost sure convergence in strong law of large numbers.

Strong Law of Large Numbers is often stated as $$\overline{X}_n\ \xrightarrow{a.s.}\ \mu \qquad\textrm{when}\ n \to \infty$$ or $$\Pr\!\left( \lim_{n\to\infty}\overline{X}_n = \mu \right) = 1$$ for $\...
6
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1answer
258 views

A consequence of the law of large numbers

Let $(X_k)_{k=1}$ be Poisson random variables with expectation $\mu$, let $Y_n = \sum_{k=1}^{n} X_k$. The weak law of large numbers states that, $$ \forall \delta>0, \forall \epsilon>0 \, \, \...
6
votes
2answers
724 views

Measuring $\pi$ by throwing darts

I want to give an approximation of $\pi$ in this way: I inscribe a circle in a square then I throw darts at random on the square from far away. If the darts falling on the square are $n$ and the ...
6
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2answers
579 views

A problem about strong law of large numbers of Shiryaev's Probability

This is a problem after the section "Strong Law of Large Numbers" of Shiryaev's Probability: Let $\xi_1,\xi_2,...$ denote independent and identically distributed random variables such thatt $E|\...
5
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4answers
924 views

What happens if I toss a coin with decreasing probability to get a head?

Yesterday night, while I was trying to sleep, I found myself stuck with a simple statistics problem. Let's imagine we have a "magical coin", which is completely identical to a normal coin but for a ...
5
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3answers
2k views

Law of large numbers for non-identically distributed Bernoulli random variables

Let $(X_n)$ be a succession of independent r.v., such that $X_n$ ~ $Bern(p_n)$. I know then that $\lim_{n \to \infty}p_n=p$ and $p_n>p>0$ for each $n \in \mathbb{N}$. I have to prove that $\...
5
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2answers
1k views

The logic behind a sequence

I am trying to get the logic behind the sequence: for $n=2,3,\ldots$ $$\left(\frac{\log (2)}{\log \left(\frac{3}{2}\right)},\frac{\log (3)}{\log \left(\frac{17}{9}\right)},\frac{\log (4)}{\log \left(\...
5
votes
1answer
107 views

Law of Large Numbers - utility/difficulty of various versions.

This may or may not be an answer to Is there an easy proof that the set of $x \in [0,1]$ whose limit of proportion of 1's in binary expansion of $x$ does not exist has measure zero?, depending on ...
5
votes
1answer
586 views

Triangular arrays and almost sure convergence of row averages

Suppose we have the triangular array $\{\{X_{in},i=1,\ldots,n\},n=1,2,\ldots\}$: $$\begin{array}{ccccc} X_{11}&&&&\\ X_{12}&X_{22}&&&\\ X_{13}&X_{23}&X_{33}&...
5
votes
1answer
270 views

Strong Law of Large Numbers for a i.i.d. sequence whose integral does not exist

Prove: Let $X_1 ,X_2 , ... , X_n , ...$ be i.i.d. random variables with $\mathbb{E}[X_1^+]=\mathbb{E}[X_1^-]=+\infty$. If $S_n=\sum_{i=1}^{n}{X_i}$, then $$\limsup_{n\rightarrow\infty}{\frac{...
5
votes
2answers
72 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(\...
5
votes
1answer
95 views

prove this martingale inequality

The problem is like this: Let $Y_1,Y_2,\ldots$ be nonnegative i.i.d. random variables with $E(Y_m)=1$. Let $X_n=\prod_{m\leq n} Y_m$, show that $\lim_{n\rightarrow \infty}X_n=0$ if $P(Y_m=1)<1$. ...
5
votes
1answer
85 views

martigale convergence theorems

Let $S_n = X_{1}+\cdots + X_{n}$ be a martingale satisfying $E[X_{k}^{2}]\leq k<\infty$, for all $k$. Show that $S_{n}$ obeys the weak law of large numbers: $$P\left(\left|\dfrac{S_{n}}{n}\right|&...
5
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2answers
1k views

Weak Law of Large Numbers for a non-iid, non-ergodic sequence

I have a somewhat open-ended question. Let's say I have a sequence of random variables $(X_n: n \geq 1)$ which are neither independent, ergodic, nor identically distributed. Normally I would say that ...
5
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0answers
235 views

Suppose $E[X_1] <\infty$. Show that $\lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s.

Let $X_1,X_2,X_3,...$ be i.i.d. with $P(X_1 >0)=1$. Define $S_n =\Sigma_{i=1}^{n} X_i$. (a) Suppose $\mathbb{E}[X_1] <\infty$. Show that $\lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s. I ...
5
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0answers
398 views

The law of large numbers with dependent random variable

Consider a sequence of i.i.d. random variables $\left\{X_i\right\}_i$, and let $Y$ be another random variable. Can we say something regard the convergence of the following series $$ \frac{1}{n}\sum_{...
4
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5answers
1k views

Law of Large Numbers, a confusion

According to Law of Large Numbers, if I throw a coin 1000 times approximately 500 will be head and 500 tail. Suppose that I throw the coin 700 times and I got 700 heads. Can I say that in the next 300 ...
4
votes
2answers
170 views

Almost sure convergence of a certain sequence of random variables

Let $(X_n)_{n\geq 1}$ be i.i.d. random variables with uniform distribution on the interval $(0,1)$. I need to prove that the following sequence of random variables $(Y_n)_n$ defined by: $$Y_n = \...
4
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2answers
314 views

Strong law of large numbers for function of random vector: can we apply it for a component only?

Consider i.i.d. random variables $\{X_1,..., X_n\}$ with well defined first moment i.i.d. random variables $\{Y_1,..., Y_n\}$ with well defined first moment By the strong law of large numbers: $$ \...
4
votes
2answers
596 views

Almost sure convergence of harmonic mean

Let $X_1,...,X_n \sim Uniform(0,1)$. Harmonic mean is defined as: $H_n = \frac {n}{\sum_{i=1}^n\frac{1}{X_i}}$ Find a.s. limit of this as $n \rightarrow \infty$ I already did the problem for both ...
4
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1answer
316 views

Variant on strong law of large numbers for infinite expectation

Let $X_0,X_1,\dots$ be a sequence of i.i.d. random variables, with $\mathbb{E}\big[|X_i|\big]=\infty$. Show that: $$ 1.\;\limsup \left(\frac{|X_n|}{n}\right)=\infty\qquad 2.\; \limsup \left(\frac{|...
4
votes
1answer
151 views

$\limsup \frac{|S_n|}{n}=\infty$

$X_n$'s are i.i.d symmetric with $E|X_1|=\infty$. Then $\limsup \frac{|S_n|}{n}=\infty$. How do I show $\limsup \frac{S_n}{n}=\infty$ and $\liminf \frac{S_n}{n}=-\infty$? My attempt: Let $c=\limsup \...
4
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2answers
822 views

Does this sequence of random variables satisfy the Law of Large Numbers?

Notation: $S_n = X_1 + \ldots + X_n$ Definition $1$ : The sequence of random variables $X_1, X_2, \ldots$ satisfies the Weak Law of Large Numbers if $$\frac{S_n - E[S_n]}{n} \overset{p}{\to} 0$$ ...
4
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1answer
110 views

Finiteness of the sum of the product of an i.i.d. sequence

Before I go to the statement of my question I just want to say a few words about the personal background of this question. I have recently taken a course on stochastic differential equations without ...
4
votes
2answers
181 views

Prove Poissons' theorem

Let $(\Omega, \Sigma, \mathbb{P})$ be a probability space. If $A_1, A_2, \ldots$ are independent events and $\bar{p}_n$ and $N$ are defined as $$ \bar{p}_n=\frac{1}{n}\sum_{i=1}^n\mathbb{P}(A_i) \quad ...
4
votes
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 ...
4
votes
1answer
192 views

Strong law of large numbers for the conditional expectation of functions of random vectors

Consider a collection of 0-1 random variables $Y_{n,N}$, for all $n$ and $N$. The random variable $Y_{n,N}$ is a deterministic function of the collection of random variables in $\mathcal{F}_n = \{(U_k)...
4
votes
1answer
688 views

Exercise about the Strong Law of Large Numbers

This is Exercise 5.3.2 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise: Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of independent identically distributed random ...
4
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1answer
72 views

Using the SLLN to show that the Sample Mean of Arrivals tends to the Arrival Rate for a simple Poisson Process

Let $N_t = N([0,t])$ denote a Poisson process with rate $\lambda = 1$ on the interval $[0,1]$. I am wondering how I can use the Law of Large Numbers to formally argue that: $$\frac{N_t}{t} \...
4
votes
1answer
551 views

what does it mean “converge in probability to a random variable”?

In statistics, a sequence of random variables $X_n$ is said to converge to a random $X$ in probability if $P(|X_n - X| > \epsilon ) \to 0 $. Also, $X_n$ is said to converge to a constant c if $P(|...
4
votes
1answer
336 views

Weak Law of Large Numbers

The Weak Law of Large Numbers is often stated with the iid assumption for the underlying RV's. However, I have seen the independence assumption being diluted to the "uncorrelatedness" assumption (e.g.,...
4
votes
0answers
72 views

Is $\frac{1}{\frac{1}{n}\sum_{i=1}^{n}\frac{1}{X_{i}}}$ a consistent estimator for $\mu$

Let $X_i \sim N(\mu,\sigma^2)$, I want to find out if $$\frac{1}{\frac{1}{n}\sum_{i=1}^{n}\frac{1}{X_{i}}}$$ is a consistent estimator for $\mu$, or not. It's easy to show, using $LLN$, that $\...
4
votes
0answers
78 views

Can one construct a monotone law of large numbers?

let $X_1, X_2, \dots$ be a sequence of IID random variables defined on a probability space $( \Omega, F, P)$ with mean $E[X_1] = \mu $, define $$\bar{X}_n = \frac{1}{n}(X_1+ \dots+ X_n)$$ then $\bar{...
4
votes
2answers
99 views

Law of large numbers problem and calculation of max expected value

We've got to prove that $\frac{M_{n}}{\ln(n)}\rightarrow 1$ a.s. Where $M_{n}=\max\left\{{X_{1},...,X_{n}}\right\}$ , with $X_{i}\sim \mathrm{exp}(1)$ i.i.d. So it is obvious that we will have to use ...
4
votes
0answers
493 views

Law of large numbers; Poisson distribution

Let $X_n$ be the numbers of job applications at a company in the year $1900+n,n\in\mathbb N$. Let $(X_n)_{n\in\mathbb N}$ be a sequence of independent, identically distributed random variables with ...
3
votes
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 $\...