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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Strong law of large numbers when sample size is a random variable

For a sequence $X_1, X_2, \ldots, X_n$ of i.i.d. random variables with mean $\mu$, the strong law of large numbers tells us that $$\sum_{i=1}^{n} \frac {X_i} {n} \xrightarrow{a.s.}\ \mu ...
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29 views

General Weak Law of Large numbers

I came across a question regarding the WLLN. Suppose for $X \geq 0$ , $\mathbb{E}[X] = \infty $ , $S_n = \sum_{i \leq n} X_i$, $X_i$ are iid copies of $X$ , and $\frac{\mathbb{E}[X \mathbf{1} _{X ...
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81 views

About strong law of large numbers

I came across a problem: Let $X_1,X_2,...$ be independent random variables with uniform(0,1) distribution from probability space $(\Omega,\cal F,P)$. Prove that: ...
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21 views

Almost Sure Convergence for Sample Mean of Bernoullis

Let {$B_i$} be a sequence of Bernoulli($\mu$) variables and $X_n$ its sample mean $X_n=\frac1n\sum_i^nBi$. Because of the Strong Law of Large Numbers, we know that $X_n$ converges almost surely to ...
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25 views

A LLN type theorem on the supremum of functions of a RV

Let $X_1,\dots,X_n$ be iid real valued random variables. Let $\mathcal{F}$ be a set of functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\mathbb{E}f(X_i) < \infty$ for all $f \in ...
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Almost sure limit of $\log(X_1 + X_2 + … + X_n) - \log(n)$

Let $X_n$ be an i.i.d. sequence of positive random variables with expectation 2 and variance 1. What is the almost sure limit of $$\log(X_1 + X_2 + ... + X_n) - \log(n)$$ as $n \to \infty$ Would it ...
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Moving Limit within Probability/Law of Large Numbers [duplicate]

Edits in Bold When I look up the strong law of large numbers it says that (Looking at Discrete Random Variables) $$P\left(\lim_{n\rightarrow\infty}\bar{X}_{n}=\mu\right)$$ That got me wondering are ...
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Is it possible to use multiple time scale algorithm here?

Suppose a random sequence is being generated (the next term generated depends on the previous term, but we don't know any distribution) until we hit some specific number. We want to calculate the ...
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36 views

Using Taylors to show convergence in probability

I'd like to show that \begin{equation} \sqrt{n} \left( (1-\frac{1}{n})^{n\bar{X}} - e^{-\bar{X}} \right) \to 0 \end{equation} in probability for a random variable with mean $\mu$ and finite variance ...
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50 views

Is the law of large numbers reflected in SO reputation system?

I have recently encountered an interesting phenomenon on SO reputation system: Let $f(n)$ denote the current score of the $n$th best user. A sample that I collected at a given moment: $f(k)=k ...
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Central Limit Theorem vs. Weak Law of Large Numbers

So, just to begin with I feel like this is a problem I am massively overthinking, and the solution is very simple. That said, it has been a while since I've taken a math class, and so some of my ...
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55 views

$\sum_{k\ge 0} e^{-an} \frac{(an)^k}{k!}f(\frac{k}{n}) = \Bbb{E}\left(f\left(\dfrac{X_1+\cdots + X_n}{n}\right)\right)$

Hello everybody i need to show following equality $$\sum_{k\ge 0} e^{-an} \frac{(an)^k}{k!}f(\frac{k}{n}) = \Bbb{E}\left(f\left(\dfrac{X_1+\cdots + X_n}{n}\right)\right)$$ Where $(X_i)_i$ are ...
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98 views

If $X_{i}$ are I.I.D and $n^{-1}\sum_{i=1}^{n}X_{i}$ converges a.s/in-distribution to a constant $a$ is it true that $a=\mathbb{E}[X_{i}] $?

The question itself is in the title. It is immediate by the strong law of large numbers that if $X_{i}$ had a finite first moment then we would have a.e convergence (and thus in probability and in ...
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38 views

Strange identity in the proof of the Strong Law of Large Numbers

Above is an extract from the proof of the strong law large of large numbers with finite fourth moment. The $X_n$ are iidrv's with $\mathbb{E}(X_n)=\mu$ and $\mathbb{E}(X_n^4)<M$ for some ...
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23 views

Law of large numbers for U-Statistics with varying kernel

Let $U_n=\binom{n}{2}^{-1}\sum_{1\leq i<j\leq n} H_n(X_i,X_j)$ be an $U$-Statistic of order $2$ and with kernel $H_n$ depending on the sample size $n$. I wonder if there exists a Strong (or Weak) ...
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Betting: Gambler's Fallacy vs. Law of Large Numbers

I know this has been asked before, but I think not in this exact way, so here goes: Suppose you're going to bet on the flip of a coin. Your bet is always "HEADS", but the amount of your bet may vary, ...
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44 views

Law of large numbers with random weights

Let $\mu_i$ be i.i.d. RVs with mean zero, and let $a_i$ be random weights that are not independent and are not identically distributed, $i=1,...,N$. $\mu_i$ is orthogonal to $a_j\;\forall j$. Is ...
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1answer
46 views

integral approximation (law of large numbers)

I am totally at a loss with this question and don't even know where to begin. Let $g:[0, 1]\rightarrow \mathbb{R}$ be a measurable and Lebesgue-integrable function. $U_1, U_2, \dots$ be a series of ...
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Answer to my question strong law of large number

i see to use the law of the iterated logarithm that tell us: $\limsup_{n\to\infty}\left|\frac{S_n}{\sqrt {n\log\log n}}\right|=\sigma\sqrt 2\text{ with probability 1,}$ where: $S_n:=X_1+\ldots+X_n$ , ...
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2answers
87 views

Strong law of large number

Let $ \{c_n\} $ be a descending positive real sequence. Let $ X_1,X_2,\cdots $ be a sequence of i.i.d random variables. Is the following equivalent? ($1$) For any i.i.d sequence $ X_1,X_2,\cdots $ ...
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48 views

In Markov inequality proof, why is $\int_a^\infty xp(x) \, dx \ge \int_a^\infty ap(x) \, dx$

Markov inequality, $$\Pr(X \ge a) \le \frac{E[x]}{a}$$ Proof $$\begin{aligned} E(X) &= \int_0^\infty xp(x)\,dx = \int_0^a xp(x)\,dx + \int_a^\infty xp(x)\,dx \\ &\ge ...
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41 views

Random variables $x_i$ with $\lim_{k\to\infty}\frac1k\sum_{i=1}^kx_i=0$

I am looking for a sequence $(x_n)_{n\in\mathbb N}$ of random variables such that the sequence hasn't any expected value and $\lim_{k\to\infty}\frac1k\sum_{i=1}^kx_i=0$. I thought about using a ...
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1answer
28 views

Convergence of random variable

I've been facing the following problem: Let $(X_k, Y_k)_k$ be a sequence of $2$-dimensional, independent random variables, each with uniform distribution over $ B(0,k) $ Verify if the following ...
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62 views

Sum of sequence of random variables infinitely often positive

Let $X_1,X_2,\ldots$ be an infinite sequence of independent (but not necessarily identically distributed) random variables with $E(X_i)=0$ for all $i$. Set $S_n=\sum_{i=1}^n X_i$. I want to show that ...
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54 views

convergence in probability of function of random variables

Suppose that $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d random variables. If we have $E(|X_1|^k) <\infty$ for some $k>0$ and $f(x)$ is a bounded continuous function on $\mathbb{R}$. Is the ...
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121 views

Laws of large numbers and independence

I just did a brief review of various sources, and they all specify that if $X_i$'s are independent, identically distributed random variables, then $S_n/n \rightarrow E(X_i)$ (with respect to various ...
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30 views

Solving this random variable problem

This is an earlier problem Proving this random variable problem but generalised, maybe you want to take a look at that one first? $X_1,X_2,X_3,\ldots$ are IID random variable taking values in ...
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Proving this random variable problem

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$. Also $t\in(0,1)$. Define random variables $Y_1,Y_2,Y_3,\ldots$ recursively like $$Y_1 = (1+tX_1)$$ $$Y_n = ...
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Law of large numbers for linear (quadratic) combinations of i.i.d. random variables

Let $(X_i)_{i\in\mathbb{N}}$ be i.i.d. real random variables with zero mean. By the law of large numbers $$\frac{1}{n}\sum_{i=1}^nX_i \to 0 \quad\text{(almost surely, in probabability...) as ...
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Show $4x^3 + y^3 = 792,864,313,578,917,724,246$ has no solution for $x, y \in \mathbb{Z}$.

I think it involves something about looking at the last digits of the number and/or modular arithmetic but I don't remember how to do this. Help?
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92 views

Khinchin's weak law of large numbers: finite variance

I have the following situation: suppose you have a sequence of i.i.d. random variables $\{X_i\}$ with mean $\mu$ and variance $1$. I would like to use Khinchin's WLLN on it, but this requires that ...
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Uniform law of large numbers for discontinuous functions?

Do you know about any Uniform Law of Large numbers (see http://en.wikipedia.org/wiki/Law_of_large_numbers#Uniform_law_of_large_numbers) that work when f is the indicator function (and thus not ...
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A basic problem on random series/ law of large numbers

Consider the following two statements : i) Suppose that $X_1, X_2, \dots$ are independent and identically distributed and $E[X_1^-] < \infty, E[X_1^+] = \infty$. Then $n^{-1} \sum_{k=1}^{n}X_k ...
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Application of the Weak Law of Large Numbers.

I have in my problem that $X_1,\ldots,X_n$ is a random sample from a distribution with probability density $f(x; \theta)=\theta x^{\theta-1}, 0<x<1$. Furthermore, $-\log X_i ...
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398 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 ...
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173 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 $u_0>0$, ...
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35 views

Can I use the law of large numbers here

Suppose I have $X_i,i=1,\dots,n$, i.i.d. finite mean random variables. Can I use the law of large numbers to get $$\frac{1}{n}\sum_{i=1}^n\frac{1}{n}\sum_{j=1}^nf(X_i)g(X_j) ...
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SLLN for nearly-identical observations

When observations $X_i\sim\mu$ are IID the usual SLLN along with separability of the underlying metric space yield that the empirical measure $\hat\mu_n=\frac1n\sum_{i=1}^n\delta_{X_i}$ converges ...
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36 views

How big is a large sample?

Short version According to the law of large numbers, how many samples do I need to take to reach the half-life of the convergence towards the mean? In other words: How big is a large sample? Long ...
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Exposition of Erdős and Rényi's 'New law of large numbers'.

Where can I find an exposition of the paper On a new law of large numbers by Erdős and Rényi? I'm reading this paper and it's rather terse, so I'd like some intuition and explanation. I did a Google ...
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Law of large numbers: second moment tends to 0, but $S_n/n$ doesn't converge a.e.

What is an example of a sequence of random variables $\{X_n\}$ on a probability space $(\Omega, \mathscr{F}, P)$ such that $E(X_n^2) \to 0$ but it is not the case that $$ \frac{S_n - E(S_n)}{n} \to 0 ...
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1answer
84 views

Sum of infinitely many i.i.d. random variables is infinite with probability 1

How do I solve this? I'm really confused. If $X_1,X_2,\ldots$ are non-negative independently and identically distributed random variables with $P(X_i>0)>0$, show that $\displaystyle ...
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1answer
59 views

Measure Theory and Law of Large numbers

If $X_1,X_2,...$ are non-negative random variables with the same distribution (but the variables are not necessarily independent) and $E[X_1]< \infty $, prove that $$\lim_{n \to ...
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How to show that $n^{-r} \sum_{j=1}^n (X_j - \mu) \rightarrow 0$ in probability

I need your help to prove the following statement. Let $X_1, \cdots X_n$ be stochasticaly independent, identically distributed random variables. Assume they have a finite expected value $\mu$ and a ...
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Is the expected value $\mu$ in the WLLN a random variable?

Am I right in thinking that the weak law of large numbers, when stating that $$\bar{X_n} \to \mu$$ in probability convergence is stating that the sequence of random variables $\{X_n\}$ tends to the ...
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1answer
55 views

Which different probabilistic bounds/inequalities apply when we are given a lower bound on the sample size

Let m be the sample size and $X_i$ be a r.v. that we sample and define a new r.v. such that: $$M_m=\frac{1}{m}\sum^m_{i=1}{X_i}$$ My question is, what type of probabilistic inequalities require some ...
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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 (MSB) bit of x (and r is a r.v. 1 or 0 that are equally likely). Let g be our guess for the MSB ...
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111 views

a.s. convergence of sum of normal random variables

From Resnick's A Probability Path, Exercise 7.7.14: Suppose $\{X_n, n \ge 1\}$ are independent, normally distributed with $E(X_n) = \mu_n$ and Var$(X_n)=\sigma^2_n$. Show that $\sum_n X_n$ ...
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42 views

law of large numbers and renewal processes

Let $(X_n)$ be an iid. sequence of real, integrable random variables with $EX_1=a>0$. Let $S_n=X_1+...+X_n$, $n=1,2,...$ and $N_t:=\sup\{n\geq 1|S_1,...,S_n\leq t\}$, $t\geq 0$ where ...
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43 views

Weak law of large numbers for dependent variables

I need to prove a version of the weak law of large numbers with the following assumptions: $X_i$ is a sequence of square-integrable real random variables with a fixed expectation $m$ and such that ...