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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Law of large numbers for moving mean

Consider the following process: For $n = 1,\ldots$ $U_n \sim U[0, 1]$, that is, uniformly distributed on $[0, 1]$, $X_n = U_n 1_{U_n > q_n}$, where $q_n = \frac{1}{n-1} \sum_{i=1}^{n-1} X_i$, ...
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1answer
31 views

limit superior and law of large numbers [on hold]

I am wondering whether the following result is true: Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. real-valued random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ ...
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0answers
19 views

Extension of erdodic theorem with WLLN

Suppose you have a ergodic (or irreducible) Markov chain $(A_t)_{t\geq0}$ in continuous time. denote by $\pi$ the invariant distribution of $A$. If $f$ is a function of $A_s$ which is integrable w.r.t....
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20 views

Can the strong law of large numbers always be applied to an IID sequence of random variables with finite mean?

According to the Wikipedia page on the law of large numbers: The strong law applies to independent identically distributed random variables having an expected value (like the weak law). This was ...
5
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1answer
142 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{...
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0answers
58 views

Law of large numbers limit dependend on a second variable. What happens when both limits are taken at once?

The question I have is as follows. I have a i.i.d. sequence of random variables $(X^\alpha_n)_{n \in \mathbb{N}}$ with a expectation $\mathbb{E}X^\alpha$ which depend on a Markov process with a scaled ...
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1answer
21 views

Showing weak law of large numbers holds

My question: $\{X_n\}$ is a sequence of random variables. Var$(X_n)\le C\ \ \forall \ n$ and $\rho_{ij}=$Cov$(X_i,X_j)\to 0 $ as $|i-j|\to \infty$ . Show WLLN holds. In my book there are 3 ...
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18 views

Weak Law of Large Numbers, biased expectation?

I want to show that: $$\hat{\sigma^2}=(1/n)\sum^{n}_{i=1} ( X_i-\bar{X} )^2$$ is a consistent estimator of $\sigma^2$. I was using the Weak Law of Large Numbers in the sense that: $$E(X_i-\bar{X })...
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0answers
54 views

Sum of correlated random variables and the Law of Large Numbers?

Suppose I have a random variable $X$ which can take values on the set $\mathcal{X}=\{1,2,\dots,m\}$ and $X$ is drawn according to the given probability mass function $\mathbf{p}=\{p_1,p_2,\dots,p_m\}$...
3
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1answer
35 views

Using the Weak Law of Large Numbers for a product or random variables?

I need to calculate the average of the following quantity: \begin{equation} S_n=\prod_{i=1}^nS(X_i) \tag{1} \label{eq:1} \end{equation} with $S(X_i):=o_{X_i}b_{X_i}$, where each $X_i\in \mathcal{X}=\...
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1answer
12 views

Asymptotic inner product of correlated random vectors

Suppose $\mathbf{x}$ and $\mathbf{y}$ are N-dimensional non-white complex random vectors independent of each other i.e., covariance matrices $\mathbf{C_{xx}}\neq\mathbf{I}$, $\mathbf{C_{yy}}\neq\...
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3answers
351 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{...
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40 views

Problem that may use Borel Cantelli Lemma

So, there is a sequence of identically distributed independent random variables taking values on the integers, and they have a positive expectation. The problem is to prove that with probability 1 the ...
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0answers
17 views

Understanding the Strong Law of Large Numbers

Strong law of large numbers (SLLN) says if $X_1, X_2, \dots$ are iid random variables with expectation $\mu$, then $\bar{X}_n \to \mu$ almost surely, or $$P(\lim_{n\to \infty} \bar{X}_n = \mu)=1.$$ ...
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1answer
32 views

Strong Law of Large Numbers and Convergence a.e.

Let $(\Omega,\mathcal F,P)$ be a probability space. A sequence of r.v.'s $X_n$ converges a.e. to $X$ if and only if there exists a null set $N$, such that: $\forall \omega\in\Omega\setminus N:\lim_{n\...
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23 views

Basic Asymptotic Theory book

I would love if someone can recommend me a book where Basic Asymptotic Theory is thoroughly covered and explained with some examples. I'm currently reading Econometric Analysis of Cross Section and ...
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1answer
25 views

Central Limit Theorem when sample size is infinity

I am studying Law of Large numbers, Central limit theorem etc. and one thought is still bugging me. According to Law of Large Numbers, when we take sample from our distribution X, which size is close ...
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0answers
43 views

Sequence of random variables, mean zero, convergence to -infinity

What would be an example of a sequence $(X_k)$ of independent random variables with zero mean such that $$\frac{1}{n} \sum_{i=1}^{n} X_{i} \xrightarrow[\mbox{almost surely}]{n \to \infty}-\infty\ ?$$...
0
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1answer
51 views

Sum of random variables goes to infinity

I'm trying to show the following: Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d random variables with $\mathbb{E}[|X_1|]<\infty$ and $\mathbb{E}[X_1]=\mu$. Consider $$S_n:=X_1+\cdots+X_n,\...
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2answers
56 views

Convergence of $\sum_{i \leq n} X_i/n$

I have a question like this: Let $(X_n)$ be an i.i.d sequence of random variables with values in $\{-1,1\}$, and define $Y_n:= \sum_{i \leq n} X_i/n$. Show that $(Y_n)$ converges almost surely and in ...
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0answers
13 views

Convergence of multiple integrals [duplicate]

The question is as follows: Let $f:[0,1]\rightarrow R$ be continuous. Prove $lim_{n\to\infty}\int_{0}^{1}\int_{0}^{1}...\int_{0}^{1}\int_{0}^{1}f((x_1+x_2+...+x_n)/n)dx_1...dx_n=f(1/2)$ I'm ...
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0answers
79 views

Monte Carlo with non uniform weighting

So, I just want to check if what is in my mind is in fact true. Assume, that we have are given a distribution $p_{z}(k)$ over the whole $\mathbb{Z}^+$. We are interested in approximating $p_v(v)$ over ...
1
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1answer
28 views

Law of large numbers for a sequence of independent random variables with different means

I have an independent sequence of random variables $(X_i)_{i\geq 1}$ such that $E[X_i] = \mu_i > 0$ and $E[X_i^2] < \infty$. I know that $\frac{1}{n}\sum_{i=1}^n\mu_i \to \mu < \infty$ as $n\...
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0answers
54 views

Can Luck Be Proven Mathematically?

Here is what I'm thinking, and it has to do with the Gambler's Fallacy and Law of Large Number. The Gambler's Fallacy states that due to the probability of an event is statistically independent and ...
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0answers
23 views

Does this context satisfy the hypothesis of the law of large numbers?

A precondition for the law of large numbers is that $X$ is a random variable with $X_1$, $X_2$, $\ldots$ being a sequence of i.i.d. random variables s.t. $E[X] = E[X_i]$. Now suppose $X$ is the ...
0
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1answer
40 views

How is this not a counter-example to the law of large numbers?

Let $\Omega = \{0,1\}$ and $X: \Omega \rightarrow \{0,1\}$ be a random variable s.t. $X = id$ with $E[X] = 0.5$ (i.e., $P(0) = 0.5 = P(1)$). Let $X_1$, $X_2$, $\ldots$ be a sequence of i.i.d. random ...
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0answers
18 views

Understanding a precondition for the law of large numbers

Discussions of the law of large numbers frequently begin like this: Let $X$ be a real-valued random variable, and let $X_1$, $X_2$, $\ldots$ be an infinite sequence of i.i.d. copies of $X$. Let $\...
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1answer
18 views

Expectation of an average (law of large numbers)

Let $X$ be some random variable; we will take different measurements of this same variable $(X_1, X_2, ... X_N)$; distribution of each random variable then is identical, but variables are not ...
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1answer
52 views

Chebyshev inequality with the weak law of large numbers

In order to estimate f. the true fraction of smokers in a large population. Someone selects n people at random. His estimator Mn is obtained by dividing Sn. The number of smokers in his sample by n, i....
3
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1answer
69 views

Does the Central Limit Theorem Imply the strong Law of Large Numbers?

Assume that $(X_{k})_{k\geq 0}$ is a stationary (or weakly stationary) process defined on the same probability space $(\Omega,\mathcal{F},\mathbb{P})$. Can we assert from the convergence in ...
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50 views

Ergodic Theorem for Markov Chain with one closed communicating class and several transient states

It is known, that if a markov chain with a finite state space has only one closed aperiodic communicating class and several transient states, then there is a unique stationary distribution for this MC....
2
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1answer
46 views

Chernoff bound - finding conclusion for fair coin

Context: We know that $u_{1},...,u_{N}$ is a sequence of iid random variables, $U(s)=\mathbb{E}_{u_{n}}(e^{su_{n}})$. We can assume that we have already proven that $\mathbb{P}[u\geq\alpha]\leq(e^{-...
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1answer
52 views

The meaning of “EXACT laws of large numbers”

I have come across various papers that consider a stronger form of probability-relative frequency convergence theorem called the 'exact law of large numbers". I note that in particular such theorems ...
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1answer
49 views

Calculating Number of possible solutions to large Integers [closed]

I was looking for the best possible mean to determine all possible integer solution sets for large number such 21527411027188897018960152013128254292577735888456 ...
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1answer
63 views

Application: Law of Large Numbers

We have two collections of random variables $X_i$ and $Y_i$. The $X_i$ are independent and identically distributed with expectation $1$, and the $Y_i$ are also independent and identically distributed ...
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1answer
47 views

Law of Large Numbers for the reciprocal

Assume $(X_i)_{i\geq1}$ and $(Y_i)_{i\geq1}$ are two independent sequences of i.i.d. random variables such that $\mathbb E[X^k]<\infty$ and $\mathbb E[Y^k]<\infty$ $\forall k\geq1$. I am ...
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1answer
28 views

Proof of strong law of large numbers: not identical case

I've seen two types of conditions for strong law of large numbers: one requires i.i.d and first order moment condition: $X_n$ i.i.d with $E|X_1| < \infty$; the other requires second order moment ...
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0answers
24 views

Almost sure convergence SLLN practice example

Incidents for product $A$ occur at time $T_1, T_2,\dots$ where $T_i=X_1+X_2+\dots+X_i$. Assume that $(X_i)_i$ are i.i.d. and let $M(t)=\max\{n:T_n \leq t\}$ the number of incidents occured at time $t$....
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0answers
50 views

If a class of functions $\mathcal{F}$ is a Glivenko-Cantelli class then it is also a Donsker class?

Definitions: Consider a random variable $X:\Omega \rightarrow \mathcal{X}$ defined on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$ with probability distribution $P$. All functions ...
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28 views

The bracketing number and entropy: how do they vary with $\epsilon$ and $r$?

I have some doubts related to the bracketing number and entropy as defined in van der Vaart "Asymptotic Statistics" p.270. Definitions: Consider a random variable $X:\Omega \rightarrow \mathcal{X}$ ...
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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(\...
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1answer
26 views

Limit of the probability of an event (sum of i.i.d. r.v.'s)

Let $X_1,X_2,\dots$ be iid random variables with mean zero and finite variance, and let $S_n = \sum_{k=1}^n X_k$. For $b>0$, find the limit $$\lim_{n\to\infty}\mathbb{P}\!\left(\left\lvert\frac{1}...
2
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2answers
57 views

AM-GM Inequality Confusing

Here is something that I find hard to make sense of. Suppose $X_1, X_2, ..., X_n$ are independent draws from some distribution. By AM-GM inequality, we have: $$ \left( X_1 X_2 .. X_n \right)^\frac{1}{...
2
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0answers
28 views

Central limit theorem for uncorrelated identically distributed random variable

I have a sum of random variables as bellow $$Y=\sum_n A_n=\sum_n B_n\times C_n$$ where $B_n$s are correlated Gaussian random variables with zero mean, variance $1$ and correlation $E\{B_nB^*_r\}=\frac{...
2
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1answer
75 views

Maximizing the probability of a poll prediction

Using the central limit theorem, I was able to find out the first part of this question. However, part b is eluding me. How do I, in general, find a value for $n$ such that we can ensure the ...
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0answers
43 views

$ \lim\limits_{n\rightarrow \infty} {\frac{B_{t}}{t}}$ Brownian Motion

I want to prove in two ways that $ \lim\limits_{n\rightarrow \infty} {\frac{B_{t}}{t}}\rightarrow 0$ almost surely, where $B_{t}$ is a standard Brownian Motion. 1) in $L^2$ Can we say $X_{t}={\...
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1answer
23 views

Reformulation of SLLN with continuous nondecreasing process as time

Given a filtered probability space $(\Omega,\mathcal{F}_{t},P)$ and a continuous nondecreasing process $U_{t}$ with $U_{0}=0$ and $U_{t}\rightarrow \infty$ $P-a.s.$ as $t$ goes to $\infty$. Given a ...
0
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1answer
61 views

Convergence of product of random variables with distribution $U(0, e)$

Let $X_1, X_2, \ldots$ be a sequence of independent random variables with uniform distribution on $[0, e]$. Let $R_n:=\prod_{k=1}^n X_k$. By Kolmogorov's zero–one law $(R_n)$ converges with ...
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0answers
40 views

SLLN when the expectation in infinite

In a Post I found it says: Whenever ${\rm E}(X)$ exists (finite or infinite), the strong law of large numbers holds. That is, if $X_1,X_2,\ldots$ is a sequence of i.i.d. random variables with ...
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1answer
39 views

How do I transform this Probability in weak law of large number form?

Let $X_{1},\dots$ be a sequence of independent random variables. Suppose, for $k=1,2,\dots$ $$P\left(X_{2k-1}=1\right)=P\left(X_{2k-1}=-1\right)=\frac{1}{2}$$ and the probability density function of $...