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 positive random variable [closed]

can someone pls help me with this problem: $X_{n}$ iid with expectation $\infty$ and $X_{n} \ge 0$ then: $S_{n}/n \to \infty$ a.s.
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1answer
50 views

Find the limit of $P(\bar{X_n}\leq 1.8)$ for i.i.d random variables $X_i$s of known distribution

Let $X_1,X_2,…$ be a sequence of independent and identically distributed random variables with $P(X_1=1)=\frac{1}{4}$ and $P(X_1=2)=\frac{3}{4}$. If $\bar{X_n}=\frac{1}{n}\sum_{i=1}^{n}X_i$, for $n=...
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1answer
26 views

Khinchin's Law of Large numbers proof unclarity.

This is the formulation: Let $X_n,n=1,2,...$ be independent, equally distributed random variables. $EX_k=a$(expectation) $k=1,2,...$. For this sequence of $X_n$ the law of large numbers applies: $$\...
2
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1answer
29 views

Bernoulli Trials: Law of Large Numbers vs Gambler's Fallacy, the N paradox

I have asked this question before but I think it wasn't clear what I implied with my succinct question, so I will be a bit more verbose this time. Lets set the following example: Bernoulli trials, K=...
2
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1answer
39 views

Convergence of $V_n=\prod\limits_{i=1}^n U_i$

I struggle to do this exercise: Let $U_1,U_2,\dots$ be a sequence of i.i.d. random variables. We define $$V_n=\prod\limits_{i=1}^n U_i$$ Show that $V_n^{1/n}$ converges almost sure and calculate the ...
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1answer
38 views

Coin throw approximation

We have coin and we want to determine the probability $p$, that this coin shows "head" after one throw. For this purpose the coin has been thrown $n$-times and $K_i$ is the event that in the $i$-th ...
2
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1answer
69 views

If $(X_n)$ are Poisson independent and $\mathbb{E}[S_n]\to+\infty$ then $\frac{S_n}{\mathbb{E}[S_n]} \rightarrow 1$ almost surely

Let $X_n$ be independent Poisson random variables with $\mathbb{E}[X_n] = \lambda_n$. Define $S_n = X_1 + \dots + X_n$. Show that if $\sum \lambda_n = +\infty$, then $\frac{S_n}{\mathbb{E}[S_n]} \...
3
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1answer
55 views

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
33 views

limit superior and law of large numbers [closed]

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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1answer
29 views

Extension of ergodic 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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0answers
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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
149 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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1answer
64 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 ...
1
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1answer
22 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
56 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
13 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
357 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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23 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\left(\lim_{n\to \infty} \bar{X}_n = \mu\...
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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
27 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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47 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 $$\frac{1}{n} \sum_{i=1}^{n} X_{i} \xrightarrow[\mbox{almost surely}]{n \to \infty}-\infty\ ?$$...
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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
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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 ...
2
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1answer
39 views

Law of Large Numbers for Martingales

The following question has me stumped: Let $X_n$ be a square integrable martingale with $E((X_n)^2)\leq n$ for all $n$. Prove that $X_n/n$ tends to $0$ almost surely. (this is in a sense a law of ...
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80 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 ...
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1answer
29 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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59 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 ...
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1answer
41 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
20 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
53 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....
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1answer
70 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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0answers
52 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....
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1answer
48 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
55 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
65 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
48 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
29 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
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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
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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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0answers
31 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}...