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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21 views

strong law of large number with Covariance

I'm supposed to prove that $ \frac{1}{n} \sum_{j=1}^n (X_j-\overline X_n)(Y_j-\overline Y_n)$ converges almost surely to $Cov(X,Y)$ assuming that $ (X_i,Y_i)$ are iid with the same distribution as $(X,...
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1answer
33 views

Strong law of large numbers and the central limit theorem

The strong law of large numbers and the central limit theorem use different modes of convergence. Is it nevertheless true that the strong law of large numbers can be shown from the central limit ...
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22 views

Empirical Distribution: problem from “All of Statistics” [duplicate]

I'm working through this textbook, and here is a problem I'm stuck on. Setting: We are given an unknown distribution $F$, observed independent data points $X_1, ..., X_n \sim F$, the empirical ...
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35 views

How to prove autocorrelation function converges in distribution to normal distribution

I am working on the proof of $n^{1/2}[\hat{\rho}(1),...,\hat{\rho}(h)]'\xrightarrow{d}(z_1,...,z_h)'$, where $\hat{\rho}(h)$ is the sample ACF of $x_t$, $\{x_t|t=...,-2,-1,0,1,2,...\}$ are i.i.d.$(0,\...
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Does the law of large numbers hold for covering numbers?

I am self-studying empirical process theory. I have encountered the covering number $N(\delta,\mathcal{G},P)$, as well as the empirical version $N(\delta,\mathcal{G},P_n)$. It seems intuitive to ...
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64 views

Finding $c$ such that $\mathbb{P}\left[ \lim_{n \to \infty}\frac{X_n}{n} = c \right] = 1$

For a given $X_n$ sequence of random variables such that for every $n$, $X_n\sim\mathsf{Geo}\left(\frac{1}{9n}\right)$. I am asked to find $c$ such that $$\mathbb{P}\left[ \lim_{n \to \infty}\frac{X_n}...
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1answer
36 views

law of large numbers for $P\{X_k = \pm 2^k\} = \frac{1}{2}$;

I want to show that the law of large numbers holds/doesn't hold for the sequence of independent random variables $P\{X_k = \pm 2^k\} = \frac{1}{2}$ A sufficient condition is $\frac{s_n}{n} \rightarrow ...
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1answer
22 views

Proof of Weak Law of Large Numbers with non-zero covariance

Let $ X_{1}, \cdots, X_{n} $ be a sequence of dependent random variables having the same finite mean $ \mu=\mathrm{E}\left(X_{1}\right), $ the same finite variance $ \sigma^{2}=\operatorname{Var}\left(...
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1answer
50 views

CLT infinite mean

I am trying to see if we can construct $(X_n)_{n\ge 1}$, a sequence of i.i.d random variables such that $E|X_n|=\infty$ while $n^{-1/2}(X_1+...+X_n)$ converges in distribution goes to a $N(0,1)$ ...
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1answer
23 views

Strong Law of Large Numbers for not identically distributed Bernoulli variables

Let $X_{n,j}$ be independent discrete random variables taking only two values. In particular, $X_{n,j}=-\mu_{n,j}$ with probability $1-\mu_{n,j}$ and $X_{n,j}=1-\mu_{n,j}$ with probability $\mu_{n,j}$,...
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1answer
45 views

Law of large number integral asymptotic results

Let $(X_n)$ be a i.i.d. random variables, following a uniform distribution. Using the weak law of large numbers, $$\lim_{n +\infty} \int_0^1 ... \int_0^1 \frac{\phi(x_1) + ... + \phi(x_n)}{\psi(x_1) + ...
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Strong law of large numbers - Continuous martingales - Reference request

Let $(\Omega, \mathcal F P)$ be a probability space with a filtration $\mathcal F_t$ and $M = (M_t,t\geq 0)$ be a real-valued continuous $\mathcal F_t$-martingale such that its quadratic variation ...
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2answers
53 views

For i.i.d. $\{X_n\}$ with non-constant $X_n > 0$ and $\mathbb{E}X_n = q \leq 1$ the product $\prod\limits_{n=1}^{+\infty}X_n \to 0$ almost surely

Suppose $\{X_n\}$ is i.i.d. with $X_n > 0$ and each $X_n$ is not constant almost surely. I want to show that if $\mathbb{E}X_n = q \leq 1$, then the product $\prod\limits_{n=1}^{+\infty}X_n \to 0$ ...
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15 views

Law of Large Number and Covariance

I understand that definition of covariance can be simplified to $$cov(X,Y)= \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]$$ For a sufficient large sample of size N, we can also conclude that $$\mathbb{E}...
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Growth in $n$ of average $\sum_i X_i/n$ for iid positive $X_i$ with unbounded expectation

I am looking for references in the growth in $n$ of the empirical average $\bar X = \frac 1 n \sum_{i=1}^n X_i$ where $X_i$ are iid non-negative random variables with unbounded expectation. For ...
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30 views

Weak Law of Large Numbers Convergence

I have the folllowing succesion of Random Variables defined as: $$\{X_n\}_{n =1}^\infty \hspace{.3cm} \text{ where } \hspace{.3cm} \mathbb{P} \left(X_n = \sqrt{\ln(n + \alpha)} \right) = \mathbb{P} \...
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1answer
24 views

Law of large numbers for non-stationary but independent process?

This question is motivated by the following: Suppose we have a conditional distribution $p(Y|X)$, denoting a “channel” (in the information theory sense). Suppose we fix a particular sequence $x_1,...$,...
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Sum of Independent, Mean Zero, Finite Variance Random Variables [duplicate]

Let $X_1, X_2, ... , X_n$ be a sequence of Random Variables: Each has Mean Zero : $\mathbb{E}[X_i] = 0$ for all $1 \leq i \leq n$. All are independent. All have finite variance uniformly bounded: $\...
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1answer
35 views

Expected value of mean of N Chi squared random variables

I understand that here they have used law of large numbers. But according to the law the result should equal the expected value of the random variable. But how is the expected value of the r.v only ...
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2answers
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proof that the sequence satisfies weak law of large numbers

strong textHow to check if this sequence satisfies the weak law of large numbers? $$P(X_n=1)= {{1}\over{3^n}} ,P(X_n=-1)= {{1}\over{3^n}}, P(X_n=0)= {1 -{{2}\over{3^n}}} $$ I calculated that: $$E(X_n)...
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1answer
52 views

A problem with a application of SLLN and CLT and distribution convergence

Let $Y_{i}\overset{i.i.d}{\sim} N(0,\sigma^{2})$ with $\sigma^{2}$ unknow. I'm trying to prove that $$ \sqrt{n}\left(\overline{Y}_{n}^{2}- \sigma^2 \right) \overset{D}{\to} N\left(0,2\sigma^4\...
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1answer
79 views

Some misunderstandings while reading the proof of Berry-Esseen Theorem

I'm reading the proof of the Berry-Esseen Theorem from Varadhan's notes and I'm having trouble with a couple of steps. The proof starts at the top of page 97. Confusion 1: On page 98 while estimating $...
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Validation of an idea about the relation of LLN and CLT

I am writing an educational paper about the Central Limit Theorem and Law of Large Numbers. The paper is meant for engineers, but with a strong emphasis on the underlying math. In the paper, I prove ...
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36 views

Law of large numbers without any assumption

I read the textbook Probability: Theory and Examples. 5th Edition by Durrett. Here is theorem 2.2.6 from the textbook. Let $\mu_n=E(S_n)$, $\sigma_n^2=var(S_n)$. If $\sigma_n^2/b_n^2 \to 0$ then $\...
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1answer
42 views

Law of large numbers without independence and identical distributed assumption

I find the theorem below on https://math.stackexchange.com/a/1366549/533565. But I cannot find it at neither the textbook I have nor the published paper in google scholar. I want to cite this theorem ...
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1answer
68 views

Proof of a corollary for the Strong Law of Large Number

Here is Theorem 1.13 from Mathematical Statistics Jun Shao. $X_i$ are i.i.d. A necessary and sufficient condition for the existence of constant $c$ for which \begin{align} \frac{1}{n}\sum_{i=1}^n X_i ...
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1answer
47 views

Show that $\mathbb{P}(\exp(\sum_{i=1}^n X_i) >1)\to\frac{1}{2}$ for $X_i\in L^2$ i.i.d. with zero mean

Let $X_1,X_2,\ldots\in L^2$ be i.i.d. random variables with $E(X_i)=0$. Show that $$\mathbb{P}\left(\exp\left(\sum_{i=1}^n X_i\right) >1\right)\to\frac{1}{2}$$ as $n$ tends to infinity. My ...
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1answer
63 views

Probability density functions that demonstrate the difference between the weak and strong forms of the law of large numbers

If $X_n$ is a sequence of iid random variables with expected value $\mu$, and the random variable $$ \overline{X}_n := \frac{1}{n} \sum_{i=1}^n X_i, \tag{1} $$ then the weak form of the Law of Large ...
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33 views

I want to find the asymptotic property for the product of a Gaussian random matrix and a related column unitary matrix.

I have known is: ${\mathbf{H}} \in {\mathbb{C}^{K \times N}}$, the entries of ${\mathbf{H}}$ are modeled as independent and identically distributed (i.i.d.) complex Gaussian random variables with zero ...
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1answer
46 views

Convergence almost surely and SLLN

I'm studying probability theory, especially about limit theorem. And I got some trouble in moving forward. The problem is: Let $(X_n)$ be i.i.d random variables with $E(X_1)=0$ and $E(|X_1|^p)<1$ ...
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0answers
23 views

Explanation of a particular step in the proof of law of large numbers

I am self-learning probability theory from Probability and Random processes by Grimmett, Stirzaker. I don't follow a particular step in the proof of the law of ...
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28 views

Almost surely convergence and SLLN

I'm studying probability theory, especially about limit theorem. When I tried to solve a question, I got some trouble. The problem is this: Let $\{X_n\}$ be i.i.d. random variables with $\mathbb E[...
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1answer
188 views

Uniform bound for law of large numbers

Let $(X_i)_i$ be a sequence of iid real valued random variables with finite variance. Is it true that given a bounded measurable function $f:\mathbb R^2\to \mathbb R$, then almost surely and uniformly ...
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31 views

Showing that the weak law of large numbers does not hold for some sequence of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of independent random variables with the following distribution: $$\mathbb{P}(X_n=-n)=\mathbb{P}(X_n=n)=\frac{1}{2\sqrt{n}},\quad \mathbb{P}(X_n=0)=1-\frac{1}...
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26 views

The convergence of the expectation for the whole sequence implies the convergence of random variable in subsequence?

I encountered such a statement in the proof of the law of large number, which states that $$\frac{E(\bar{S_n})}{n} \rightarrow \mu \Rightarrow \frac{\bar{S_{n_k}}}{n_k}\rightarrow\mu \,\,a.s.$$ where $...
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2answers
22 views

Why WLLN implies this convergence?

I'm trying to understand this theorem which shows the convergence of the sample standard deviation to the standard deviation parameter. I didn't understand why $\frac{1}{n}\sum X_i^2\to \sigma^2+\mu^2$...
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1answer
52 views

Proof check for almost sure convergence

I encountered this problem from Durrett (Exercise 2.5.1): Suppose $X_1, X_2, \dots$ are i.i.d. with $EX_i = 0$, $var(X_i) = C< \infty$. Use Kolmogorov's maximal inequality to prove that if $S_n = \...
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1answer
85 views

Product and sum of iid random variables converges a.s.

The set of independent and identically distributed r.v. $Y_{1},Y_{2},...,Y_{n}$ has a distribution $Pr(Y_i=1)=Pr(Y_i=1/2)=1/2\ \forall i>0$ . If $Z_n=\prod_{i=1}^{n}Y_i$ and $S_n=\sum_{i=1}^{n}Z_i \...
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1answer
30 views

Simple example to apply the SLLN

I'm trying to find concrete examples of the SLLN theorem. Before, let's see the statement of this theorem precisely from this book, page 81: Definition: We say that $X_n$ converges almost surely to $...
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1answer
132 views

A generalization of SLLN: Convergence of combination of iid variables

Problem: Let $(\varepsilon_i)_{i\geq 1}$ be a sequence of iid random variables. Let $(x_i)_{i\geq 1}$ be a sequence of real numbers. Assume that: $\mathbb E[|\varepsilon_1|]<\infty$ and $\mathbb E[...
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0answers
14 views

How can the Law of Large Numbers give approximations of the expected value?

Consider $10000 \ N(0, 1)$-distributed r.v.’s and let $Z_n = \frac{1}{n} \sum_1^n X_i^2$ Calculate $E(X^2_1 )$. How does this relate to the statement of the Law of Large Numbers? Explain how this ...
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31 views

Does the weak law of large numbers hold for any sequence of random variables with pairwise negative covariances?

Let $\{X_n\}$ be a sequence of random variables with $EX_n = \mu$, $\text{Var}(X_n) < \infty$ for every $n\in\mathbb{N}$ and $\text{Cov}(X_n, X_m) < 0$ for any $n\neq m$. Does the Weak Law of ...
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1answer
33 views

Sample mean converges to 1/2 with error less than 0.01 and probability 0.99

The sample mean random variable of $N$ IID random variables with $X_i$ ~ $U(0, 1)$ will converge to 1/2. How many random variables need to be averaged before we can assert that the approximate ...
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1answer
34 views

SLLN for iid $X_n$

For i.i.d $X_n$'s with $E[(X_1)_-]<+\infty$ and $E[(X_1)_+]=+\infty$, I want to prove that $$\frac{1}{n}\Sigma_{i=1}^nX_i\xrightarrow{\text{a.e}} +\infty$$ as $n\xrightarrow{}+\infty$. I know that ...
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1answer
67 views

Proof of more general strong law of large numbers for dependent random variables

I want to show the following version of the strong law of large numbers. This is the longest proof I've ever attempted, and I feel a bit overwhelmed. Let $X_i$ be a sequence of real-valued random ...
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32 views

Law of Large Numbers for Uniform Discrete Distribution

Suppose X is uniform discrete distribution from a set of (1,2,3,...,m). How do i investigate the law of large numbers for this? I thought of doing this by maybe setting m as 10 and having sample of ...
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1answer
18 views

Generalisation of strong law of large numbers for kth moments

I have a quick question regarding the SLLN, and haven't been lucky enough to find the answer to this after searching on the internet but will ask on this forum: I am aware that the SLLN states that ...
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1answer
21 views

Limiting Value Sum $c^X$ for Uniform $RV$

I have $n$ draws $X_i$ from a Uniform r.v. on $[0,1]$ and want to find the limit of $\frac{1}{n} \sum_{i=1}^n c^{X_i}$ where $c$ is a known constant in $[0,1]$. My thought is to transform this ...
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1answer
115 views

If $X$ is a Lévy process, can we show $\frac1tX_t\xrightarrow{t\to0+}\operatorname E\left[X_1\right]$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a $\mathbb R$-Banach space, $(X_t)_{t\ge0}$ be an $E$-valued Lévy$^1$ process on $(\Omega,\mathcal A,\operatorname P)$ and $\...
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0answers
64 views

Does The Law of Large Numbers Apply to All Events?

My school has a research program and I potentially want to use this topic but I'm unsure if it can be answered with a simple response or if the question merits any research for that matter. I want to ...

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