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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Deciphering proof of SLLN

I was looking at a proof of the string law of large numbers, and am having trouble finding where the proof uses the assumption that the random variables are identically distributed. I'll reproduce the ...
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1answer
46 views

Expectation and Convergence of Sum of Random Variables [closed]

Let $X_1, X_2, ...$ be a sequence of independent random variables with $$\mathbb{P}[X_i=1]=\mathbb{P}[X_i=-1]=\frac{1}{2}$$ Let's now consider the sum $S_n=\sum_{k=1}^{n} X_k$. I need to show three ...
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1answer
57 views

Limit of a sequence of random variables

Suppose $Z_n$ is a sequence of independent random variables, which are uniformly picked from the interval $[1,2]$. Show that: $$ \lim_{n_\rightarrow \infty}P\left(\left|\sqrt[n] {Z_1 Z_2\cdots ...
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0answers
19 views

When can weak law of large numbers be turned into a strong law?

I am trying to do the following exercise from Chung. I would like a hint if possible, rather than a full solution. Let $(X_n)$ be a sequence of random variables (not necessarily independent or ...
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22 views

Law of Large Numbers - IID Assumption

My first question here! I was doing some probability review and was just wondering why exactly we need the IID assumption for the law of large numbers to work? Intuitively it makes sense of course, ...
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1answer
41 views

Find a sequence of random variables $(X_n)$ with $\lim E(X_n^2) = 0$ but not obeying SLLN

I am looking for some sequence of random variables $(X_n)$ such that $$ \lim_{n \rightarrow \infty} E(X_n^2) = 0 $$ but such that the following almost sure convergence does NOT hold: $$ \frac{S_n ...
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40 views

Strong law of large numbers with changing expected value

Question: Suppose that $X_1^n,...,X_n^n \sim^{iid} X^n$ and $X^n \rightarrow X$ in distribution/weakly. Is it true that $\frac{1}{n} \sum_{i=1}^{n} X_i^n \rightarrow \mathbb{E}[X]$ almost surely? ...
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1answer
20 views

checking if consistent estimator using LLN

$X_1,X_2, \cdots , X_n$ be an iid sample from an exponential distribution with unknown parameter $\theta$ I need to show that $lim _{n \to \infty}$ Pr$(| (1/ \bar{X_n} ) - \theta | \ge \epsilon) = ...
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2answers
29 views

Large numbers and CLT: confusion over the behavior of the sum of iid random variable

In a nutshell I am confused about the fact that the fluctuations of the sum behave as $ \sqrt n $ but the empirical mean converges (fluctuations here behave as $ \frac {1}{\sqrt n} $). Below my ...
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2answers
52 views

Show that $\limsup|Y_{1}+…+Y_{n}|/n = \infty$ almost surely

Can someone help me with part c) of question 2.8 located here (a 2005 probability course from Warwick University): https://homepages.warwick.ac.uk/~masgav/teaching/pm05_sheet2.pdf The question is: ...
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23 views

Random variables converge question

Let $X_1, X_2, \ldots$ be random variables that are independent and identically distributed. with $E[X_i] = 0, V[X_i] = 1$. Then there exist a random variable $Z$, that $(X_1 + X_2 + \cdots + ...
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35 views

Law of large numbers for a continuum of random variables

Consider a continuum of random variables such that each takes the value $1$ with probability $p$ and $0$ with probability $1-p$. The random variables should be essentially pairwise independent. Sun ...
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19 views

Do higher order sample moments converge to the distributional mean?

The Methods of moments estimation is based on the law of large numbers, which says that the sample means of i.i.d. random variables from any distribution converge to the distributional mean as the ...
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1answer
47 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 ...
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24 views

Confused about strong convergence

If $X, X_1, X_2, \ldots$ are real random variables defined on a probability space $(\Omega, \mathcal{A}, \mathbf{P})$, we say $X_n$ converges almost surely to $X$, if ...
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Practical applications of SLLN where WLLN does not suffice?

Are there any practical applications of the Strong Law of Large Numbers for which the Weak Law of Large Numbers would not suffice? When, in practice, is the result $$\lim_{n\rightarrow\infty} ...
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1answer
24 views

Showing Stronger result of Weak Law of Large Numbers

So, Khintchine's form of the Weak Law of Large Numbers asserts that $i) E(X_1)=0 \Rightarrow (S_n/n) \rightarrow 0$ The stronger result is: $ii) E(X_1)=0 \Rightarrow E(\|S_n\|)=o(n)$ Now ii) is ...
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1answer
33 views

Is there a version of WLLN with weaker conditions than SLLN?

Any proof of the weak law of large numbers i know requires conditions under which one can also proofe the strong law of large Numbers. Is there a version of the weak law with conditions so decend, ...
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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 ...
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1answer
36 views

Does the Strong Law of Large Numbers imply the following?

The Strong Law of Large Numbers in my Probability textbook is given as follows. Let $X_n$ be a sequence of identically distributed pairwise independent $\mathbb{R}$-valued random variables. Let $S_n$ ...
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1answer
51 views

Geometric mean with the law of large numbers

I'm currently studying some probability and I'm stuck with this question. Let $R_1, . . . , R_n$ be independent continuous uniform over [0, 1] random variables. The geometric mean of $R_1, . . . , ...
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Is the large-$n$ limit of scatter matrix always equal to the underlying covariance matrix?

While trying to answer this question scatter versus covariance the following occurred to me: Suppose $f(\vec{x})$ is a probability distribution function for which the mean and the covariance matrix ...
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1answer
58 views

Weak/strong law of large numbers for dependent variables with bounded covariance

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of $L^2$ random variables with expected value $m$ for all $n$. Let $S_n=\sum_{i=1}^n X_i$ and $|\mathrm{Cov}(X_i,X_j)|\leq\epsilon_{|i-j|}$ for finite, ...
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Unclear of details of limit for a Taylor expansion of moment generating function - test this Wed!

I keep coming across these limits - the context is moment generating functions and the Central Limit Theorem, but I'm guessing it's a more general question - here is one example (from the proof for ...
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1answer
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Approximate normal distribution(this is different from what I asked earlier $\log(n)$ is replaced by $\sqrt{\log{n}}$)

Let $ X \sim N (0, 1)$. For $x$ large enough, the tail of the distribution of $X$ may be approximated as $$P(X > x) \sim e^{-x^2/2}/(x\sqrt{2\pi})$$ Consider a sequence of independent r.v. all ...
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1answer
35 views

Central limit theorem in the setting of Poisson variables

Setting Given $S_{\lambda} \overset{d}{\sim} \operatorname{Poisson}(\lambda)$. Let $G_{\lambda}(t)$ be the distribution function of $\frac{S_{\lambda}}{\lambda}$. I need to determine ...
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55 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 $P(Y_m=1)<1$. ...
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1answer
44 views

Can you prove the Law of Large Numbers?

So clearly it is not hard to experimentally prove that the more times something is done, say rolling a die, the closer your experimental results come to your theoretical likelihoods, but is there a ...
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44 views

Stronger version of strong law of large numbers

Let $(X_i)_{i\in\mathbb{N}}$ be pairwise independent random variables where $E[X_i]=0$ for all $i\in\mathbb{N}$ and $\sup_{n}E[X_n^2]\lt\infty$. Then for $S_n=\sum_{i=1}^n X_i$ and ...
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2answers
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Almost surely convergence of the sequence

Let ${X_n}$ be a sequence of independent and identically distributed, square integrable random variables. Write $ u = E(X_n)$. Study the almost sure convergence, as $n \rightarrow \infty$, $$S_n ...
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1answer
68 views

Simplified Strong Law of Large Number by Using Truncating Function

Consider $X_1,X_2,...$ be i.i.d. random variables with $E|X_i| <\infty$ and let $EX_i := \mu$ and $S_n := \sum_{i=1}^n X_i$. Now, consider the corresponding truncated random variables $Y_k := ...
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Almost sure convergence of Chi-Squared variable

Setting $$X_1,X_2,\ldots \overset{d}{\sim} \mathcal{N}(0,1)$$ $$S = X_1^2 + \ldots + X_n^2$$ I would like to show $\frac{R_n}{\sqrt{n}} \rightarrow 1$ almost everywhere. I am using Borel-Cantelli ...
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1answer
38 views

Show almost everywhere convergence for variable with Chi distribution

Setting $$X_1,X_2,\ldots \overset{d}{\sim} \mathcal{N}(0,1)$$ $$R_n = \sqrt{X_1^2 + \ldots + X_n^2}$$ I would like to show $\frac{R_n}{\sqrt{n}} \rightarrow 1$ almost everywhere. I have tried to set ...
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1answer
66 views

Example for which weak law of large numbers holds, but strong LLN does not

Let $(X_n)_{n\gt 2}$ be independent, $P(X_n=n)=\dfrac{1}{n\log n}, P(X_n=0)=1-\dfrac{1}{n\log n}$. I want to show that this sequence obeys the WLLN, but not the SLLN. I am trying to prove the ...
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1answer
31 views

Expected value of the negative portion of sum of poisson random variables

Setting: Defn: for every $x \in \mathbb{R}$ define its negative part by $x^{-} = -x$ if $x \leq 0$, and $x^{-} = 0$ if $x > 0$ Let $\{X_j, j \ge 1\}$, $X_j \overset{d}{\sim} Poisson(1) = \Pr\{X = ...
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2answers
62 views

Proof of the law of large numbers for higher moments

Let us work on some probability space $<\Omega,\mathscr{A},\mathbb{P}>$: I'm looking for (independent) proofs of two proofs, of the generalised weak and strong law of large numbers ...
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1answer
9 views

Asymptotic distribution of ratio / multiplication of two variables

Suppose $\rightarrow_D $ denotes convergence in distribution. If we know $$ f_1 \rightarrow_D W_1 $$ $$ f_2 \rightarrow_D W_2 $$ Can we say something about the convergence of $$ f_1 f_2 ...
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1answer
54 views

Law of large numbers, problem

I have a specific problem to solve using strong law of large numbers. Let $X_k$ be independent uniform random variables on interval $(0,k)$. Let $Y_n ={1 \over n^2}\sum\limits_{k=1}^n {X_k^3 \over ...
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3answers
54 views

Is it possible for a reality to exist where the law of large numbers does not apply? [closed]

Being more specific, is the law of large numbers more empirical than it is rational? That is, is it more a feature of the observable universe that it is something that is true based on our definition ...
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1answer
27 views

Notation with random variable $\overline{X}_{n^2}$ in Strong Law of Large Numbers proof.

I'm reading the proof for the strong law of large numbers. It says: Let $X_1,X_2,\ldots$ be a sequence of independent and i.i.d. random variables with finite mean $\mu$ and finite variance ...
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Applying the Law of Large Numbers recursively

If I want to apply the LLN for an estimator that uses another estimator, can I apply the LLN inside the summation and after it simplify the outer summation by using the expected value of the inner ...
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1answer
37 views

Recurrent Set and i.i.d. random sequence

Consider an i.i.d. discrete random sequence $\{X_i\}$, suppose $EX_1 \neq 0$ and define $R:=\{x: \text{ $x$ is recurrent value for $S_n$}\}$. I was trying to show the set $R = \emptyset$ where $S_n ...
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2answers
124 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 ...
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1answer
75 views

Rate of convergence for 'Law of large numbers'

Consider the following question: A coin has the probability of landing of head equal to 1/4 and is flipped 2000 times. Use the law of large numbers, find a lower bound to the probability ...
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30 views

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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1answer
44 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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1answer
99 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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2answers
38 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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1answer
30 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 ...