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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A detail on a proof of the strong Law of Large Numbers.

In the following blog post https://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/ one is presented with a nice account of the LLN. Suppose that I have shown that if $(n_j)$ is a ...
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Finiteness of the sum of the product of an i.i.d. sequence

Before I go to the statement of my question I just want to say a few words about the personal background of this question. I have recently taken a course on stochastic differential equations without ...
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Is there a connection between uniform law of large number and Ibragimov's conjecture?

In limit theorems, one of the biggest problem is to give an answer to Ibragimov's conjecture, which states the following: Let $(X_n,n\in\Bbb N)$ be a strictly stationary $\phi$-mixing sequence, ...
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Prove that the product of 2 vectors Normally distributed converges for large dimensions to the full zero matrix

Let $\mathbf{x}, \mathbf{y}$ $\in C^{M \times 1}$ are two i.i.d. vectors with distribution $\mathcal{CN(0,1)}$. How we can prove by the strong law of large numbers that: $\lim_{M\rightarrow \infty} ...
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1answer
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Law of large numbers for nonnegative random variables [closed]

I'm struggling with specific variation of Strong Law of Large Numbers. Suppose $X_1,X_2,\ldots$ are independent, identically distributed, nonnegative random variables and $\mathbb{E} X_1 = \infty $. ...
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28 views

What is the proportion of edges with a certain capacity among possible edges?

Assume you have a graph with $n$ nodes/vertices and we can assign to each node a "type" : type $0$ or type 1. The types are independent. The probability of type $0$ is $$1 - \lambda \in (0,1)$$ and ...
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Law of Large Numbers once again

$(\phi_n)$ - independent random variables with Poisson distribution with parametr $\lambda=2$ , $(\psi_n)$ - gaussian random variables with $\text{E} \psi_n=0$ and $\text{Var}\psi_n=\frac{1}{n^2}$. ...
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How to apply strong law of large numbers for non-constant mean and variance

Consider $X_1, ..., X_n,...$ independent random variables that satisfy: $E[X_i]=1+\frac{1}{1+i^2}$ and $Var[X_i]=\sqrt{i}$. Show that the chance of $\frac{1}{n}\sum\limits_{i=1}^n X_i$ converges to ...
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49 views

A dice and Law of Large Numbers

The dice A has 2 red faces and 4 green, and the dice B conversely: 4 red and 2 green. We toss a symetric coin; if come up heads, we choose the dice A, otherwise - the dice B. Next we execute a series ...
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Show that $\frac{X_1+\dots+X_n}{n}$ converges a.s. for $X_n \sim U([0,1-2^{-n}])$ independent

Let $(X_{n})$ be a sequence of independent random variables and let $X_{n}$ have a uniform distribution on $[0, 1-2^{-n}]$. Prove that the sequence: $$\frac{X_{1}+X_{2}+\dots+X_{n}}{n}$$ converges ...
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Weak Law of Large Numbers and Central Limiting Theorem problem

From past experience, a teacher knows that the result of an exam is a random variable, with average $75$ and standard deviation $8$. How many students must take the exam to guarantee, with a ...
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empirical estimation of Bernoulli distribution (lower bound)

Let $X_i$ be an i.i.d. Bernoulli distributed sequence, with probability $p$ being 1. Now consider an empirical estimation of $p$ with $l$ samples and I am looking for a lower bound for following ...
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1answer
31 views

Probability that sum of independent random (non iid) variables exceeds the variance infinitely often

Let $X_1,X_2,\ldots$ be independent random variables, with $\Pr[X_i=\sigma_i]=\Pr[X_i=-\sigma_i]=1/2$ ($\sigma_i\geq 0$). Suppose that: (a) There is an upper bound $B$ such that $\sigma_i\leq B$ for ...
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martigale convergence theorems

Let $S_n = X_{1}+\cdots + X_{n}$ be a martingale satisfying $E[X_{k}^{2}]\leq k<\infty$, for all $k$. Show that $S_{n}$ obeys the weak law of large numbers: ...
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1answer
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Proof of (a step in the proof of) the Law of Large Numbers

Theorem: Let $f:[0,1] \to \mathbb R$ be a measurable function bounded by $c$. Let $U_1,U_2,\ldots,U_n$ be i.i.d. and Uniform$(0,1)$. Then: $$ P \left( \left\lvert ...
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32 views

bound on expectation of a two-variable function under an independent distribution

Consider a probability distribution $P(x)$, a set observed samples $S = \{x_1,\cdots, x_n\}$ where $x_i \sim P(x)$ for $i \leq n$, and a symmetric function $h(x,y)$. How can one efficiently ...
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Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables such that $$P(X_n=n+1)=P(X_n=-(n+1))=\frac{1}{2(n+1)\log(n+1)}$$ $$P(X_n=0)=1-\frac{1}{(n+1)\log(n+1)}$$ Prove that $X_n$ ...
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43 views

Reducing the Weak Law of Large Numbers criterion

I'm having a really tough time wrapping my head around the weak and strong laws of large numbers especially regarding modes of convergence. So a the WLLN critierion I was given was $$ \lim_{N \to ...
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Law of large numbers - almost sure convergence

Quoting Wikipedia may be considered not a very nice partice, but: When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to 1/2. Therefore, ...
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inferring parameters from limting relative frequencies

I refer to my previous question concerning what i call the converse strong law of large numbers (instead o the normal SLLN given the probability=p that with prob1, the limiting relative frequency=p; ...
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1answer
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Weak Law of Large Numbers for uniformly integrable, independent random variables

On page 58-59 of the notes by Knill (found here :http://www.math.harvard.edu/~knill/teaching/math144_1994/probability.pdf ) there is a version of the WLLN whose proof I have trouble understand. On ...
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Question about Strong law of large numbers

I am confused by this problem. Let $F$ be a distribution function with $F(0-)=0$ and $F(1)=1$ and let $\mu$ be the associated law. Let $m_k=\int_{[0,1]}x^k dF(x)$. Define \begin{array}{c c c} ...
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Sequence of finite, positive and i.i.d random variables and limit of $\frac{S_{n+1}}{S_{n}}$

Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of finite, positive and i.i.d random variables and let's call $\mu:=E(X_{1})>0$ and $S_{n}:=\sum_{i=1}^{n}X_{i}$. We know that ...
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35 views

Does the law of large numbers pin down the distribution of an infinite sample?

Imagine you draw (independently) an infinite amount of draws from a random variable with infinite support, and the strong law of large numbers applies. We know the average for sure will be equal to ...
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1answer
29 views

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
54 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
69 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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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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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
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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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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
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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
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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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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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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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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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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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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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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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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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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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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
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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
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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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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 $P(Y_m=1)<1$. ...