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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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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31 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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33 views

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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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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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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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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28 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
52 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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68 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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43 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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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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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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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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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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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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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 $P(Y_m=1)<1$. ...
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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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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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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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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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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
76 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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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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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
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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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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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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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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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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 ...