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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Application of Law of Large numbers (1)

If we have an i.i.d random variable $X_i$ with mean variance $(\mu, \sigma^2)$. By Law of Large number, we have $\bar{X}\rightarrow^p \mu$. But can we use Law of large number as well and claim that ...
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Law of Large Numbers - utility/difficulty of various versions.

This may or may not be an answer to Is there an easy proof that the set of $x \in [0,1]$ whose limit of proportion of 1's in binary expansion of $x$ does not exist has measure zero?, depending on ...
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37 views

Something like law of large numbers

$X_n$ is an increasing sequence of non-negative random variables with $\mathbb{E}(X_n)\sim an^\alpha$ and $\text{Var}(X_n)\sim bn^\beta$ with $a,b,\alpha>0$ and $\beta<2\alpha$. Show that ...
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Show that the likelihood ratio converges to $0$ a.s.

Let $S$ be a finite set, for simplicity assume $S=\{1,2,...,m\}$. Let $f_0$ and $f_1$ be two non-equal probabilities defined on $S$, with $f_0(j)=P_0(X=j)$ and $f_1(j)=P_1(X=j)$, such that ...
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$\limsup \frac{|S_n|}{n}=\infty$

$X_n$'s are i.i.d symmetric with $E|X_1|=\infty$. Then $\limsup \frac{|S_n|}{n}=\infty$. How do I show $\limsup \frac{S_n}{n}=\infty$ and $\liminf \frac{S_n}{n}=-\infty$? My attempt: Let $c=\limsup ...
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Find the limit distribution of $Y_n =\frac{\Sigma_{i=1}^{n}X_i}{\sqrt{\Sigma_{i=1}^{n}|X_i|^2} }$.

Let $X_1,X_2,X_3,...$ be i.i.d. with uniform distribution on $[-1,1]$. Find the limit distribution of $Y_n =\frac{\Sigma_{i=1}^{n}X_i}{\sqrt{\Sigma_{i=1}^{n}|X_i|^2} }$. I think this is just a direct ...
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reference to iterated logarithm law and Smirnov law of empirical CDF

I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws. Let ...
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Intuition behind almost sure limit of $\frac{|S_n|}{n^{1/p}}$

Suppose $X_i$'s are non-degenerate i.i.d. Then (1) If $E|X_1|^p=\infty$ we've $\limsup_{n\rightarrow \infty} \frac{|S_n|}{n^{1/p}}=\infty$. And this is true $\forall p>0$ (2) However for $p=2$ ...
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Law of large numbers for arbitrarily dependent random variables

Consider a sequence of random variables $X_1,X_2,...,X_n$. No assumptions abou independence is made. Only joint probability density function is known, i.e. $f(x_1,...,x_n)$. Then Markov's theorem ...
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Strong law of large numbers

Suppose $X_i\in\mathcal{L}^2$ with expectation $0$ such that $\sum_{i=1}^\infty \mathbb{E}[X_i^2]/i^2<\infty$ and suppose they are pairwise non correlated. Does then the SLLN still hold?
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WLLN and a asymptotic property of the survival function

Let $X$ be a nonnegative random variable with right continuous distribution function $F$. Let $\bar{F}(x)=1-F(x)$, which is called the survival function. In an article by Hall and Wellner (pdf), on ...
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Kolmogorov's sufficient and necessary conditon for SLLN - What about pairwise uncorrelated RV?

Kolmogorov proved, that, as one considers independent (not necessary equally distributed) Random Variables: $\{X_n\}_{n\ge0}\subseteq \mathcal L^2$ With $\mathrm{Var} (X_n)=\sigma^2_n$ and without ...
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Check if the weak law of large numbers holds true for the following sequence of random variables

Suppose we have $n$ independent discrete random variables, whose distribution is as follows: $X(k)$, where $k$ is any integer from $1$ to $n$, can take any of three values: $-\sqrt{k}$ with a ...
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Strong law of large numbers using Fatou's lemma?

Let $X_n$, $n \in \mathbb{N}$, be a sequence of i.i.d random variables with $\mathbb{E}|X_1| < \infty$. I've been thinking about proving the strong law of large numbers using the following ...
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Sequence of random variables that follow WLLN but not SLLN

I have to contruct a sequence of random variables that follow Weak Law of Large Numbers, but don't follow Strong Law of Large Numbers. Can canyone give me any hint please? Basically i need to choose ...
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Strong and weak laws of large numbers

Let $X_1,X_2,\ldots$ be a sequence of random variables. Weak (strong) law of large numbers states that: If $X_1,X_2,\ldots$ are i.i.d. RVs and they have finite expectation $m$, then ...
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On proving that a infinite intersection of truth sets is empty and on the usefulness of almost surely.

I am trying to solve the exercise at the end of this page, the framework is that of measure theory where we are tossing a coin infinitely often so we are working with a probability triple $( \Omega, ...
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The law of large numbers - limits of $\max$ vs $\max$ of a limit.

Assume that $X_{1,1}, \dots , X_{1,n}, X_{2,1},\dots, X_{2,n}, \dots ,X_{n,1}, \dots , X_{n,n}$ are i.i.d. random variables, and that $\mathbb EX_{i,j}$ exists and is finite. From the strong law of ...
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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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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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29 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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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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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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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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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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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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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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58 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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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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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 ...