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).

learn more… | top users | synonyms

0
votes
0answers
13 views

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 ...
0
votes
1answer
29 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 ...
5
votes
1answer
44 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 $P(Y_m=1)<1$. ...
-1
votes
1answer
38 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 ...
3
votes
0answers
29 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 ...
3
votes
2answers
38 views

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 ...
0
votes
0answers
32 views

Weak Law of Large Numbers for asymptotically uncorrelated random variables [on hold]

$X_n$ is a sequence of random variables with $Var(X_n)\le{c} \space \forall \space n$ where $c \in (0,\infty)$ and $$Corr(X_i,X_j)\rightarrow 0 \space \space \text{if} \space \space ...
2
votes
2answers
58 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 := ...
0
votes
0answers
9 views

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 ...
0
votes
1answer
25 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 ...
0
votes
1answer
25 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 ...
0
votes
1answer
28 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 = ...
0
votes
2answers
28 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 ...
0
votes
0answers
30 views

Applicability of Law of Large numbers for a given sequence of random variables

What are the general methods of showing that law of large numbers doesn't hold for a sequence of random variables?
0
votes
0answers
1 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 ...
2
votes
1answer
39 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 ...
0
votes
3answers
48 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 ...
1
vote
1answer
19 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 ...
1
vote
0answers
24 views

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 ...
1
vote
1answer
34 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 ...
2
votes
1answer
61 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 ...
1
vote
1answer
51 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 ...
0
votes
0answers
23 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 ...
3
votes
1answer
34 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 ...
2
votes
1answer
88 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: ...
0
votes
2answers
29 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 ...
1
vote
1answer
27 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 ...
2
votes
0answers
44 views

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 ...
1
vote
2answers
64 views

Moving Limit within Probability/Law of Large Numbers [duplicate]

Edits in Bold When I look up the strong law of large numbers it says that (Looking at Discrete Random Variables) $$P\left(\lim_{n\rightarrow\infty}\bar{X}_{n}=\mu\right)$$ That got me wondering are ...
0
votes
0answers
10 views

Is it possible to use multiple time scale algorithm here?

Suppose a random sequence is being generated (the next term generated depends on the previous term, but we don't know any distribution) until we hit some specific number. We want to calculate the ...
2
votes
1answer
38 views

Using Taylors to show convergence in probability

I'd like to show that \begin{equation} \sqrt{n} \left( (1-\frac{1}{n})^{n\bar{X}} - e^{-\bar{X}} \right) \to 0 \end{equation} in probability for a random variable with mean $\mu$ and finite variance ...
2
votes
1answer
51 views

Is the law of large numbers reflected in SO reputation system?

I have recently encountered an interesting phenomenon on SO reputation system: Let $f(n)$ denote the current score of the $n$th best user. A sample that I collected at a given moment: $f(k)=k ...
1
vote
3answers
113 views

Central Limit Theorem vs. Weak Law of Large Numbers

So, just to begin with I feel like this is a problem I am massively overthinking, and the solution is very simple. That said, it has been a while since I've taken a math class, and so some of my ...
3
votes
1answer
58 views

$\sum_{k\ge 0} e^{-an} \frac{(an)^k}{k!}f(\frac{k}{n}) = \Bbb{E}\left(f\left(\dfrac{X_1+\cdots + X_n}{n}\right)\right)$

Hello everybody i need to show following equality $$\sum_{k\ge 0} e^{-an} \frac{(an)^k}{k!}f(\frac{k}{n}) = \Bbb{E}\left(f\left(\dfrac{X_1+\cdots + X_n}{n}\right)\right)$$ Where $(X_i)_i$ are ...
3
votes
1answer
105 views

If $X_{i}$ are I.I.D and $n^{-1}\sum_{i=1}^{n}X_{i}$ converges a.s/in-distribution to a constant $a$ is it true that $a=\mathbb{E}[X_{i}] $?

The question itself is in the title. It is immediate by the strong law of large numbers that if $X_{i}$ had a finite first moment then we would have a.e convergence (and thus in probability and in ...
0
votes
1answer
43 views

Strange identity in the proof of the Strong Law of Large Numbers

Above is an extract from the proof of the strong law large of large numbers with finite fourth moment. The $X_n$ are iidrv's with $\mathbb{E}(X_n)=\mu$ and $\mathbb{E}(X_n^4)<M$ for some ...
0
votes
0answers
29 views

Law of large numbers for U-Statistics with varying kernel

Let $U_n=\binom{n}{2}^{-1}\sum_{1\leq i<j\leq n} H_n(X_i,X_j)$ be an $U$-Statistic of order $2$ and with kernel $H_n$ depending on the sample size $n$. I wonder if there exists a Strong (or Weak) ...
1
vote
2answers
60 views

Betting: Gambler's Fallacy vs. Law of Large Numbers

I know this has been asked before, but I think not in this exact way, so here goes: Suppose you're going to bet on the flip of a coin. Your bet is always "HEADS", but the amount of your bet may vary, ...
0
votes
0answers
47 views

Law of large numbers with random weights

Let $\mu_i$ be i.i.d. RVs with mean zero, and let $a_i$ be random weights that are not independent and are not identically distributed, $i=1,...,N$. $\mu_i$ is orthogonal to $a_j\;\forall j$. Is ...
1
vote
1answer
50 views

integral approximation (law of large numbers)

I am totally at a loss with this question and don't even know where to begin. Let $g:[0, 1]\rightarrow \mathbb{R}$ be a measurable and Lebesgue-integrable function. $U_1, U_2, \dots$ be a series of ...
1
vote
0answers
29 views

Answer to my question strong law of large number

i see to use the law of the iterated logarithm that tell us: $\limsup_{n\to\infty}\left|\frac{S_n}{\sqrt {n\log\log n}}\right|=\sigma\sqrt 2\text{ with probability 1,}$ where: $S_n:=X_1+\ldots+X_n$ , ...
2
votes
2answers
101 views

Strong law of large number

Let $ \{c_n\} $ be a descending positive real sequence. Let $ X_1,X_2,\cdots $ be a sequence of i.i.d random variables. Is the following equivalent? ($1$) For any i.i.d sequence $ X_1,X_2,\cdots $ ...
1
vote
1answer
49 views

In Markov inequality proof, why is $\int_a^\infty xp(x) \, dx \ge \int_a^\infty ap(x) \, dx$

Markov inequality, $$\Pr(X \ge a) \le \frac{E[x]}{a}$$ Proof $$\begin{aligned} E(X) &= \int_0^\infty xp(x)\,dx = \int_0^a xp(x)\,dx + \int_a^\infty xp(x)\,dx \\ &\ge ...
1
vote
1answer
42 views

Random variables $x_i$ with $\lim_{k\to\infty}\frac1k\sum_{i=1}^kx_i=0$

I am looking for a sequence $(x_n)_{n\in\mathbb N}$ of random variables such that the sequence hasn't any expected value and $\lim_{k\to\infty}\frac1k\sum_{i=1}^kx_i=0$. I thought about using a ...
1
vote
1answer
31 views

Convergence of random variable

I've been facing the following problem: Let $(X_k, Y_k)_k$ be a sequence of $2$-dimensional, independent random variables, each with uniform distribution over $ B(0,k) $ Verify if the following ...
3
votes
0answers
67 views

Sum of sequence of random variables infinitely often positive

Let $X_1,X_2,\ldots$ be an infinite sequence of independent (but not necessarily identically distributed) random variables with $E(X_i)=0$ for all $i$. Set $S_n=\sum_{i=1}^n X_i$. I want to show that ...
0
votes
1answer
60 views

convergence in probability of function of random variables

Suppose that $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d random variables. If we have $E(|X_1|^k) <\infty$ for some $k>0$ and $f(x)$ is a bounded continuous function on $\mathbb{R}$. Is the ...
3
votes
3answers
134 views

Laws of large numbers and independence

I just did a brief review of various sources, and they all specify that if $X_i$'s are independent, identically distributed random variables, then $S_n/n \rightarrow E(X_i)$ (with respect to various ...
0
votes
1answer
32 views

Solving this random variable problem

This is an earlier problem Proving this random variable problem but generalised, maybe you want to take a look at that one first? $X_1,X_2,X_3,\ldots$ are IID random variable taking values in ...
1
vote
2answers
51 views

Proving this random variable problem

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$. Also $t\in(0,1)$. Define random variables $Y_1,Y_2,Y_3,\ldots$ recursively like $$Y_1 = (1+tX_1)$$ $$Y_n = ...