Questions tagged [law-of-large-numbers]

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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Tail Probabilities of a martingale difference sequence

I'm currently facing the problem, that I can neither prove nor find a counterexample for the following statement. Let $q \in (1,2)$ and let $(D_n)_{n \in \mathbb{N}}$ be a martingale difference ...
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some variant of the law of large numbers?

Let $\{X_i\}$ be iid random variables on $\mathbb R$ for some probability measure $\mu$, $X_i\sim X$, $f: \mathbb R\to \mathbb R$ is a continuous function and expectations $\mathrm EX$ and $\mathrm ...
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Convergence of sum of MLE's over all possible $0/1$ sequences

Fix $N\in\mathbb N$ and take $\Theta=\{\frac1{N+1},\ldots,\frac N{N+1}\}$. For $y^T\in\{0,1\}^T$, let $\hat\theta(y^T)$ be the number in $\Theta$ that is closest to $\frac1T\sum_{i=1}^Ty^T_i=:k(y^T)$. ...
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Convergence in probability of subsets of random variables

Let $\{X_k\}_{k\geq 1}$ be random variables that satisfy all requirements for the applicability of the Weak Law of Large Numbers apart from having the same expected value, and such that: $E[X_k] = 1$ ...
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Law of large numbers over supremum of functions

Let $(X_i)_{i=1}^n\subset [0,1]$ be iid random variables and let $\mathcal{F}$ be a functionclass, containing bounded functions i.e. $f:[0,1]\rightarrow [-a,a]$. I need to show, $$ \lim_{n\rightarrow \...
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How do I prove that the weak law of large numbers holds?

We have given $X_1,X_2,...$ an i.i.d. sequence of random variables with $$\Bbb{P}(X_1=1)=\Bbb{P}(X_1=-1)=\frac{1}{2}$$ From class we know that then the characteristic function is $\Phi_{X_i}(t)=\cos(t)...
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(strong law of large numbers) We played a game in a casino. $X_i$ the money we won or lost the i-th time....

>We played a game in a casino. $X_i$ the money we won or lost the i-th time. Each time that we win, we take 1 dollar. When we lost, we lost 1 dollar. If p is the probability of winning and q the ...
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Why does the uniform law of large numbers hold with non-i.i.d. random variables in Bayesian experimental design?

This paper, Asymptotic theory of information-theoretic experimental design, studies Bayesian experimental design where in each round $n$, the experimenter selects a stimuli $X_n$ that maximizes mutual ...
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Finding covariance matrix for bootstrapped errors in OLS

Let's say we have matrix $x \in \mathbb{R}^{n \times k}$, $y \in \mathbb{R}^n$ and $\beta^*$ vector, which $\beta^* = \arg\min_\phi\sum_i (y_i - x_i\phi)$, i.e. we have classic regression problem and $...
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Proving That A Sequence Does Not Obey The Weak Law Of Large Numbers

I have a sequence $f_n$ of random variables such that $\mu( \{ f_n = n \}) = \mu( \{f_n = -n \}) = \frac{1}{2}$, and need to prove that it doesn't satisfy WLLN. Clearly, $E(f_n) = 0$, so I have to ...
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Show a Function Obeys the Strong Law of Large Numbers

Let $f_n$ be a sequence of random variables such that $\mu ( \{f_n = 0 \}) = 1 - \frac{1}{n^2}$ and $E(f_n) = 0.$ I have to prove that $f_n$ obeys the Strong Law of Large Numbers. In order for a ...
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random variables and weak law of large numbers

Good morning, I'm having some trouble understanding the weak large number theorem that says: Let $(X_n)_{n\in{\mathbb{N}}}:\Omega \to \mathbb{R}$ be a sequence of real random variables with $X_n\in{L^...
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Law of Large Numbers for Martingale Difference Sequences (in probability)

I've been told in class that, given a Martingale Difference Sequence (MDS), $(X_t)_{t\geq0}$, if $\mathbb{E}|x_t|^p < \infty$ for some $p> 1$, then $$ \frac{1}{T}\sum_{t=0}^T X_t \overset{P}{\to}...
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"CLT implies LLN?"

By the central limit theorem we have that for a an iid sequence $X_i$ with mean $\mu$ and variance $\sigma^2$ that, $$\sqrt{n}\frac{\overline{X}_n-\mu}{\sigma}\rightarrow_d N(0,1)$$ as $n\rightarrow\...
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Gap between two consecutive order statistics under arbitrary distribution.

Consider an arbitrary distribution $\mathcal{D}$ supported on $[a,b]$ with density function $\phi(x)\in[\gamma, \Gamma]$ where $\Gamma\geq \gamma>0$. M i.i.d samples $\{d_i\}_{i=1}^M$ are drawn ...
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3 votes
1 answer
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Law of large numbers with incomplete observation

I am currently reading the book Introduction to Reinforcement Learning by R. S. Sutton and A. G. Barto. The authors often reason with the LLN. In particular, at one point there is an expression like ...
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maximal increment of renewal process and partial sums when only second moment exists

I am working on a problem that needs to bound the increment of the absolute value of centered partial sum process and its associated renewal process. the iid partial sum process $S(t)=\sum_{i=1}^{\...
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Statistics transforming vectors to summation notation

I have this question where I would like to get from one form to the following below. $\sqrt(N) ( x'x + x' \epsilon + \epsilon ' x + \epsilon ' \epsilon)^{-1} (x ' u + \epsilon ' u)$ to $ \frac{1}{\...
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Sample Mean of Correlated/Dependent Random Variables...

Suppose $\bar{U}$ and $\bar{V}$ are sample means from a highly correlated or highly dependent random process (e.g., waiting times from a queueing process) That is, let $$ \bar{U} = \frac{1}{n}\left(...
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How to prove law of large numbers doesn't hold

I'm reading Feller's An introduction to Probability theory and Its Application. I don't know how to prove when lln doesn't hold. I just know that the sufficient condition for the lln to hold is that $\...
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2 answers
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Prove weak law of large number doesn't hold for $S_n$ [closed]

$X_k$ is random variable. $S_n$ = $\Sigma$$X_k$. Prove that: If $|S_n|$ < $cn$ and Var($S_n$) > $\alpha$$n^2$, then law of large number does not apply to {$X_k$}. I don't know how to solve this ...
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2 votes
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Consistently estimate the covariance matrix with weakly correlated observations

Suppose there are T k-dimensional observations following the generating process: $Y_t = \mu + \epsilon_t$, where $\mu$ is the mean and $\epsilon$ is a weak stationary error with zero mean and time-...
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2 answers
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With $Y_n$ i.i.d, $EY_n = 1$, $X_n = \prod_{k=1}^n Y_k$, use Strong Law of Large Numbers to show $\frac{\log(X_n)}{n} \to c < 0$

Let $Y_n$ be a sequence of non-negative i.i.d random variables with $EY_n = 1$ and $P(Y_n = 1) < 1$. Consider the martingale process formed by $X_n = \prod_{k=1}^n Y_k$. Use the strong law of large ...
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Show that $E \sup_n |\frac{\xi_n}{n}|<\infty \Leftrightarrow E|\xi_1| \log^+ |\xi_1|<\infty$.

Problem: Let $\xi_1,\xi_2,...$ be a sequence of independent ientically distributed random variables. Show that $E \sup_n |\frac{\xi_n}{n}|<\infty \Leftrightarrow E|\xi_1| \log^+ |\xi_1|<\infty$. ...
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1 answer
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Convergence almost sure and strong law of large number

Let $\{B_n\}$ and $\{X_n\}$ random variables i.i.d., at $(\Omega, \mathcal{F}, P)$. (a) Suppose that $P(B_1 = 1) = p = 1 - P(B_1 = 0).$ Define $$\hat{p}_n(\omega):=\frac{card\{j | 1 \leq j \leq n , \ ...
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1 answer
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$\{T_n\}$ r.v. independent and equally distributed then $P\left[\sum_{n=1}^{\infty}T_n < \infty\right]=0$

Let $\{T_n\}$ random variables independent and equally distributed, at $(\Omega,\mathcal{F}, P)$, with $0 < m :=E(T_1) < 1$. (a) Let $0 < \epsilon < m$. Prove thath for almost every $\...
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1 vote
1 answer
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A question about probability convergence (using the Law Of Large Numbers)

I think this problem will help me learn some convergence techniques. Given $(\xi_j)_{j=1}^{\infty} \sim \xi$. Supposse that in probability $$\frac{1}{n}\sum_{j=1}^n \exp ( i s \xi_j ) \to E[\exp ( i ...
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3 votes
2 answers
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A possible misconception on the weak law of large numbers

Let $\{X_n\}_{n\geq1}$ be a sequence of i.i.d. random variables having common probability density function $f(x)=xe^{-x}, x\geq0$ Let $\bar{X_n}=\frac{1}{n}\sum_{i=1}^nX_i$ Then, $\lim_{n\to\infty}P(...
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1 answer
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Does the law of large numbers apply for a single iteration? [closed]

I argued with my brother, a math teacher. He presented a situation where a doctor would perform a procedure on you with a 50% chance of success, but has done it 20 times in the past and has a 100% ...
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3 votes
0 answers
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Rick Durrett Probability Theory 5th edition exercise 4.6.1

Ex 4.6.1: Let $Z_1,Z_2, ...,$ be independent and identically distributed with $E|Z_i|<\infty$, let $\theta$ be an independent random variables with finite mean, and let $Y_i=Z_i+\theta$. If $Z_i$ ...
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1 vote
0 answers
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Harris chain converges to true probability

It is well-known that if a Harris chain $\Phi$ has an invariant probability $\pi$, then we have law of large number, i.e. $$ \lim_{n\rightarrow\infty}\frac1n \sum_{t=1}^n f(\Phi_t)={\rm E}_{\pi}[f(\...
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1 answer
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Find the limit of a series of interdependent bets

Description In a game, you can bet on the outputs 1, ... . , m, which are drawn with the probabilities p1, p2, ... , pm. If output i is drawn, the stakes on i are paid back m-fold, all the other ...
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2 votes
0 answers
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Continuity check in the application of Dini's Theorem for Riemann Sums and Glivenko-Cantelli Theorem.

I am unable to understand the following applications of Dini's theorem. In particular, verification of continuity of the sequence of functions. To prove the Riemann sum uniformly converges to the ...
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Probability to get a chocolate snowman or a chocolate reindeer

$N\in \mathbb{N}$ christmas presents will be distributed. In every gift there is additional either a chocolate snowman or a chocolate reindeer. It will be independent of each other and with the same ...
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-1 votes
1 answer
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Empirical means in the weak law of large numbers

I'd like to know why in the weak low of large number we talk about empirical means (plural) $\overline{X}_n$ (I'm considering a series of $n$ random variables $X_i$). Are we using the plural because ...
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5 votes
2 answers
88 views

Does the sum of random variables sampled with/without substitution differ for large populations?

We have a population of $N$ different balls. Half the balls are red, and half the balls are blue. We perform $N$ trials. In trials $i = 1,\cdots,N$ we pick a ball $B_i$ randomly. First, we pick the ...
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1 vote
1 answer
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Given $n$ iid random variables $X_1, ..., X_n$ with mean $\mu$, then $X_1+...+X_n = n \mu + o(n)$ a.s.

Here's my attempt, is it correct? Let $S_n = X_1+...+X_n$. Since $S_n / n = \mu$ almost surely as $n \rightarrow \infty$, then $S_n = n \mu + o(n)$ almost surely as $n \rightarrow \infty$.
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2 votes
1 answer
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Stable laws of probabilty and convergence.

I'm working on a probabilty exercise and i'm stuck at some point. I did the first 3 questions without trouble. Here is what is says : We consider $(\Omega,\mathcal{A},\mathbb{P}) $ a probability ...
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Almost sure convergence of modified random variables

Let $\{X_{n}\}_{n \geq 1}$ be a sequence of i.i.d random variables with mean $\mu$. Prove that: $$ \lim_{n \to \infty}\frac{1}{\log n}\sum_{i=1}^{n}\frac{X_{i}}{i}= \mu \quad a.s $$ I was thinking ...
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2 votes
0 answers
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Proving the Weak LLN formulation (Feller)

In chapter VI, Introduction to Probability theory and its applications, Feller obtains an upper bound for the the right tail and left tail of the binomial distribution. Theorem. If $r \geq np$, the ...
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1 vote
1 answer
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Expected Value Gambling Qn

Question: If a video poker player has found a 0.1% edge on a game and their goal is to make as much money from this play as they possibly can, with an average hourly EV of 20 dollars per hour. If they ...
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2 votes
2 answers
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Law of large numbers for partial sums of order statistics

Let $X_1, X_2,\cdots$ be independent and identically distributed random variables supported on the nonnegative integers with finite mean $\mu$. Denote the $i$th order statistic of the sub-collection $...
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1 vote
1 answer
60 views

The Equivalent Condition of the Weak Law of Large Numbers When Random Variables Are Uniformly Bounded

When the random variable $\{X_n,n\ge1\}$ satisfies the uniformly bounded condition, why does $$ \frac{1}{n^2}\operatorname{Var}\left(\sum_{k=1}^{n}X_k\right)\rightarrow0 $$ become a necessary and ...
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1 vote
1 answer
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Proving $\frac{1}{n}(\sum_{k=1}^{n}a_k\xi_{k}+\sum_{k=1}^{n}\eta_{k})\stackrel{P}\to 0 \iff \frac{1}{n^2}\sum_{k=1}^{n}a_k^2\to0$

A problem related to the weak law of large numbers,$\{\xi_k\}$ and $\{\eta_k\}$ are independent of each other and $\{\xi_k\}$ and $\{\eta_k\}$ are all independent sequence. They all obey the standard ...
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1 vote
1 answer
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Limiting Distribution for Squared Sum of i.i.d RVs divided by Sum of Square iid RVs

I'm facing some difficulties in determining the limiting distribution of $$\frac{\left( \sum_{i=1}^n X_i\right)^2}{\sum_{i=1}^n X_i^2}$$ as $n \to +\infty$ for a sequence of i.i.d. random variables $\...
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1 vote
0 answers
38 views

The law of large number of minimum i.i.d variables random

Note that from https://projecteuclid.org/journals/annals-of-probability/volume-21/issue-3/Limit-of-the-Smallest-Eigenvalue-of-a-Large-Dimensional-Sample/10.1214/aop/1176989118.full Let $\{x_{ij}\;|\;i,...
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1 vote
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30 views

Bounded first moment vs. bounded absolute moment?

I notice a couple different versions of the law of large numbers, one that assumes finite mean, another that assumes finite absolute first moment. Are these equivalent? I think we can argue one ...
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What happens when you add n Cauchy random variables that have a positive correlation?

The reason the law of large numbers fails to apply for the Cauchy distribution is that the distribution of $X_1+X_2+X_3+\dots X_n$ is the same as $X_1+X_1+\dots X_1 = n X_1$. This is billed a curious ...
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1 vote
1 answer
63 views

Law of Large Numbers and Convergence in Probability of Quadratic Form

Let $\{\mu_N\}_N$ be a sequence of $N$ dimensional vectors, and $\{\Lambda_N \}$ a sequence of $N \times N$ symmetric positive definite precision matrices. I drop the $N$ subscripts for simplicity of ...
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-1 votes
1 answer
89 views

Brownian motion running maximum [closed]

Could someone please explain how to find the limit as $t \rightarrow \infty$ of the running maximum of a brownian motion $B_t =\max W_t-ut$? Is there a way to calculate the limit itself and not just ...
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