Questions tagged [law-of-large-numbers]

Several theorems stating that sample mean converges to the expected value as $n\to\infty$. There is a weak law and a strong law of large numbers.

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70 views

Limiting distribution of $\sum_{j=1}^{p}\lambda_j U_j$

Assume $U_j$ are $\chi^2(1)$ random variables and $\lambda_1, \ldots, \lambda_p$ are the eigenvalues of a covariance matrix $\Sigma = (r^{|i-j|})_{ij}$ with a Toeplitz-type structure (for some fixed $|...
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Law of Large Numbers for geometrically decaying sequences

Let $(X_n)_n$ be a sequence of i.i.d. random variables, and let $\rho \in (0,1)$. Is there any asymptotic Theorem for the following random variable: $$ Y = \lim_{n\rightarrow \infty}\sum_{i=0}^n \rho^...
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Sum of squared normals: $ \sum_{k=1}^{n}U_k^2 = \frac{1}{n}\sum_{k=1}^{n}a_k^2 \cdot \Gamma\left(\frac{n}{2},\frac{1}{2}\right)$

Assume $U_k \sim \mathcal{N}(0,a_k^2)$, where $a_k \rightarrow c > 0$ as $k \rightarrow \infty$. It follows that $U_k^2 \sim \Gamma(\frac{1}{2}, \frac{1}{2a_k^2})$. I'm interested in the exact and ...
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42 views

Does central limit theorem help in making inference only about the population mean and not other parameters?

As per the weak law of large numbers, if your sample size is large, your mean of the sample is likely to be closer to the population mean than in a smaller sample. Additionally the CLT tells us the ...
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37 views

Central Limit Theorem Approximation and Relation to Law of Large Numbers

Assume the Linberg-Levy CLT to where we know $$\sqrt{n}\frac{\bar{X}_n-\mu}{\sigma}\xrightarrow{d}N(0,1).$$ I feel like I commonly see then that $$\bar{X}_n\approx N(\mu,\frac{\sigma^2}{n}),$$ but ...
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Weird behavior in simulating LLN for standard deviations in R

I am simulating the law of large numbers and how it applies to standard deviation, as well. I wrote this code that works well but there is something that I am having a hard time understanding. ...
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80 views

Confused about conditions of the weak and strong laws of large numbers

I am a little confused by what conditions need to hold for the weak law of large numbers (WLLN) and the strong law of large numbers (SLLN) to be true. It seems different sources give me different ...
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23 views

Terminology - population and sample in repeated rolling dice experiments

The the law of large numbers wikipedia page has an example on fair, six-sided dice: According to the law of large numbers, if a large number of fair six-sided dice are rolled, the average of their ...
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Finite sample behavior of learning algorithms

In machine learning, given a joint distribution $\mathbb P_{X, Y}$, where $Y=\{0, 1\}$ making it a binary classification problem, and $N$ iid training samples $\{(x_i, y_i)\}_{i=1}^N$ and iid test ...
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Bernoulli distribution and Strong Law of Large Numbers

I'm trying to find concrete examples of the SLLN theorem. Before, let's see the statement of this theorem precisely from this book, page 81: Definition: We say that $X_n$ converges almost surely to $...
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671 views

Law of Large Numbers

Suppose X is uniform discrete distribution from a set of (1,2,3,...,m). How do i investigate the law of large numbers for this? I thought of doing this by maybe setting m as 10 and having sample of ...
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21 views

Convergence of Normalised Sum of IID Random Variables

I have a Markov chain X that starts from the stationary distribution. Let define $S_n = X_1 + \cdots + X_n.$, where $X_i$ is the state of the Markov chain. Let's have 3 states. I wanted to prove the ...
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26 views

If $X_n-X=o_p(N^{-\alpha})$, $f(\cdot)$ is smooth, do we have $f(X_n)-f(X)=o_p(N^{-\alpha})$?

If $X_n-X=o_p(N^{-\alpha})$ with $\alpha>0$, $f(\cdot)$ is smooth, do we have $f(X_n)-f(X)=o_p(N^{-\alpha})$? I guess this is true as $f(X_n)-f(X)=f'(X)(X_n-X)+o_p((X_n-X))$, which has the same ...
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Is there a law or concept related to unchanging average?

I have a survey comprising 30 questions with 50k respondents where respondents mark each answer on a 5 point scale. An average of all the questions for all the respondents is metric we track monthly. ...
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Weak Law of Large Number to Central Limit Theorem [closed]

Consider a distribution with unknown mean μ and population standard deviation σ=30. Using the Weak Law of Large Numbers, what is the minimum sample size in order to attain a probability of at least 99%...
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35 views

Law of large numbers applied to $\overline{X}$ [duplicate]

The law of large numbers states that if we have $X_1,\ldots X_n$ independent variables which are identically distributed, then: $$\overline{X}:=\frac{1}{n}\sum_{i=1}^nX_i \longrightarrow \mu \quad \...
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62 views

Difference between uniform laws of large numbers and law of large numbers

How to understand the difference between uniform laws of large numbers and law of large numbers? In particular, what does the word "uniform" mean?
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37 views

Are this simple claim and its proof correct?

Suppose random sequence $\{X_{i}(N)\}_{i=1}^{N}$ is a row-wise i.i.d. triangular array, where $N$ is sample size. This means for any given $N$, $X_{i}(N),\dots,X_{N}(N)$ are i.i.d. following ...
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108 views

Why taking an average makes convergence to zero faster?

Let $f(x,y)$ be some density, and let the leave-one-out Nadaraya-Watson estimator $\widehat{f}_{-i}(x,y)$ be defined as follows: $\widehat{f}_{-i}(x,y)=\frac{1}{(n-1)h^2}\sum_{j=1,j\neq i}^nK(\frac{(...
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aquiring different results for a limit using CLT and LLN

I'm having some trouble with convergence in distribution and convergence in probability, mainly because I'm getting different results that seem to contradict each other using the Central limit theorem ...
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58 views

Convergence of bootstrap standard error estimate (one of the problems from Efron's book)

I am trying to solve problem 2.5 from the Efron/Tibshirani book, An Introduction to the Bootstrap (page 16). The problem asks to show that by applying the weak law of large numbers, the bootstrap ...
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Proving Identifiability Using Law of Large Numbers? [closed]

Well normally proving identifiability follows by showing that $p_{\theta}(x)=p_{\theta'}(x)$ implies $\theta=\theta'$. Usually this proceeds by showing that a function dependent on $\theta$, such as ...
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2answers
62 views

Does random walking have a memory?

I do not know if this is a good question, but I have not found an answer for it anywhere, does a binomial random variable have memory, [simple logic says it can have no memory, but statistics show ...
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42 views

Law of large numbers vs Short events vs Probability

:STATEMENT: Suppose an individual plays a gambling game where it is possible to lose $1.00$, break even, win $3.00$, or win $10.00$ each time she plays. The probability distribution for each outcome ...
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Central limit theorem implications about $\bar{X}$

The central limit theorem (CLT) and law of large numbers (LLN) look to make the same claim. $$\text{CLT: }\sqrt{n}\big(\bar{X}_n-\mu\big) \rightarrow N(0,\sigma^2) $$ $$\text{LLN: }\bar{X}_n \...
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47 views

Linear regression with one generated regressor

Suppose I have the regression model: $Y_i=T^{\top}_{i}\beta_0+e_{i}$ with $E(e_i|X_i)=0$, where we have two regressors $X_i,\ E(D|X_{i})$ so that $T^{\top}_{i}=[X_i,\ E(D|X_{i})]$. $X_{i}$ is a ...
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Can someone provide a proof for this theorem that was used in the proof of law of large number?

Let $X_1,X_2,X_3,...$ be $i.i.d.$ with finite mean $\mu$. Then let $Y_i=X_i1_{\{|X_i|<i\}}$ There will be only finitely many terms such that $Y_i\neq X_i$ While that lecture notes did not ...
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Is this proof of convergence in probability to zero correct?

I want to show that $A=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(\widehat{B}_{i}-B_{i})X_i$ converges in probability to 0, where $B_i=E(C_i|Z_i)$ and $C_i$ is i.i.d. binary and $Z_i$ is a discrete random ...
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452 views

Showing the weak law of large number of non-IID sequence of random variables

Common proofs on the law of large numbers usually assume a sequence of IID random variables. If $X_1\dots X_n$ has a common expected value $\mu$, finite but not necessarily common variance (hence not ...
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260 views

Should bootstrapping and collecting sample means from a series of binomial distributions result in standard normal?

I’m trying to better understand several statistical concepts (bootstrapping, central limit theorem, and confidence intervals) by applying them to a binomial distribution (you can think of it as a coin ...
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2answers
76 views

Why do typical sequences have probabilities $\sim2^{-nH(p)}$?

I've been reading a bit about typical sequences (in particular from these notes (pdf alert), pages 3 and 4). Let us focus on the case of binary sequences for simplicity. As far as I understand the ...
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30 views

Law of Large Numbers

How do i proceed with this? any hints would help
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1answer
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Posterior mean with MCMC

Let's say we have a posterior distribution: $$\pi(\theta_1, \theta_2, \theta_3 | \bf{y})$$ and that we've run an MCMC algorithm to approximate this distribution. I know that there is a Markov chain ...
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Expectation of 500 coin flips after 500 realizations

I was hoping someone could provide clarity surrounding the following scenario. You are asked "What is the expected number of observed heads and tails if you flip a fair coin 1000 times". Knowing that ...
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Binomial Distribution:equivalency between the average and proportion?

Another user and I are having a debate regarding properties of a binomial distribution, most notably the relationship between the proportion and the mean. The central prompt is in quotations: “I ...
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2answers
207 views

Optimal way to estimate the irreducible error E [ V [Y | X] ]?

I was reading about conditional variance in Wikipedia and then the following property showed up $$E[V[Y \mid X]]=E[(Y-f(X))^2]-E[(E[Y \mid X]-f(X))^2]$$ Which i interpret as an irreducible error, ...
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Beyond the CLT: guarantees on the shape of the sample mean distribution?

If the L.L.N. tell us where our sample mean is going, and the C.L.T. "extend it" telling us how fast the variance is decaying, do we have an other tool telling us how fast the shape (or the further ...
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Work flow question regarding sample theory

The question is pulled below from a book. The solution to this problem is also posted below. I am able to get the result before taking the expectation by applying the sum of a finite geometric ...
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1answer
72 views

Flattening of sums (Law of Large Numbers or Central Limit Theorem)

If we add $N$ curves with independent random phases, the shape of the sum tends to flatten out as $N\to \infty$. That is, the fluctuation depth decreases as $N$ gets larger. Is this phenomenon a ...
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1answer
88 views

random walk on Z towards the origin

Consider a random walk on $\mathbb{Z}$ with rate $a>0$ (begin no origin). The r.w. jumps one step towards the origin with probability $p$ or one step away from the origin with probability $1 −p$. ...
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1answer
327 views

Expressing Law of Large numbers in terms of binomial probabilities

Suppose I let $p \in (0,1)$. Then for $n \ge 1$ and $0 \le k \le n$ let $${P}(k;n,p) = \dbinom{n}{k} p^k(1-p)^{n-k}$$. How can I express the law of large numbers in terms of the probabilities $P(k;n,...
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Acceptance-Rejection using Functional

Setup Let $X\in L^1(\Omega,\mathcal{F},\mathbb{P})$. As far as I've seen, Monte-Carlo methods generate $x_1,\dots,x_n$ from the distribution of $X$ and uses the Glivenko-Cantelli theorem to conclude ...
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2answers
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Asymptotic Expectation of Ratio of Sample Averages

I have two random variables: $X$ and $Y$. I know that: \begin{equation} E[X]=E[Y]=\mu>0 \end{equation} I know that variance of both can be bounded: \begin{equation} \operatorname{Var}[X]<k, \...
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67 views

Prove convergence of a sum of random variables

I am trying to grab on to some intuition about the area where random variables start looking a bit more like calculus. I've learned about random variables and the weak law of large numbers, but seem ...
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Law of Large Numbers under Transformation

The Law of Large Number, draw i.i.d examples of a random variable y, then with propability of 1 the average of y_1, ... y_n goes to the expected value of y. When i apply a function to the y_i's, does ...
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Residuals and the law of large numbers

Can I apply the law of large numbers (LLN) to $ê$ (the observed residuals)? For example, can I use LLN for ($1/n)∑ê$ and say that converge to $E[e]$ (where $e$ is the true error)? I think that I can'...
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785 views

Convergence in probability of $\frac{1}{n}\sum_{i=1}^n X_i^2$ when $X_i$'s are i.i.d $N(0,1)$

Question: My approach: And after this I am stuck..How do I put the modulus over here and how do I determine the appropriate value of "k" ? (here k signifies the value of convergence in probability ...
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1answer
597 views

Law of Large Numbers for Covariance Stationary Processes… Difference and Relationship between LLN and Ergodicity

We have a covariance stationary time series. We must assume that the time series was produced by an ergodic process if we are to make the bridge between the realization of the time series that we ...
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1answer
38 views

Can we conclude, with the strong law of large numbers, that $n$ random variables are independent? [closed]

Suppose we have a sequence of identically distributed random variables $X_1, \ldots, X_n$, and that we know $(X_1 + \ldots + X_n)/n$ converges almost surely to $\mu = E[X]$ as $n$ approaches infinity. ...
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1answer
406 views

Sampling from finite population with replacement

Suppose we have a population of size 10,000. We choose 1000 persons from the population, uniformly and independently, and ask them some question(s). The answers are random variables $X_i$. Since we ...