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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Law of large numbers and standard deviation

In wikipedia, the law of large numbers is defined as follows : "The average of the results obtained from a large number of trials should be close to the expected value and tends to become closer ...
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How does the law of large numbers relate to regression to the mean?

Intuitively, these two important statistical principles appear to describe two facets of the same phenomenon, namely that in the long run, any extreme occurrences get counter-balanced, and things tend ...
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Does coin tossing obey mean reversion? [duplicate]

If a coin comes up Heads 90 times out of the first 100 tosses, should one expect Tails to make a comeback over the next 100? Reference from this page: https://www.financialwisdomforum.org/gummy-stuff/...
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Range of $a$ such that $w \leftarrow w-a x \langle w, x \rangle$ converges almost surely?

Edit Sep 19 this answer on Mathoverflow matches simulation results Suppose $x_i$ come from 2-d standard Normal centered at 0. What is the range of $a$ for which the following iteration converges ...
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Doesn't the CLT contain the law of large numbers?

Im sorry for asking a newbie question. The Central limit Theorem (CLT) states that when sample size tends to infinity, the sample mean will be normally distributed, and the variance is decreasing ($\...
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Best approximation for the size of a test

Let $X \sim \mathrm{Bernoulli}(\vartheta)$ for some unknown $\vartheta \in (0,1)$, and let $(X_1, …, X_n)$ be a moderately large IID sample for $X$. Let $\vartheta_0 \in (0,1)$. I want to test $H_0 \...
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Does convergence of $\sqrt{n}X_n$ to $N(0,1)$ in distribution implies $X_n \rightarrow 0$ in probability?

This question stems from the WLLN and the Central Limit Theorem. Suppose we have $n$ iid random samples $X_1,\ldots,X_n$ with common mean $\mu$ and finite variance $\sigma^2$. Then the sample mean $\...
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Law of large numbers for transformed and non-transformed random variables

Law of large numbers states that: If $X_1, \dots X_n \sim p(x)$ are IID, then $ \frac{1}{n} \sum_{i=1}^{n} X_i \rightarrow \mathbb{E}_{p(x)}\{X\}= \mu$, where $X \sim p(x)$. Below is what I'm having ...
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Strong Law of Large Numbers related proof

I am trying to prove the following: So far I have used Kronecker's Lemma as such: \begin{equation} \tag{1} \text{Since } \sum_{i=1}^{\infty} \frac{\sigma_i ^2}{B_i ^2} < \infty, \text{ then, } \...
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Probability of future independent events given past and expected distribution with board games

This question is specifically in reference to how statistics apply to Catan. For background, two fair dice are rolled and rolling a 7 is a bad event. Given that the average number of turns for a game ...
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Using the WLLN for deriving an OLS estimator

I am reading an introductory econometrics book and I am having trouble understanding how they "directly applied the law of large numbers". Basically, they consider the case of simple linear ...
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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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Limit of the sum of independent identically distributed random variables

The question I'm working on says: Let $X_1, X_2, \cdots$ be iid random variables each with mean $\mu$ and variance $\sigma^2$. a) Determine $$ \lim\limits_{n \to \infty} \frac{X_1^2 + \cdots + X_n^2}{...
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Why should the frequency of heads in a coin toss converge to anything at all?

Suppose we have any kind of coin. Why should the relative frequency of getting a heads converge to any value at all? One answer is that this is simply what we empirically observe this to be the case, ...
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$L^2$ convergence of inverse

Let $h$ be some bounded non-negative function. Assume that some random quantity $\mu^N (h)$ be some random quantity with almost sure limit $\mu(h) > 0$. For instance we could have $\mu^N(h) = N^{-1}...
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Sufficient condition for ergodicity in the first-order moment

We have a WSS process $X_n$ with expectation $\mu_X$. We assume the covariance function converges to zero as $k$ increases. I feel like its a simple question that I don't understand, If you could give ...
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Is this point estimate for mean biased?

I was wondering if this point estimate for mean: $\frac{1}{n+1}\sum_{i = 1}^{n}x_i$ is biased? My first thought was that $\frac{1}{n+1}\sum_{i = 1}^{n}x_i \neq \frac{1}{n}\sum_{i = 1}^{n}x_i$, so then ...
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Law of Large Numbers for whole distributions

I'm aware of the Law(s) of Large Numbers, concerning the means. However, intuitively, I'd expect not just the mean, but also the observed relative frequencies (or the histogram, if we have a ...
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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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The law of large numbers and coin flip probability

I really appreciate some help with the mathematics behind an example I came across: Suppose a fair coin flip where you are getting a 50% chance for heads. If you want to be 98% certain to make a ...
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Central limit theorem and strong law of large numbers

I had a question in my mind , if a i.i.d distribution function follows central limit theorem , does that mean it will follow Strong law of large numbers also ?? Since in both cases sample means tends ...
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Convergence to law of large numbers without using borel cantelli lemma

I am studying convergence in probability ... almost sure , in mean , And I have not studied borel cantelli lemma . Now in most of the answers on such questions this lemma is quoted . Since it is ...
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MGF of sample mean of poisson distribution

Let $X_1,X_2,\dots,X_n\stackrel{iid}{\sim}Poiss(\lambda)$ Let mgf of $X_1$ is given by $M_X(t)=e^{\lambda(e^t-1)}$ and let $\bar{X_n}=\frac{1}{n}(X_1+X_2+\dots+X_n)$ Then, by Weak Law of Large Numbers ...
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How can I edit this code to determine the accuracy of data? [closed]

My employer has asked me to perform a few analysis on a data about wood piles which contains their diameter and bark thickness. I am a beginner in R and started with some basic descriptive analysis ...
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The law of large numbers when using the midrange estimator for expectation

I have a growing sequence $A_1 \subseteq A_2 \cdots$ of finite sets that contain samples of $X \sim U([a, b])$. From this sequence I construct a sequence of estimates of $\mathbb{E}[X]$ using the ...
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Invertibility of $E(X^\top X)$ and $\tilde{X}^\top \tilde{X}$

Consider the linear regression model with 3 regressors $$ Y=\beta_1 Q+\beta_2 W+\beta_3 Z+\epsilon $$ Let $X\equiv (Q, W, Z)$ and $\beta\equiv (\beta_1,\beta_2,\beta_3)^\top$. Also suppose that $Q,W,...
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What does "Expectation with respect to true unknown parameter" mean?

I am trying to study the asymptotic properties of MLE, but I am having trouble understanding an expression that seems to be consistently used in all lecture notes available online (page 93,page 18,...
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What causes probability distributions to equalize?

To my understanding: Flipping a coin has a discrete 1/2 probability to be heads or tails, and every iteration of that trial resets the probability back to 1/2. So, it could be heads every time, or, ...
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strong law of large numbers for V statistic

Recently, I encountered a problem regarding the a.s.-limit of $\frac{1}{n^2} \sum_{k, \ell = 1}^n ||\boldsymbol X_k - \boldsymbol X_\ell||_2$, where $\boldsymbol X_i$ are $i.i.d.$ sample following ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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