# 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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### Are uniform LLNs preserved under monotone transformations

A simple question that I'm trying to answer is the following if I have a uniform LLN for a sequence of random vector; namely \begin{equation} \sup_\beta \left\| \frac{1}{n} \sum_n^NX_n(\beta)\right\| ...
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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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### 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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### 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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### 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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### Main differences (problems and mathematics) between traditional statistics and high dimensional statistics

High dimensional statistics seems to be hot nowadays. What are the main differences, in terms of questions and problems it tries to solve, as well as the mathematical tools used, between "traditional"...
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### 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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### 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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### 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 ...
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### instrumental variables property

How can I show that the instrumental variables (IV) estimator is consistent from this equation using the two stage least squares method? Where does this equation come from?
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### De Finetti: equivalence a.s. according to which measure

In Zens answer at What is so cool about de Finetti's representation theorem? that is concerned with De Finetti's 0 -1 representation theorem, he says that "De Finetti's law of large numbers" ...
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### Derivation of law of averages in Grimmet and Stirzaker

I am working through Probability and Random Processes by Geoffrey Grimmet and David Stirzaker, and got stumped following their proof concerning the law of averages. I will start by posting the proof ...
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### Convergence in probability of a multinomial sample correlation coefficient

This problem is from a Ph.D Qualifying Exam on mathematical statistics(also related to probability theory). Let $(X_1,\cdots,X_k)$ be a random vector with multinomial distribution of $n$ trials and ...
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### A special case of the law of large numbers for (possibly) dependent, identically distributed random variables

I want to show that $\dfrac{1}{N}\sum\limits_i^N x_iw_i \to 0$ as $N\to\infty$, where $w_i\sim \mathcal{N}(0,\sigma^2)$, $\mathbb{E}[x_i] = 0$ and $\mathbb{E}[x_i^2] = \rho$. The $x_i$s are ...
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### Law of Large Numbers Convergence for Different Means

Considering the LLN stating that: If we have $\{X_i\}$ $iid$ then as $n \rightarrow \infty$ it follows that $\bar{X} \rightarrow \mu = E[X_i]$, almost surely In my case, we have a logistic ...
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### Limiting fraction of number of arrival while another event is occurring

A light bulb burns for an amount of time having distribution F with mean μF then burns out. A janitor comes at times of a rate Poisson process to check the bulb and will replace the bulb if it is ...
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### “Law of large numbers” for distribution with infinite variance?

This is a purely explorative question. I asked a question here about a "central limit theorem" for random variables with infinite variance. I did not expect it, but it turns out that even some ...
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### Have i done this right?

The question is based on weak law of large numbers.So the given pdf here is a gamma distribution with population mean=2.The answer given to the question is 0.Shouldn't be the probability 1 according ...
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### Any convergence results other than to a mean?

I am wondering whether there can be a statistic that does not converge to its mean but does converge to some other quantities such as the mode. Here convergence is convergence in probability (or a.s. ...
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### Example for which the CLT holds but the LLN doesnt

I am currently thinking about the relationship between the law of large numbers and the central limit theorem and I was wondering whether someone can give me an example of a familiy of random ...
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### Probability Distribution Approximation problem

My problem is 7.6 from A First Course in Probability and Statistics. The answer is provided in the book but not how to arrive at the solution. I thought I understood the chapter fairly well, but I can'...
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### Why does the law of large number hold for pseudorandom numbers

i`m wondering why the law of large numbers holds for pseudorandom numbers. Is this a empirical observation such as that the law of large numbers holds for coin tosses, or is this somehow programmed ...
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### Fair coin tosses simulation and Law of Large Numbers

My question is related with a simulation Professor John Guttag did in the 15th lecture of the course MIT Opencourseware "Introduction to C.S." The program simulates multiple fair coin tosses ...
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### Can someone please explain ‘why’ one standard deviation of the number of heads/tails in reality is actually proportional to the square root of N?

I can grasp that the LLN is basically saying that deviations grow slower then the total number etc, but what I can't intuitively grasp is “why” 1 standard deviation would actually “be” proportional to ...
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### (Nomenclature) Are there two different Weak Laws of Large Numbers?

Question: Are there two different Weak Laws of Large Numbers in the statistics literature? I have found many sources which claim that the weak law of large numbers requires existence of finite second ...
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I have a question on how to interpret Law of Large Numbers. Its meaning seems to me clear when we are in the i.i.d. case: Consider $(Y_i)_{i=1}^n$ i.i.d. random variables with $\mu\equiv E(Y_1)<\... 0answers 389 views ### Prove that OLS residuals have 0 expectation even when the linear model is false For the sake of this question, I will consider a linear model as a statistical model such that$\mathbb{E}[y|\mathbf{x}]=\mathbf{x}\boldsymbol{\beta}$. I know that usually some more assumptions go in ... 1answer 5k views ### Difference between the Law of Large Numbers and the Central Limit Theorem in layman's term? [duplicate] Almost all books teach Law of Large Numbers first then the Central Limit Theorem one next. But what are the relationship and differences between two theorem? My attempt: Here is my understanding (... 1answer 311 views ### How does Stigler derive this result from Bernoulli's weak law of large numbers? In Stephen Stigler's History of Statistics, there's a section on Jacob Bernoulli and his attempt to formalize uncertainty about an unknown proportion given an accumulation of evidence, leading ... 0answers 141 views ### An example for using the strong law of large numbers Let$X_i$'s be independent random variables such that$X_i$'s are symmetric about 0 and$\mathrm{Var}(X_i) = 2i - 1$, for$i \ge 1$. Then, $$\lim_{n\to\infty} P(X_1 + X_2 + \dots + X_n > n \log n)... 1answer 42 views ### Question about law of large numbers derivation I am struggling with a small part of the proof of the law of large numbers. I understand from Markov's inequality:$$P(X\ge t) \le \frac{E(X)}{t}$$and therefore if$ X = (\bar{Y} - E(Y))^2$... 1answer 158 views ### Application of the law of large numbers In the book by Christian Robert and George Casella on Monte Carlo Statistical Methods, they use an argument of LLN on pages 551 and 552. I'm attaching the argument in this screen shot$t$is ... 1answer 334 views ### How to analyze random variables with non normal distribution I'm wondering how random variables can be analyzed using parametric methods if the distribution is not normal. For example if a variable Y is normal distributed but I'm interested only on values Y2 ... 0answers 65 views ### Convergence of$\frac{1}{n}\sum_{i=1}^n a_i X_i$Assume that$\{X_1,X_2,\ldots\}$is a sequence of i.i.d. random variables with$EX_i=0$and$VX_i = \sigma^2 < \infty$. What are the conditions on a sequence of real numbers$\{a_1,a_2,\ldots\}$... 1answer 60 views ###$n^{-1}\sum_{i=1}^{n}(1-R_i)\overset{p}{\to}P(R_i=0)$, where$R_i$is a binary variable Say,$R_i(i=1,2,\ldots,n)$is a binary variable. I have an estimator$n^{-1}\sum_{i=1}^{n}(1-R_i)$. How can I show$n^{-1}\sum_{i=1}^{n}(1-R_i)$converges in probability to$P(R_i=0)$? That is,$...
Suppose $W_i=\frac{\delta_i}{\pi_i} S^\prime_i(y_i,\vec\theta)$, where $\delta_i$ is iid Bernoulli r.v, $y_i$ is fixed variable and $\vec \theta$ is a vector of parameters. \$S^\prime_i(y_i,\vec\theta)=...