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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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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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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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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 Zen`s 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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Consistency of estimators (linear regression) [closed]

I need to show that (in a linear regression model under assumptions MLR1 - MLR5), $$\tilde\beta_1 = (\frac{1}{100}\sum_{i=1}^{100} x_ix_i')^{-1} *( \frac{1}{100}\sum_{i=1}^{100} x_iy_i)$$, assuming $n ...
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Monte Carlo - Importance sampling using normal distribution as sampling distribution

Suppose that I want to approximate an integral over finite range, say for example 0 to 10 using the Monte Carlo method. Can I choose a normal distribution as the sampling distribution even though the ...
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Law of “large but not infinite” numbers?

Say there is a population of size $N$, e.g. the population of human beings in China. The distribution of a given characteristic of these individuals, e.g. height, follows a distribution $X$, with mean ...
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Law of large numbers with population data

The LLN refers to the convergence of the sample average to the population mean when the sample is iid. This is, for a random variable $X$ with $E(X)=\mu$, we know that in a sample of size N, and with ...
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Reference request: law of large numbers

I am writing a paper, and I need to cite the law of large numbers. Precisely, I want to use the statement that if $\{x_{1},...,x_{n}\}$ are independent random samples generated from a distribution ...
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Monte Carlo simulation of integral diverges

If we want to do a simple Monte Carlo evaluation of the expectation of a rv, say X, which is a definite integral, and we find when we evaluate this definite integral analytically that it is infinite. ...
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Looking for a law of large numbers

I am not familiar with laws of large numbers (LLN) and I have a question on whether some LLN is applicable to the following setting: Assumption (*): Let $i\in I\equiv \{1,...,n\}$ and $j\in J\equiv \{...
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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 doesn`t

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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Clarification of LLNs when non i.i.d. observations

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)<\...
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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 ...
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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 (...
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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 ...
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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)...
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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$ ...
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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 ...
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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 ...
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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 $\{a_1,a_2,\ldots\}$ ...
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$n^{-1}\sum_{i=1}^{n}(1-R_i)\overset{p}{\to}P(R_i=0)$, 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, $...
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Uniform Law of Large numbers for non-iid matrices

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)=...
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Can we prove Weierstrass Approximation using Strong Law of Large Numbers

I have only been able to the prove point wise convergence till now. I am not sure if the answer to the question is yes or no. If Yes (can somebody prove it or give a slight hint). See Question 17 [...
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Durret 2.2.1 and the statement “every convergent sequence is bounded”

Actually I asked this question on Math Stack Exchange (here) but it did not generate much interest. Since it is a question from probability theory, perhaps the crowd on cross validated will be more ...
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home work question ,central limit theorem / law of large numbers

please reffer to this image , the entire question was tried earlier during an assignment , i have mentioned everything that i tried here in the image