"Given certain conditions, the mean of a sufficiently large number of iterates of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed" ([Wikipedia](http://en.wikipedia.org/wiki/Central_limit_theorem)).

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7
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65 views

A dynamical systems view of the Central Limit Theorem?

(Originally posted on MSE.) I have seen many heuristic discussions of the classical central limit theorem speak of the normal distribution (or any of the stable distributions) as an "attractor" in ...
4
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0answers
43 views

Central Limit Theorem when the dimension size increases with the sample size

Let $X_1, X_2,\ldots, X_n \in \mathcal{R}^d$ and be zero-mean, unit variance random variables. Here the dimension ($d$) is a function of the sample size($n$) i.e, $d=f(n)$. For example $d = \sqrt{n}$. ...
0
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1answer
27 views

Understanding formulas for the sampling distribution of the mean

In the passage below, what does $k_c$ mean, and why (in "$σ_0/\sqrt n$") is $\sigma$ being divided by the square root of $n$? I got this form the book Principles of statistical inference.
0
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1answer
32 views

Basic question about central limit theorem and sampling distributions

The CLT states that "if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population, then the distribution of the sample means will be ...
1
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0answers
22 views

Normality test in panel data analysis

Is it necessary to perform normality test such as Jarque Bera, Kolmogrove etc in panel data analysis? Assume I have 3000 obervation..Since the observation size is big, can I just assume that the data ...
12
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2answers
214 views

Independent samples t-test: Do data really need to be normally distributed for large sample sizes?

Let's say I want to test if two independent samples have different means. I know the underlying distribution is not normal. If I understand correctly, my test statistic is the mean, and for large ...
10
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3answers
221 views

Yet another central limit theorem question

Let $\{X_n:n\ge1\}$ be a sequence of independent Bernoulli random variables with $$P\{X_k=1\}=1-P\{X_k=0\}=\frac{1}{k}.$$ Set $$S_n=\sum^{n}_{k=1}\left(X_k-\frac{1}{k}\right), \ ...
5
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3answers
147 views

In CLT, why $\sqrt{n}\frac{\bar{X}_n-\mu}{\sigma} \rightarrow N(0,1)$ $\implies$ $\bar{X}_n \sim N(\mu, \frac{\sigma^2}{n})$?

Let $X_1,...,X_n$ be independent observations from a distribution that has the mean $\mu$ and variance $\sigma^2 < \infty$, when $n \rightarrow \infty$, then $$\sqrt{n}\frac{\bar{X}_n-\mu}{\sigma} ...
3
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0answers
47 views

Convergence of Monte Carlo - Why square root n for rate?

Let $\theta_n = \frac{1}{n} \sum_{i=1}^n X_i$ be the Monte Carlo estimator for $E(X)$. Letting $\sigma^2 = \operatorname{Var}(X)$, by the CLT, $$ \sqrt{n}(\theta_n - E(X)) \xrightarrow d ...
0
votes
1answer
24 views

E[M(cells)/2] where F(t) is the probability that cells die in t

Lets m be a population of cells in $t=0$ (very big population) and $M(t)$ the number of alive cells a time t, so $M(0)=m$. Each cell has a probability $F(t)$ to have die in $(0,t)$ (independent to the ...
3
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0answers
19 views

Alternative distribution of $T^2$ statistic without Gaussian assumption

Background Let $p(x)$ be an arbitrary distribution defined on $\mathbb{R}^d$. Define $\mu = \mathbb{E}[x]$. Given an i.i.d. sample $x_1, \ldots, x_n \sim p(x)$, consider the following $T^2$ statistic ...
4
votes
1answer
93 views

Using the central limit theorem

Use the central limit theorem to show that for $x>0$, $$\lim_{n \rightarrow \infty} \frac{1}{3^n} \sum_{k:|3k-2n| \leq \sqrt{2n}x} \binom{n}{k} 2^k = \int^{x}_{-x} ...
3
votes
1answer
51 views

Lindeberg condition example for a sequence of independent discrete random variables

Let $X_1,X_2,...$ be independent and for any n $\ge 1$ and $\alpha>0$ $$X_n = \left\{ \begin{array}{rl} n^\alpha & \text{with } Pr(X_n= n^\alpha) = \frac{1}{2n^{2\alpha}},\\ -n^\alpha ...
2
votes
1answer
84 views

What is wrong with these statements? [closed]

Explain what is wrong in each of the following statements. (a) For large sample size n, the distribution of observed values will be approximately Normal. (b) The 68-95-99.7 rule says that $\bar x$ ...
5
votes
0answers
30 views

Asymptotic distribution of a weighted sum of chi squared variables beyond CLT?

I have a sum $$ S = \sum_{i=1}^{n} d_i X_i^2, $$ where $X_i$ are independent standard normals, and $d_i > 0$ are fixed real numbers, for example $d_i = i$. The asymptotic distribution of this sum ...
0
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0answers
6 views

Gaussian approximation for general Covariance matrices

There are quite a few results in the literature that provide Berry-Esseen type bounds for the convergence of a standardized sum $S_n:=\frac{1}{\sqrt{n}} \sum_{i=1}^n X_i$ of iid random vectors in ...
0
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0answers
48 views

Sampling log files and the central limit theorem

I have a few large log files from different computer systems at my disposal, and would like to classify the log entries in these log files in order to evaluate an algorithm that I have worked on. The ...
0
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0answers
27 views

Central limit theorem for maximum likelihood estimators when modelling assumptions are violated

Lehman's Element's of Statistical Learning Theory gives in Theorem 7.5.2 a central limit theorem for multiparamter maximum likelihood estimators. (Many other sources provide similar theorems.) The ...
0
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0answers
10 views

The case for variable transformation (parametric model)

Hi :) I have data from my own full randomized experiment (2 groups). Of course I want to conduct simple parametric tests on it. There are many decision trees for which tests to apply. Each test has ...
1
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2answers
85 views

Role of Central Limit Theorem in one-way ANOVA

Background: It has been shown and widely referenced (applets even exist, etc.) that for even a highly-skewed numeric variable, a sample size of $n\ge{}$30 is often "large enough" for the Central Limit ...
3
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2answers
254 views

Why is this definition of the Central Limit Theorem not incorrect?

I found the following definition of the Central Limit Theorem from book Probability and Statistics by Degroot (also from Wikipedia). It simply states the CLT as $$ \lim_{n\to\infty} ...
0
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0answers
13 views

Distribution of samples from a uniform distribution [duplicate]

Let's say we are taking $n$ samples from a uniform distribution, that spans from $0$ to $1$. According to the central limit theorem, the mean of the $n$ samples will follow a normal distribution with ...
7
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3answers
353 views

Central limit theorem and the Pareto distribution

Can somebody please provide a simple (lay person) explanation of the relationship between Pareto distributions and the Central Limit Theorem (e.g. does it apply? Why/ why not?)? I am trying to ...
1
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1answer
52 views

Central Limit Theorem for Wilcoxon signed rank tests

Let $X_i, i=1,2,...,n$ be a set of iid observations, assumed symmetric about $\mu$. Let $R_i$ be the rank of the absolute deviations from some $\mu_0$, i.e. $R_i=\text{rank}(|X_i-\mu_0|)$. Let ...
1
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0answers
43 views

Relationship between CLT for estimator of sample and number of combinations for subsample?

What is the relationship between the Central Limit Theorem as applied to the expected value of an estimator of a parent sample and the number of possible combinations of a subsample used to calculate ...
8
votes
1answer
157 views

Two Sample chi squared test

This question is from Van der Vaart's book Asymptotic Statistics, pg. 253. #3: Suppose that $\mathbf{X}_m$ and $\mathbf{Y}_n$ are independent multinomial vectors with parameters $(m,a_1,\ldots,a_k)$ ...
5
votes
1answer
184 views

Exercise on Chebyshev inequality compared to the Central Limit Theorem

Problem Take this (easy) problem as an example: An astronomer is interested in measuring the distance, in light-years, from his observatory to a distant star. Although the astronomer has a measuring ...
4
votes
1answer
53 views

Is finite and strictly positive variances all that is needed for the CLT Lindeberg condition?

I cannot understand what I am doing wrong here, so please somebody, point it out from me. The issue? I keep finding that the Lindeberg sufficient condition for the Central Limit Theorem for ...
1
vote
1answer
40 views

Is there a central limit theorem for i.n.i.d. variables when normalised by inconsistent variance estimate?

I am wondering whether there exists a central limit theorem for the following situation. Consider the sum of normally distributed variables $\epsilon_i$ with unequal variances according to ...
6
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0answers
124 views

Topologies for which the ensemble of probability distributions is complete

I have been struggling quite a bit with reconciling my intuitive understanding of probability distributions with the weird properties that almost all topologies on probability distributions possess. ...
2
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1answer
144 views

Convergence to normal distribution

Let $X_{1},X_{2},...$ be independent random variables such that $X_{k}$ is Po(k)-distributed for k=1,2... Show that: $$Z_{n}=\frac{1}{n}\sum_{k=1}^{n}\left(X_{k}-\frac{n^{2}}{2}\right)$$ ...
2
votes
1answer
32 views

distribution of $T^2$ statistic without Gaussian assumption

I would like to know if the Gaussian assumption is needed on $x_i$ in deriving the asymptotic distribution of the $T^2$ statistic. Here is the presentation sequence I got from the Wiki page on ...
1
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0answers
19 views

Confidence intervals and Central Limit Theorem with only one sample

I know that to erect confidence intervals, standard errors must be calculated, a process which in turn makes use of the CLT (but I am not clear how). I also understand that, very generally, the CLT ...
0
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0answers
10 views

Central Limit Theorem- observations vs samples [duplicate]

I'm confused with the concept of central limit theorem and I need some clarification about it, hope you guys can help! Now suppose there is a population in form of numbered balls contained in a box. ...
1
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0answers
29 views

Difference between asymptotic normalities of OLS and MLE

Let's see the comparison below Asymptotic normality is given by CLT for both cases. In MLE case, a variance of $\hat{\theta}$ is in distribution as $\frac{1}{I(\theta)}$, but in OLS case ...
0
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0answers
23 views

Fisher information of multiple samples and the standard error

I'm curious on whether the two following results are related (or indeed just the same result). First result The Fisher information is some measure of how well one can estimate the value of a ...
1
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1answer
110 views

Central Limit Theorem for R coding assignment

I had a homework assignment asking us to code some basic things in R. Here are the tasks: Generate a Gamma(1,2) population of size N=5000 and show its density plot and descriptive statistics. Create ...
5
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2answers
75 views

Sum of random variables without central limit theorem

I know that using central limit theorem we approximate sum of random variables into Gaussian distribution. Is the any other approximation method available for finding the probability distribution ...
1
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0answers
41 views

Finding n using central limit theorem

Problem 3: Central Limit Consider tossing a fair coin $100n$ times, where n is geometrically distributed with mean 6. Due to a counting error, it is reported that the fraction of heads is in ...
0
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1answer
67 views

Do we draw samples with or without replacement when we state the central limit theorem

In central limit theorem (in its most basic flavor), , we say that we draw a large number of samples from an unknown distribution and each time we calculate the mean. Then, for a large sample size, ...
0
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2answers
72 views

Clarification on central limit theorem

I found this definition of central limit theorem in the book: Intro to probability and statistics using R: I thought the sample mean $\bar X$ itself followed the normal distribution and not the ...
1
vote
1answer
34 views

Can I use sub-sample means to verify that the sample mean is approximately normal?

I have a large (N=10^6) Monte Carlo ensemble of runs of my engineering code. I want to compute a confidence interval for the means of two output variables, and falsify the hypothesis that they have ...
10
votes
0answers
165 views

Assessments of “Approximately Normal” for t-tests

I am testing equality of means using Welch's t-test. The underlying distribution is far from normal (more skewed than the example in a related discussion here). I can obtain more data but would like ...
1
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2answers
72 views

Required number of random numbers for using Central Limit Theorem

I wanted to know how many i.i.d random variables have to be summed in order to be able to use Central Limit Theorem. I know it varies depending on the distribution, but does there exist any number ...
4
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4answers
216 views

The role of variance in Central Limit Theorem

I've read somewhere that the reason we square the differences instead of taking absolute values when calculating variance is that variance defined in the usual way, with squares in the nominator, ...
2
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0answers
49 views

Local Version of Bernstein Von-Mises Theorem?

The Bernstein-Von Mises theorem says that, under reasonable conditions, the posterior distribution $p(\theta | x_{1},\ldots,x_{n})$ converges weakly to the normal distribution after suitable ...
3
votes
2answers
193 views

Why is there a need for a 'sampling distribution' to find confidence intervals?

I understand the key principles behind confidence intervals, but there's something I want a bit of clarification on. Let's say I have a basket of apples that I picked at the orchard. The weight is ...
0
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0answers
71 views

Conjecturing asymptotic normality for a sum of dependent random variables

I am hunting for the asymptotic distribution of a scaled partial sum of pair-wise equi-correlated, identically distributed continuous random variables $$W =n^{-\delta}\sum_{i=1}^nY_i(n),\;\; ...
2
votes
3answers
537 views

Central Limit Theorem for Normal Distribution of Negative Binomial

The question is: Explain why the negative binomial distribution will be approximately normal if the parameter k is large enough. What are the parameters of this normal approximation? I have ...
0
votes
1answer
59 views

Central Limit Theorem Equation

I was wondering what the difference is between $Z_n=\frac{S_n-n\mu}{\sigma \sqrt{n}}=\frac{X_1+\ldots+X_n -n\mu}{\sigma \sqrt{n}}$ and $Z_n=\frac{X-n\mu}{\sigma/\sqrt{n}}$? What is the benefit of ...