Questions tagged [central-limit-theorem]
This tag should be used for each question where the term "central limit theorem" and with the tag (tag:probability-limit-theorems). The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.
1,589 questions
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Central Limit Theorem via Fixed Point Theorem and Entropy
I'd like to work out the details of a proof of the Central Limit Theorem that utilizes the Banach Fixed Point Theorem and possibly also entropy. The rough idea is:
The average $\bar{X} = \frac{1}{n} \...
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Pass the limit in a sum
I am currently studying the derivation of the distribution of the maximum of the Wiener process ($W$) using the ** Donsker Theorem** (Functional Central Limit Theorem) and the Reflection Principle on ...
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“Central limit theorem” for symmetric random variables with no finite mean
Let $(X_n)_{n\in\mathbb N}$ be a sequence of independent random variables with the same distribution. The common distribution $\mu$ is such that it is symmetric, that is, $\mu((-\infty,x])=\mu([-x,\...
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Distribution of sum of product [closed]
Let $x_n\overset{p}{\to}c$ and $x_n\overset{d}{\to}N(0,\sigma^2)$ denote convergence in probability to a constant $c$ and convergence in distribution to a random normal variable (with some abuse of ...
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Theorem 5, Chapter 7 in Chung (leading up to De Moivre-Laplace theorem)
I'm studying the below theorem from Elementary Probability Theory With Stochastic Processes by Chung. The proof is also shown in this answer. Cf. also a proof on Wikipedia. The proof in the book is ...
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MGF of standardized $\bar{X}$ when $X_i$ follows exponential distribution [closed]
When $X_1, X_2, ... ,X_n$ follow an exponential distribution, whose $\theta$ is 2,
mgf of $W = \frac{\bar{X}-\mu}{\sigma/\sqrt{n}}$ will be $$\frac{e^{-t\sqrt{n}}}{(1-t/\sqrt{n})^n}$$
I understand the ...
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Uniform convergence in the central limit theorem
I was reading the following notes, https://www.cs.toronto.edu/~yuvalf/CLT.pdf, on the central limit theorem. I am a little confused about what the author says on page two,
"The exact form of ...
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Confidence Interval for Reliability Weighted Samples
I'm trying to do statistical inference on a home poker game. I have calculated the winnings per hour, and I want to create a confidence interval for the variable winnings per hour, in say dollars.
The ...
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Approximating a Gaussian like n-integral by a univariate integral using CLT
For $n \in \mathbb{N}$, let $f:[0,1]^n \to \mathbb{R}_+$ continuous. For some $t \in \mathbb{R}$ we want to compute:
$$
I(t) = \int_{[0,1]^n} dx \cdot e^{-t^2 \cdot f(x)}
$$ One can have the following ...
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A Berry-Esseen-type inequality for uniform distribution
Suppose that $X_1, X_2, \cdots ,X_n$ are i.i.d with $X_i \sim U([-\sqrt3,\sqrt3])$, $\Phi(t)=(2\pi)^{-\frac{1}{2}}\int_{-\infty}^{t}e^{-\frac{x^2}{2}}dx$. Show that there exists $C>0$ such that
$$\...
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A Statistics Problem using the Central Limit Theorem
Below is a problem I did. I got the answer in the back of the book but I feel my style and/or notation may not be right. Therefore, I would like somebody knowledgeable in this area to check it over.
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Central limit theorem (Berry-Esseen theorem) for sum of a random number of random variables - from centred to non-centered variables?
The following Berry-Esseen theorem was obtained by Stein's method:
Theorem (Chaidee and Keammanee, 2008, Theorem 2.1). Let $X_1, X_2, \dots$ be independent, identically-distributed random variables ...
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Conditions for convergence in distribution for a sum of random variables
I have an $N \times N$ matrix $Q$ of independent Bernoulli-weighted exponential random variables with means $0\leq\lambda_{ij} < \infty$,
$$
Q_{ij} \sim \textrm{Bernoulli}(\mu_{ij}) \times \textrm{...
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On the convergence of moments in the central limit theorem.
I'm trying to solve a problem from the book Shiryaev A.N. Problems in Probability.
Problem 3.4.22. (On the convergence of moments in the central limit theorem.)
Let $\xi_1, \xi_2, \ldots$ be any ...
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Central limit theorem for dependent Bernoullis on regular graphs
I am trying to determine whether a set of (slightly) dependent negatively correlated Bernoulli random variables satisfy a Central Limit Theorem (CLT).
Let $\mathcal{G}_{reg}$ denote the uniform ...
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