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).

820 questions
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Convergence in distribution of conditional expectations

I was just reading this question, which is about how the classical central limit theorem can be interpreted as giving a rate of convergence for the law of large numbers for iid random variables. I was ...
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Rate of convergence in the central limit theorem (Lindeberg–Lévy)

There are similar posts to this one on stackexchange but none of those seem to actually answer my questions. So consider the CLT in the most common form. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence ...
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On the central limit theorem

The Central Limit Theorem states for a sequence of i.i.d. random variables $\{X_i\}$, $$\frac{\overline{X} - \mu}{\sigma/\sqrt{n}} \to N(0,1)$$ in distribution as $n \to \infty$. I saw in some ...
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Intuition for $N(\mu, \sigma^2)$ in terms of its infinite expansion

To gain deeper insight to the Poisson and exponential random variables, I found that I could derive the random variables as follows: I consider an experiment which consists of a continuum of trials ...
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Applying Central Limit Theorem to show that $E\left(\frac{|S_n|}{\sqrt{n}}\right) \to \sqrt{\frac{2}{\pi}}\sigma$

In the book Probability Essentials, by Jacod and Protter, the following question has bugged me for a long while and I'm wondering if it is bugged. The question is an application of Central Limit ...
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Checking the Lindeberg condition (central limit theorem)

Problem. Let $W_1, W_2,...$ be independent and identically distributed random variables such that $E(W_1)=0$ and $\sigma^2 := V(W_1) \in (0,\infty)$. Let $T_n = \frac{1}{\sqrt{n}} \sum_{j=1}^n a_j W_j$...
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Convergence in Distribution of Sums of Random Variable

Suppose I have $X_1,X_2,...,X_n$ random variables that are independent and identically distributed, from ANY distribution. Suppose that $E(X_i)=\mu$ and $V(X_i)=\sigma^2$. Suppose I define the ...
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Applications of the CLT for the Sample Mean

Related to Confusion of central limit theory, and the fact that I just finished my first course in (Master's-level) graduate-level probability, which relates to this material. The Central Limit ...
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Taking limit of a probability distribution

I have a probability distribution of the form $$p_{m+1}(s)= \frac {(bs)^m}{b(m!)} e^{-bs}$$ I want to show that under the limit $m \to \infty$, it will becomes a Gaussian. I applied Stirling's ...
The problem at hand is to show that for $X_j$ iid holds \lim_{n \to \infty}\frac{P(\bar{X}_n \leq x)}{\frac{1}{ \sqrt{2\pi(\sigma^2/n)}} \int_{-\infty}^x \exp\Big(\frac{-(t-\mu)^2}{2(\sigma^2/n)}\...
My text defines the weak law of large numbers: If $X_1,\ldots,X_n$ are IID, then $\overline{X} \overset{P}{\to} \mu$. And the CLT as: Let $X_1,\ldots,X_n$ be IID with mean $\mu$ and variance \$\...