# 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.

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### Using Central Limit Theorem to estimate deviation of mean estimator of sampled Bernoulli r.v.

I am answering the below question from recitation 20, this course. The question is: "In your summer internship, you are working for the world’s largest producer of lightbulbs. Your manager asks ...
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### Laplace distribution CLT

I'm really stuck with this problem that my statistics professor gave me: n = 81 measurements X = (X1,…,X81) were made according to a Laplace(θ), $f_{\theta}(x) = \frac{1}{2} \theta e^{-\theta|x|}$. ...
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### "It is no restriction to assume, w/o loss of generality, that"

In this proof for CLT, the author said "it is no restriction to assume that $\mu=0, \sigma=1$". This is not clear to me. Can anyone share your thoughts?   My attempt Note that \begin{...
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### Is there a CLT-type result for $S^2$?

The subject line is essentially the question: it there a theorem, like the CLT for the sample average, that allows us to say anything useful about the distribution of the sample variance as the sample ...
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### Proving Convergence to the Lower Bound and Upper Bound for Monotonically Decreasing and Increasing Sequences

Let, a sequence $a_n$, where Minimum value of $a_n=b$ and Maximum value of $a_n=c$. If $a_{k+1}<a_k$, Then, it has a limit $L_1=b$ If $a_{k+1}>a_k$, Then, it has a limit $L_2=c$ It is quite ...
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### How CLT states the sampling distribution of one large sample while sampling distribution needs multiple samples?

I have a bit confusion of understanding. The central limit theorem in statistics states that, given a sufficiently large sample size, the sampling distribution of the mean for a variable will ...
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### Help understanding CLM equation "If x∼N(μ,σ) then x¯∼N(μ,σn‾√)" [closed]

Stats n00b. What does N(μ, σ) mean? I'm trying to piece it all together, but can't fill in all the gaps (search engines don't recognise it and I can't seem to find anywhere that uses this same formula ...
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### Limit central theorem - binominal distribution inequality ( Independant binominal Random variables )

$$\lim _{n\to \:\infty }\mathbb{P}\left(\sum _{n=1}^{\infty }\:X_n>\frac{pn^2}{2}\right)=?$$ When $p\in (0,1)$ , $\left\{X_n\right\}^{\infty }_{_{_{n=1}}}$, $X_n~Bin\left(n,p\right)$, as they are ...
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### central limit theorem for random field

Let $x\in(1,\infty)$ and let $Z(x)=\sum_{i\ge1}Z_i(x)$ be a sum of independent random variables $Z_1(x),Z_2(x),\dotsb$ such that each random variable is bounded as $Z_i(x)\in[-1/i,1/i]$. Moreover, it ...
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### On the distinction between “Pairwise independent” and “Mutually independent” random variables

I have familiar with the fact that for $N$ events, we have the concepts of pairwise independent and mutually independent. From this I am interested in extensions of this notion to random variables. ...
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### How does the Radial Basis Functions transform multimodal distributions to normal distributions?

In the book Hands-On Machine Learning by Aurélien Géron, in chapter two, the author states: Another approach to transforming multimodal distributions is to add a feature for each of the modes (at ...
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### Uniform convergence in distribution implies convergence of moments

I am reading a paper in which the author wants to prove the convergence of the moments. He transforms the object of interest $\varepsilon^{-1} (\vartheta_\varepsilon^*-\vartheta_0)$ into \begin{align*}...
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### An intuitive derivation of a central limit theorem for renewal processes

I am reading "Introduction to Stochastic Processes" by Lawler, and I am having trouble understanding his intuitive explanation of a CLT for renewal processes (Chapter 6, p.134). Here is his ...
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### CLT Problem of $N_t=\sup\{n\geq 1 | S_n \leq t\}$

Let $(Y_k)$ be i.i.d denote $\mu =\mathbb{E}[Y_1],\ \sigma ^2 = Var[Y_1]$ and there exists $c>0$ s.t $P(Y_1\geq c)=1$ $$S_n=Y_1+...+Y_n,\ \ \ N_t=\sup\{n\geq 1 | S_n \leq t\}$$ for $t>0$ a) ...
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### $\frac{X_1+...+X_{n-1}-\log n }{\sqrt{\log n} }\rightarrow N(0,1)$ where $X_n\sim \operatorname{Ber}(\log \frac{n+1}{n})$

Prove : $\frac{X_1+...+X_{n-1}-\log n }{\sqrt{\log n} }\rightarrow N(0,1)$ where $X_n\sim \operatorname{Ber}({\lambda}_n)$ and $\lambda _n = \log \frac{n+1}{n}$ I tried using a similar proof to CLT ...
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