Articles of central limit theorem

Central Limit Theorem for exponential distribution

Suppose that $X_1$ ….. $X_n$ are a random sample from a population having an exponential distribution with rate parameter $\lambda$. Use the Central Limit Theorem to show that, for large n, $\sqrt{n}(\lambda\bar{x}-1) \sim Normal(0,1)$ My attempt: honestly I am really not understanding what this question is asking. I can see that for an exponential distribution, […]

How common are probability distributions with a finite variance?

It’s always very surprising to learn that some of the entities one has been assiduously studying actually represent negligibly tiny minorities (e.g. continuous functions vis-à-vis all functions)… Now, the Central Limit Theorem, for one, holds only for probability distributions with a finite variance. How common are such distributions in the space of all probability distributions? […]

$W_n = \frac{1}{n}\sum\log(X_i) – \log(X_{(1)})$ with Delta method

Note: $\log = \ln$. Suppose $X_1, \dots, X_n \sim \text{Pareto}(\alpha, \beta)$ with $n > \dfrac{2}{\beta}$ are independent. The Pareto$(\alpha, \beta)$ pdf is $$f(x) = \beta\alpha^{\beta}x^{-(\beta +1)}I(x > \alpha)\text{, } \alpha, \beta > 0\text{.}$$ Define $W_n = \dfrac{1}{n}\sum\log(X_i) – \log(X_{(1)})$, with $X_{(1)}$ being the first order statistic. I wish to show $$\sqrt{n}(W_{n}^{-1}-\beta)\overset{d}{\to}\mathcal{N}(0, v^2)$$ as $n \to […]

Central limit theorem and convergence in probability from Durrett

I saw following exercise from Durrett’s probability theory book and I managed to solve the 1st part, but couldn’t get the 2nd part. Let $X_1, X_2, \dots$ be i.i.d samples with mean $0$, and finite non-zero variance. Denote $S_n = X_1 + X_2+\dots + X_n$. Use central limit theorem and Kolmogorov $0-1$ law to show […]

A conditional normal rv sequence, does the mean converges in probability

$X_1, X_2, \dots, X_n$, are $n$ mutually independent r.v.s. $Y_1,\dots,Y_n$ are another set of mutually independent r.v.s. $X_k\mid Y_k=y_k\sim N(y_k,y_k^2)$ and $Y_k\sim\text{uniform}(-k,k)$ for $k=1,\dots,n$. Define $\bar{X}_n=\frac{\sum_{i=1}^nX_i}{n}$ the sample mean of $X_i$’s. I’ve calculated E$[\bar{X}_n]=0$ and Var$(\bar{X}_n)=\frac{n(n+1)(2n+1)}{9n^2}=\frac{(n+1)(2n+1)}{9n}$. How can we prove that $\bar{X}_n$ doesn’t converge to E$[\bar{X}_n]$ in probability? Thanks!

Convergence in distribution for $\frac{Y}{\sqrt{\lambda}}$

Given a sequence of independent r.v’s $\{X_n\}_{n\geq 1}$ such that $P(X_n=x)=\frac{1}{2}$ if $x=-1$ and/or $x=1$ Let $N\in Po(\lambda)$ be independent of $\{X_n\}_{n\geq 1}$ and we set that $Y=X_1+X_2+\dots+X_N$ One have to show that $\frac{Y}{\sqrt{\lambda}}$ converges in distribution to $N(0,1)$ as $\lambda$ goes to infinity. How can this be shown? I have been thinking of some […]

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 of i.i.d. random variables with $X_1 \in L^2(P)$ and $\mathbb{E}[X_i]= \mu$ and $\mathbb{V}ar[X_i] = \sigma^2>0$. Denote with $\widehat{X}:= \frac{(X_1+\dots+X_n)}{n}$. Then […]

Distribution of sum of iid cos random variables

$X_i(1 \leqslant i \leqslant M)$ is i.i.d distribution of $\mathcal{U}\left({-\pi,\pi}\right)$, $\mathcal{U}$ is uniform distribution. $Y_M = \sum_{i=1}^{M}\cos\left(X_{i}\right)$. When $M$ is large, I want to get a distribution of $Y_M$. Although according to the central limit theory, it turns to be a normal distribution. But based on the simulation result, I think there should be a […]

Does really convergence in distribution or in law implies convergence in PMF or PDF?

Ref :Introduction to Mathematical Statistics-Prentice Hall (1994) by Robert V. Hogg, Allen Craig. Now , in the above problem it has been shown that a sequence converges to a random variable X in distribution but the sequence of PMF doesn’t converge to the PMF of X. but we know that “a sequence {Xn} with PDF/PMF […]

Point of maximal error in the normal approximation of the binomial distribution

I am sorry for the long question! Thanks for taking the time reading the question and for your answers! Context: Let $B_n\sim\text{Binomial(n,p)}$ be the number of successes in $n$ Bernoulli trials of probability $p\in(0,1)$. Let $$\tilde B_n=\frac{B_n-np}{\sqrt{np(1-p)}}$$ be the standardized random variable and let $N\sim\text{N}(0,1)$ have the standardized normal distribution. Let $\epsilon_n(x)$ be the error […]