Articles of central limit theorem

How to prove the continuity theorem for moment-generating functions?

I’m attempting to understand a proof of a weak version of the central limit theorem. I got through the proof, but it relied on a lemma that was left without a proof nor without any motivation. It is of course the lemma in the picture below. How does one prove this? Does anyone perhaps know […]

Local central limit theorem for sum of random variables (size of unrooted trees) with infinite variance

Following set up: We have $i.i.d.$ integer valued random variables $\tau_1,\dots,\tau_n$ with \begin{align*} \mathbb{P}(\tau_1=k)=2k^{k-2}\frac{e^{-k}}{k!} \end{align*} for $k\in\mathbb{N}$. It holds that \begin{align*} \mathbb{E}[\tau_1]=2,\quad \mathrm{Var}(\tau_1)=\infty. \end{align*} The aim is to investigate the asymptotic behaviour of \begin{align*} \mathbb{P}\left(S_{n/2}=n\right), \end{align*} where $S_{n/2}:=\sum\nolimits_{i\leq n/2}\tau_i$, as $n$ tends to infinity. Obviously $\mathbb{E}[S_{n/2}]=n$, but as the variance is not finite, there is […]

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 […]