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 […]

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 […]

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, […]

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? […]

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 […]

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 […]

$X_1, X_2, \dots, X_n$, 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!

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 […]

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 […]

$X_i(1 \leqslant i \leqslant M)$ 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 […]

Intereting Posts

What is known about Collatz like 3n + k?
Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$
How to fit $\sum{n^{2}x^{n}}$ into a generating function?
Definite integral of an odd function is 0 (symmetric interval)
Prove that a simple graph with $2n$ vertices without triangles has at most $n^2$ lines.
Doubt in Application of Integration – Calculation of volumes and surface areas of solids of revolution
Why is the permanent of interest for complexity theorists?
cup product in cohomology ring of a suspension
Number of permutations of thet set $\{1,2,…,n\}$ in which $k$ is never followed immediately by $k+1$
Finding limit points in lexicographic order topology
When Schrodinger operator has discrete spectrum?
Why isn't $\mathbb{C}/(xz-y)$ a flat $\mathbb{C}$-module
Sort-of-simple non-Hopfian groups
Fractal derivative of complex order and beyond
The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms.