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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).

12
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2answers
439 views

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 ...
11
votes
2answers
1k views

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 ...
10
votes
1answer
301 views

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 ...
10
votes
1answer
268 views

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 ...
8
votes
3answers
745 views

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 ...
8
votes
2answers
1k views

Why aren't the strong LLNs and CLT contradicting each other?

Given $n$ i.i.d. random variables $\{X_1, X_2, \dots , X_n\}$, each with mean $M$ and variance $V$, both strong and week LLNs seem to say that the average of the $n$ random variables, $S_n = \frac{X_1 ...
7
votes
1answer
583 views

Convergence of maximum of iid random variables in distribution

Given $X_i, i \geq 1$ iid random variables, with mean zero and variance one, I would like to show that $$ M_n = \max_{1 \leq k \leq n}\left\{\frac{|X_k|}{\sqrt n}\right\} \xrightarrow{d} 0$$ as $n \...
7
votes
1answer
2k views

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 $T_n = \frac{1}{\sqrt{n}} \sum_{j=1}^n a_j W_j$...
7
votes
2answers
315 views

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 ...
7
votes
1answer
133 views

Convergence of sample mean using CLT

Assume $X_i$s are i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. Prove: $$\lim_{n\to\infty}n^2\mathbb{P}\left(\left|\frac{\sum_{i=1}^{n} X_i}{n}-\mu\right|>n^{-1/4}\right)=0. \...
7
votes
0answers
551 views

Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
6
votes
2answers
1k views

Higher Order Terms in Stirling's Approximation

Some websites and books give stirling approximation as $$n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \left( 1 + O \left(\frac{1}{n} \right)\right)$$ However when I check their derivations most ...
6
votes
1answer
205 views

Central Limit Theorem for Lévy Process

I am reading a book, which uses the Central Limit Theorem of Lévy Processes $X_t$ without mentioning the exact theorem. Due to the infinite divisible property I can write $X_t$ as a sum of $N$ iid ...
6
votes
1answer
100 views

Find the limit of $\sum\limits_{r=\lfloor an \rfloor}^{\lfloor bn \rfloor} {n \choose r } p^r (1-p)^{n-r}$ using the central limit theorem

Let $p \in(0,1)$. What is the distribution of the sum of $n$ independent Bernoulli random variables with parameter $p$? Let $0 \leq a < b \leq 1$. Use approprtiate limit theorems to determine how ...
6
votes
1answer
928 views

Proof of Central Limit Theorem via Fourier Transform: Case of non-zero mean and non-unit variance

Apologize in advance for use of informal language and non-rigorous presentation. I don't have formal background in Probability Theory, but am exposed to Fourier Transforms (FT). When I learned that ...
6
votes
1answer
398 views

Central limit theorem for random variables with exactly a 2nd moment

Consider $X_m$ which are independent, identically distributed random variables which have moments exactly up to order $2$ and no higher. This can be done in numerous ways. One is the following: let $...
5
votes
7answers
193 views

Why does $e^{-(x^2/2)} \approx \cos[\frac{x}{\sqrt{n}}]^n$ hold for large $n$?

Why does this hold: $$ e^{-x^2/2} = \lim_{n \to \infty} \cos^n \left( \frac{x}{\sqrt{n}} \right) $$ I am not sure how to solve this using the limit theorem.
5
votes
3answers
310 views

Use De Moivre–Laplace to approximate $1 - \sum_{k=0}^{n} {n \choose k} p^{k}(1-p)^{n-k} \log\left(1+\left(\frac{p}{1-p}\right)^{n-2k}\right)$

I am trying to use De Moivre–Laplace theorem to approximate $$1 - \sum_{k=0}^{n} {n \choose k} p^{k}(1-p)^{n-k} \log\left(1+\left(\frac{p}{1-p}\right)^{n-2k}\right)$$ The idea of an approximation is ...
5
votes
2answers
320 views

Show CLT for Poisson random variables, using no generating function

Question is as following: $X\sim Po(\lambda)$ $$\frac{X-\lambda}{\sqrt{\lambda}} \,{\buildrel d \over \rightarrow}\, N(0,1)$$ as $\lambda \rightarrow \infty$. Obs. One is asked not ...
5
votes
1answer
103 views

Show that $X_n$ follow somewhat a CLT

Let $X_i$ be i.i.d. $f_X(x)=\dfrac{1}{|x|^3}1_{\{|x|>1\}}$. Show that $\dfrac{\sum_{i=1}^nX_i}{\sqrt{n\log n}}\to N(0,1)$. I realized that $Var(X_i)=\infty$ and $E(X_i)=0$. I cannot apply ...
5
votes
1answer
2k views

central limit theorem for a product

Given $-1\leq x_i\leq 1$ identically distributed random variables for $i=1,2,\dots n$. What is the distribution function of their product? Is there a central limit theorem for products if $n$ is large?...
5
votes
3answers
218 views

Contradiction when applying CLT to Poisson random variables?

We know that if $X\sim \mathrm{Pois}(\lambda)$ and $Y\sim \mathrm{Pois}(\mu)$ are independent then $X+Y\sim \mathrm{Pois}(\lambda+\mu).$ This means that if $X_{1},\ldots,X_{n}$ are independent with ...
5
votes
1answer
489 views

CLT for independent, but non-identically distributed exponential variables

This problem is practice for my qualifying exam and comes from Resnick, chapter 9. Could anyone comment on my solution(s)? Problem Suppose ${e_n, n\ge 1}$ are independent exponentially distributed ...
5
votes
1answer
106 views

About a factor 2 in Donsker's CLT normalization

I have a question about the following paper: Donsker, Monroe D. "An invariance principle for certain probability limit theorems." Mem. Amer. Math. Soc 6.1951 (1951): 12. There is a factor of two ...
5
votes
1answer
165 views

Explain how if $X_i$ are independent, then $(S_n/\sqrt{n})$ will (often) diverge almost surely

Suppose $X_i:\Omega \to \mathbb{R}$ are independent. Consider a function $f(n)$ which goes to $+\infty$ as $n\to \infty$, and so $(\sum_{i=1}^n X_i)/f(n)$ is a generalized average. My understanding ...
5
votes
1answer
256 views

Deriving the asymptotic distribution of a two-stage estimator

Suppose $X_i \stackrel{iid}{\sim}F_\theta$, where $F_\theta$ is a probability distribution parameterized by a finite-dimensional vector $\theta$. Let $\hat{\theta}_n$ denote the maximum likelihood ...
5
votes
1answer
250 views

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 ...
5
votes
1answer
840 views

Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
5
votes
1answer
2k views

Central Limit Theorem for uncorrelated (non-independent) but bounded random variables

Given uncorrelated, discrete random variables $X_i$ that are bounded, e.g., they can only take on values $|X_i| \leq 4$, then is there a form of the central limit theorem that one can apply to the sum,...
5
votes
1answer
284 views

Central Limit Theorem exercise question

Let $ (X_n)_{n \in \mathbb{N}}$ be i.d.d. random variables with $E{X_1}=0$, $Var(X_1)=1$ and $ S_n = X_1 + X_2 +...+ X_n $. Calculate $ \lim_{n \to +\infty}\Pr(S_n>\sqrt{n})$. On the back ...
4
votes
2answers
145 views

Show $X_1$ and $X_2$ have a common Gaussian distribution

Anyone has any idea about the following question? Let $\Bbb E(X_1^2)$ and $\Bbb E(X_2^2)$ be finite. Show that if $X_1$ and $X_2$ are independent and likewise $X_1+X_2$ and $X_1-X_2$, then both $...
4
votes
1answer
858 views

Rigorous, real analysis, proof of De Moivre–Laplace theorem

The undergraduate books in probability theory that I know do not provide a rigorous proof of even a weak version of the central limit theorem. Instead, they rely on Lévy's continuity theorem, whose ...
4
votes
3answers
198 views

Lindeberg condition fails, but a CLT still applies

I'm having difficulty with this old qualifying exam problem. Suppose we have a sequence of independent R.V's $\{X_n\}_{n\in\mathbb{N}}$ satisfying, $$ \mathbb{P}(X_n = \pm n^2) = \frac{1}{12n^2}, \;\...
4
votes
3answers
239 views

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 ...
4
votes
2answers
72 views

Tilted sum of independent random variables

Let $(X_i)_i$ be a sequence of centered i.i.d. random variables with finite variance. Is it true that $$\frac{\sum_{i=1}^{\lfloor n^{0.6} \rfloor}X_i}{\sqrt{n}}\stackrel{\mbox{a.s.}}{\longrightarrow} ...
4
votes
1answer
112 views

The converse on the Central Limit Theorem

The problem is the following: let $X_n$ be i.i.d r.v's and $\{b_n\}$ be a sequence of positive real numbers s.t. $b_n \longrightarrow \infty$ and $$ (I) \qquad \frac{\sum_{k=1}^n X_k }{b_n} \...
4
votes
1answer
397 views

Central limit theorem — Is it about sums, averages, or both? [closed]

The central limit theorem is, as far as I know, a statement about the probability distribution of the average ${\sum (X_i)_n\ /\ n}$ of $n$ iid random variables $(X_i)_n$, where $(X_i)_n:=(X_1, \ldots,...
4
votes
2answers
306 views

Asymptotic Distribution of Sample Mean

Suppose I have a discrete random variable $X$ which follows a geometric distribution on $x=0,1,2,...$ and I take a random sample from this distribution of size $n$. What is the asymptotic ...
4
votes
2answers
1k views

Central limit theorem: Poisson equals Normal? Tell me where I'm wrong

We just covered the Central Limit theorem in class, and I stumbled upon the following reasoning that makes me think I am missing some key part of... well, something. So, here goes: Let $X_1$ and $X_2$...
4
votes
1answer
441 views

Sample proportion and the Central Limit Theorem

Suppose that $ (\Omega,\Sigma,\mathsf{P}) $ is a probability space and that $ (X_{k})_{k \in \mathbb{N}} $ is a sequence of i.i.d. Bernoulli trials on $ (\Omega,\Sigma,\mathsf{P}) $, each with ...
4
votes
1answer
318 views

Calculation problem with Central limit theorem

Let $X_1,X_2,\dots\,$ i.i.d random variables with mean zero and variance $1$. Let $S_n=\sum_{i=1}^n X_i\,,n\in \mathbb N.$ Compute the weak limes $\lim_{n\to\infty} \frac1n \sum_{i=1}^n \frac{S_i}{\...
4
votes
1answer
88 views

Let $P(X_j=j)=P(X_j=-j)=1/2j^{\beta}$ and $P(X_j=0)=1-j^{-\beta}$ where $\beta\in(0,1)$, then $S_n/n^{(3-\beta)/2)}\Rightarrow c\chi$

Suppose $P(X_j=j)=P(X_j=-j)=1/2j^{\beta}$ and $P(X_j=0)=1-j^{-\beta}$, where $\beta>0$. Show that: (i) If $\beta>1$ then $S_n\to S_\infty$ a.s. (ii) If $\beta\in(0,1)$ then $S_n/n^{(3-...
4
votes
1answer
50 views

Calculate weak limit of $S_n/\sqrt{n}$

Let $(X_n)_{n\in \mathbb N}$ be independent continuous random variables with cdf $$f_n(x) := f_{X_n}(x) = \frac{n+1}{2}\lvert x \rvert ^n \mathbb 1_{[-1,1]}.$$ Let $S_n := \sum_{k=1}^nX_k$ and ...
4
votes
1answer
98 views

$\lim_{n \to\infty} E(|S_n|)= \infty$ for $(X_n)_{n \geq 1}$ i.i.d. real RV with Var$(X_1)=1, E(X_1)=0$

Problem: For $(X_n)_{n \geq 1}$ i.i.d. real RV with Var$(X_1)=1$ and $E(X_1)=0$ and $S_n$ denoting the partial sum of the RVs we have $$\lim_{n \to \infty} E(|S_n|)=\infty $$ My Approach: I have ...
4
votes
1answer
845 views

Central limit theorem and Poisson distribution

$X$ is the sum of $n$ independent Poisson random variables with parameter 1. Therefore $X$ has a Poisson distribution with parameter $n$. Use the central limit theorem to show that $P(X≤n)→(1/2)$. I ...
4
votes
1answer
410 views

Example 19.6 in van der Vaart: show that the class of indicator functions is P-Glivenko-Cantelli and P-Donsker

I have some doubts related to example 19.6 in van der Vaart "Asymptotic Statistics" which applies Theorem 19.4 (Glivenko-Cantelli) and Theorem 19.5 (Donsker) to the distribution function. Definitions:...
4
votes
1answer
340 views

Confusion about the proof of the central limit theorem

My book is trying to prove this version of the central limit theorem: The proof starts like this: I am totally confused about this proof. I have two questions: Question 1: Why must there exist $N(0,...
4
votes
1answer
69 views

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 ...
4
votes
1answer
104 views

Why can we apply the central limit theorem in this case?

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)}\...
3
votes
0answers
134 views

Law of Large Numbers contradicts Central Limit Theorem?

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 $\...