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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).

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1answer
15 views

What are mean and variance of $W_i$, given that $Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1)$? [on hold]

Let $$Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1),$$ where $W_i=X_i-\mu$. What are the mean and variance of $W_i$?
2
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0answers
12 views

Emergence of Lognormal distribution for the concentration of chemical compounds

I'm currently reviewing the literature about lognormal distributions describing/approximating the variability of a given chemical compound across different cells/ samples etc etc. The main argument ...
0
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0answers
14 views

Central Limit Theorem with linear transformation?

Suppose $Z=(z_{1},...,z_{m})^T$ is m dimensional vector, each $z_{i}$ is independent identical distribution with mean 0. If we do linear transformation like: $$X = \Gamma Z$$ where $\Gamma$ is $p*m(p&...
3
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0answers
23 views

Central limit theorem for square integrable martingales

Let $(M_t)_{t \in \mathbb{R}^*}$ be a square-integrable martingale. I am looking for a reference for the following convergence result : $$\frac{M_t}{\sqrt{\langle M_t \rangle}} \overset{d}{\to} \xi$$ ...
0
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1answer
36 views

Normal approximation of MLE of Poisson distribution and confidence interval

Let $(X_1,\ldots,X_n)$ denote a random sample from a Poisson distribution with parameter $\lambda$. Maximum Likelihood Estimate of $\lambda$ is given by $\hat{\lambda} = \bar{X} = \frac{1}{n} \sum\...
0
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1answer
19 views

How do you use the central limit theorem to approximate the sample mean when pdf f(x)=2x

Let $X_{1},\ldots, X_{n}$ be a collection of independent identically distributed (iid) samples from a population with pdf $$ f(x) = \begin{cases} 2x,&\text{if } 0 \leq x \leq 1\\ 0,&\text{...
1
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1answer
23 views

a transformation $g$ such that $g(S^2)$ has asymptotic distribution that depends on $\beta_2$

Let $X_1, X_2,\dots,X_n$ be i.i.d. RVs with $E|X_1|^4 < \infty$. Let $var(X_1) = \sigma^2$, $\beta_2 = \mu_4/\sigma^4$. (a) Using CLT for i.i.d. RVs, show that $\sqrt{n}(S^2-\sigma^2)\...
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0answers
16 views

Equidistributed sequences satisfying the central limit theorem

For a sequence (x_n) of reals that are equidistributed modulo one (such as (prime-) multiples of irrationals, or most geometric sequences) does the (properly scaled) sum of the remainders converge to ...
1
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1answer
25 views

Central limit theorem for weighted average

Let $(a_i)_{i\ge1}$ be a bounded positive sequence and $X_i$ be iid random variables with mean $0$ and finite variance. Let $s_n=\frac{\sum_{i=1}^n a_i X_i}{\sqrt{\sum_{i=1}^n a_i}}$. If $a_i=1$ for ...
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1answer
29 views

Central Limit theorem with extremely skewed population

If $X$~$N(1,0.1)$, and $Y=X^n$ where $n$ is very large (e.g. 200). Due to the asymetric distribution, $E(Y)$ will be skewed very far towards the extreme upper end of the distribution, dominated by ...
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0answers
46 views

Variance of sum of $m$ dependent random variables

Let $X_1,X_2,...$ be a sequence of identically distributed and $m$-dependent random variables with $\mathbb{E}[X_i]=0$, $0<Var(X_i)<\infty$ ($m$-dependent means that each $X_i$ is independent of ...
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0answers
22 views

How to show that there is a number at which the truncated Taylor Series is exact?

So, I'm reading an "easy" proof of the Central Limit Theorem in the book Probability and Mathematical Statistics, by Sahoo, and there is a point where the moment generating function is expanded in a ...
3
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2answers
87 views

A question about the central limit theorem

The question is: $g:R\rightarrow R$ has at least three bounded continuous derivatives and let $X_i$ be $iid$ and in $L^2$. Prove that: i) $\sqrt{n}[g(\overline{X_n}) - g(\mu)]\xrightarrow{w} N(0,g^{...
4
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2answers
67 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} ...
0
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1answer
16 views

Is Confidence Interval taken on one Random Sample or A Sampling Distribution

I am stuck at CI. Intuitively, a confidence interval (CI) is a type of interval estimate, computed from the statistics of the observed data ( random samples), that might contain the true value (mean) ...
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0answers
37 views

Doubts regarding Central Limit Theorem

All, I am studying CLT and I had a doubt. I understand that sampling distribution has a mean same as population mean and std.dev as 6/sqrt(n) where n is sample size. Each point on X axis of ...
0
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1answer
33 views

How to compute $\lim_{n \to \infty}P(C_n>C_0)$?

The unit price of a certain commodity evolves randomly from day to day with a general downward drift but with an occasional upward jump when some unforeseen event excites the markets. Long term ...
2
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1answer
29 views

Convergence (distribution)

$X_1, X_2, X_3....$ are independent random variables. $P(X_n=0)=P(X_n=2)=1/4, P(X_n=-1)=1/2$. Find the limit of: $\frac{4\sqrt{n}(X_1+X_2+...+X_n)-7n}{n+(X_1+X_2+....+X_n)^2}$. I computed: $EX_n=...
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0answers
32 views

Central Limit Theorem- Lapunow, Linderberg

I have a task: $(X_n)_{n>=1}$ are independent. $P(X_n=0)=1/n$ and $P(X_n=2n)=1-1/n$. Check the weak convergence $\frac{X_1+X_2+X_3+....+X_n}{n}-n$. I tried use the Lapunow theorem or Linderberg ...
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1answer
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Problem with Linderberg condition

I am trying to prove that $$\frac{S_n=\sum_{k=1}^n X_k^2-n}{\sqrt{n}}$$ converges in distribution, where $X_i$ are iid. and are normally distributed with mean 0 and variance 1. I defined $X_{n,k}=\...
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0answers
50 views

How to handle little $o$ in the central limit theorem

I am having some trouble understanding a couple of lines in the proof of the central limit theorem using characteristic functions: https://en.wikipedia.org/wiki/Central_limit_theorem#...
1
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1answer
24 views

Lindeberg CLT condition on Discrete Uniform independent sequence of random variables

There is a worked exercise in my book, however there is a line that I am not sure sure about. I understand all of the work before and after this line to finish the proof. Here is what we are given: ...
2
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1answer
29 views

Proving IID Central Limit Theorem using Lindeberg Conditions.

The goal is to prove the IID Central Limit Theorem through Lindeberg's Condition. Suppose that $X_1,X_2,\ldots\displaystyle\sim\text{i.i.d.}$ with $E[X_i]=\mu$ and $Var[X_i]=\sigma^2<\infty$. ...
1
vote
1answer
18 views

Trimmed second moment going to zero by LDCT

We have $0<\sigma<\infty$ and $\epsilon >0$ and $X_1, X_2,...$ iid. The argument involves CLT and it continues on until the line below: That $\frac{1}{\sigma^2}E\big[(X_1-\mu)^2\mathbb{1}\...
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1answer
24 views

limiting distribution of Xbar considering an autoregressive process

Here is the question: Consider the stationary Gaussian autoregressive process of order 1, $X_{i+1} − μ = ρ(X_i − μ) + \sigma Z_i$, where $Z_i$ are iid N(0, 1). Find the limiting distribution of $\...
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1answer
51 views

A die is thrown until the first time the total sum of the face values of the die is 700 or greater. What is the probability for $n$ tosses?

Estimate the probability that, for this to happen, more than 210 tosses are required less than 190 tosses are required between 180 and 210 tosses, inclusive, are required We're supposed to be using ...
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2answers
41 views

Prove Rate of Convergence of Monte Carlo

Let $X_1, X_2, \ldots$ be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. How does \begin{equation} \mathbb E\left[\,\left|\frac{1}{N} \sum_{i=1}^n X_i - \mu\, \right|\,\right] \to O\...
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1answer
22 views

Limit of a Symmetric Random Walk

I'm given a probability space of ($\Omega$, $\mathcal{F}$, $\mathbb{P}$) and am asked to look into a symmetric random walk with its n-step defined as $$ X_k = \Bigg\{ \begin{matrix} 1 & \text{...
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0answers
29 views

central limit theorem: what is the variance?

This is a very basic question that I'm pretty sure I understand but I wanted to double check. Given a regression model of: $$y_t = \mathbf{x_{t}^{\prime}}\beta + u_t$$ We can use one of the CLT ...
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0answers
52 views

Prove the central limit theorem for a sequence of i.i.d. Bernoulli($p$) random variables

Prove the central limit theorem for a sequence of i.i.d. Bernoulli($p$) random variables, where $p\in(0,1)$. I am trying to do this by computing the moment generating function of the object I want ...
1
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1answer
29 views

Generating a number belonging to N(0,1) using *m* numbers from U(0,1) using central limit theorem

I was going through a blog which details how to generate a multivariate Gaussian vector, given a mean vector μ and co-variance matrix σ. As a starting point, author uses generated uniform ...
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1answer
25 views

Probability dice flip Central Limit Theorem problem

Suppose you flip a fair dice 300 times. Let $X$ be the number of times a $6$ was thrown. What is the probability that $X$ is greater than 60? How can I start with such a problem?
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1answer
46 views

Can the Central Limit Theorem be applied here?

My problem statement is to identify in a healthcare organisation, which of it's doctors are lagging in providing proper care to their patients. My Random Variable X is defined as {0 if the patient ...
0
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1answer
23 views

Inverse Gaussian Distribution and the Central Limit Theorem

Let the random variables $Y_1,\ldots,Y_n$ be independent and identically distributed (i.i.d.) (standard) Inverse Gaussian random variables with parameters $\mu$ and $\lambda$. Then, let the random ...
2
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1answer
32 views

Probability of making it on the train if $100$ people stand in line in front of you and each person takes time according to exponential law

The train is leaving in $10$ minutes and there are $100$ people standing in line before you to buy tickets. Each person buys $1.85$ tickets on average with a standard deviation of $1.5$ and $9$ people ...
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0answers
29 views

General Central Limit Theorem for Binomial Random Variables

Question Let $(X_n)_{n\geq 1}$ be a sequence of arbitrary binomial random variables such that $EX_n\to \infty$ and $\text{Var}(X_n)/EX_n^2\to 0$ as $n\to \infty$. Then show that $$ Z_n=\frac{...
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0answers
80 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 $\...
3
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1answer
78 views

My simulation of the Central Limit Theorem does not converge to correct value

The Lindeberg–Lévy CLT states: Assume $\\{ X_1, X_2, \dots \\}$ is a sequence of i.i.d. random variables with $\mathbb{E}[X_i] = \mu$ and $\text{Var}[X_i] = \sigma^2 < \infty$. And let $S_n = \...
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1answer
53 views

What's wrong with my simulation of the Central Limit Theorem?

While there is code in this question, I suspect the answer will be mathematical. I am trying to create a numerical simulation of the Central Limit Theorem (CLT). My understanding is that if $S_n = \...
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0answers
30 views

How does the CLT justify statistical models which are not modeling our data as a sum of random variables?

Wikipedia says (emphasis mine): The Central Limit Theorem states "that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal ...
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27 views

Let $X_1,…, X_n$ be a sample from the normal mixture $(1-p)N(0, 1)+pN(\theta, 1)$

Let $Z_i(\theta)=e^{X_i \theta-\frac{1}{2}\theta^2}-1$. Is $\sum Z_i(\theta)=O_p(\sqrt(n))$, $\sum Z^2_i(\theta)=O_p(1)$ and $\sum Z^3_i(\theta)=O_p(1)$ and why?
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1answer
33 views

Proving that CLT doesn't hold for a given sequence.

During examination of compound Poisson process, with log-normal distribution I came across to the following problem. I have examined the following form $$L=\sum_{i=1}^{N}X_{i}$$ And $X_{i}\sim LogN(\...
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1answer
56 views

Show $Z_{n} \xrightarrow{d} \mathcal{N}(0,1)$

Let $(X_{k})_{k \in \mathbb N}$ a sequence of independent random variables and $F_{k}=F_{X_{k}}$ the respective cdf functions of $(X_{k})_{k \in \mathbb N}$ that are both continuous and strictly ...
3
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1answer
55 views

If $X_n$ is Gamma $(n,\lambda)$ distributed then $(\lambda X_n -n)/\sqrt n\to N(0,1)$

Let $X_n$ be Gamma $(n,\lambda)$ distributed, and $Y_n = \dfrac{\lambda X_n -n}{\sqrt{n}}$. Show that $Y_n \rightarrow N(0,1)$. My idea to prove this is to use Lévys theorem with the ...
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0answers
27 views

Domain of Central Limit Theorem

The central limit theorem says that if you take infinite number of samples ( > 30) from a population, compute their mean values, and collect them, you will reach normal distribution. Is this valid for ...
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0answers
13 views

An example where Lyapounov Condition is much easier than the Lindeberg?

Currently studying the Lindeberg-Levy-Feller and Lyapounov's Central Limit Theorems, and was wondering - for sake of justification - are there any examples where verifying Lyapounov's condition is ...
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1answer
46 views

A central limit theorem for dependent random variable.

Suppose that $u_{j}$ is a sequence of iid standard Gaussian random variable, i.e. $$ u_j\stackrel{d}{=}\text{N}(0,1). $$ Call $\mu_r=\mathbb{E}[|u_j|^r]$. I need to find the asymptotic distribution ...
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0answers
29 views

CLT for decaying random variables

Suppose $X_1,X_2,...$ are bounded random variables with compact support, and $\frac{X_1+...+X_n}{\sqrt{n}}\overset{d}{\longrightarrow}N(0,1)$. Is there neccessarily a central limit theorem for the ...
0
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1answer
38 views

Central limit theorem … need help

So far I have used Chebychev's inequality to calculate n = 157, and my initial thinking for applying the central limit theorem is -1.2815 = (5-0n)/(n* sqrt(391/n)) because P(Z > -1.2815) = 90%, ...
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0answers
28 views

Proof that if you take enough steps of equal magnitude on a plane, you'll always end up at the starting point

Is the claim in the title true? If yes, is the following sufficient to justify it intuitively? Assume for simplicity that the steps are performed with magnitude 1 in a Cartesian plane. To construct a ...