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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
45 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 ...
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0answers
19 views

Does Central Limit Theorem imply finite mean and variance?

Given is a statistic $\hat{\theta}_n(X_1,...,X_n)$, where $X_i$ i.i.d. and have finite mean and variance. Also let $\sqrt{n}(\hat{\theta}_n(X_1,...,X_n) - \theta) \overset{d}{\to} N(0, \gamma^2)$ ...
2
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1answer
34 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
26 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
8 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 ...
1
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1answer
30 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
25 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 ...
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1answer
36 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
27 views

Poisson Process For Large n

A supermarket has two entrances, the main entrance and the side entrance. The arrival of customers through the main entrance follows a Poisson process with rate 3 per minute and the arrival of ...
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0answers
25 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 ...
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2answers
36 views

Probability limits of random variable sums

I have $X_1, X_2, X_3, \cdots$ which are independent random variables with the same non-zero mean ($\mu\ne0$) and same variance $\sigma^2$. I would like to compute $$\lim_{n\to\infty} P[\frac{1}n\sum^...
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1answer
16 views

Central limit theorem. Calculating probability P(N≤49)

Here is the problem: Apples are being packed in a box. One apple weight is expected to be 200 g with a dispersion of 20 g. Packing is stopped as soon as the total weight is 10 kg or more. Calculate ...
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0answers
50 views

$Z_i = \min\{X_i, Y_i\}$, central limit theorem [closed]

Suppose $X_i, Y_i$ are two random variables independently taken from the uniform distribution $(0, \sqrt{i})$, i=1,2...n. Let $$Z_i = \min\{X_i,Y_i\}.$$ Assume further that the pair $X_i,Y_i$ is ...
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1answer
17 views

A Central Limit Theorem simple example

A disscusion in the book: Let $(X_n)_{n=1}^\infty$ a sequence of i.i.d random variables such that $\mathbb{E}[X_n]=60, \operatorname{Var}[X_n]=25$. Let $S_N= \sum_{i=1}^NX_i$. By the central limit ...
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0answers
24 views

Normal approximation of sum of uniform independent RVs using CLT

Let $X_1$, $X_2$, ... $X_{16}$ and $Y_1$, $Y_2$, ... $Y_{16}$ be independent uniform random variables over the interval [-1,1] and let: $$ W = \frac{(X_1 + .... + X_{16}) + (Y_1 + .... + Y_{16})}{16} ...
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1answer
30 views

Central Limit Theorem for not identical distributed but independent centered random variables with variance one.

so let's assume we have independent random variables $X_1,X_2, X_3, \ldots$ with $$\mathbb{E}[X_k]=0 \mbox{ and } \mathbb{Var}[X_k]=\sigma_k^2=1 \quad \forall k\in\mathbb{N}. $$ We define $$s_n^2:= ...
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2answers
50 views

Does a sequence of random variables constructed in a certain manner converge in distribution to a Gaussian?

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of of IID random variables taken for simplicity with mean zero and variance one. The Central Limit Theorem give us that $$ \frac{X_1 + \dots + X_n}{\...
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1answer
22 views

using Central Limit Theorem to approximate a probability with large 'n'

For $i>1$ , let $X_i$ ~ $G_{1/2}$ be distributed Geometrically with parameter $1/2$. $$ Y_n= \frac{1}{\sqrt n} \sum_{r=1}^n X_r-2 $$ Approximate $P(-1\le Y_n\le 2)$ with large n.($Y_n$ is not ...
4
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1answer
79 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-...
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0answers
35 views

Solving CLT with Exponential Distributions

I am working on a CLT problem and am a bit stuck. The problem: N1 = 10,000 with claims X1 ~ Exp() with mean = 100 N2 = 3000 with claims X2 ~ Exp() with mean = 200 N3 = 1000 with claims X3 ~ Exp() ...
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1answer
33 views

Central Limit Theorem using sample standard deviation

Let $X_1, X_2,..$ be iid random variables with mean $\mu$ and variance $\sigma^2$. Show $$ \displaystyle \frac{\sum_{i = 1}^n (X_i - \mu)}{\sqrt{\sum_{i = 1}^n (X_i - \overline{X}_n)^2}} \to N(0,1) $...
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2answers
27 views

Question about limiting distribution for standard normal distribution

I am studying CLT and meet a question but I have no idea how to solve it, could you please show me how to prove this question (b)? Thank you so much! Wang
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0answers
33 views

Central Limit Theorem Bounds

I am working on a Central Limit Theorem problem and am a bit stuck. The problem: N = 10000 Xi = X ~ Exp(1) Yi = Y ~ Exp(2) All independent, find P(total amount is between 7400 and 7560) What I ...
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1answer
67 views

Central limit theorem for sequence of Gamma-distributed random variables.

Suppose that $X_ n \sim \text {Gamma}\ (n\alpha , \lambda)$ for all $n \ge 1$, for fixed $\alpha,\lambda >0.$ Show that $$\frac {1} {\sqrt n} \left (X_n - \frac {n \alpha} {\lambda} \right ) \...
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1answer
25 views

Central Limit Theorem and convergence of transformed

I have the following exercise to solve: Let $X_n, n \geq1$ be a sequence of i.i.d random variables where each $X_n$ is a discrete random variable with distribution $P(X_n=1)=1-p$ and $P(X_n=2)=p$, ...
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1answer
37 views

Central Limit Theorem - Different Forms

Given the following function: $W_n = \frac{1}{\sqrt{n}}\Pi_{k=1}^{\infty}\log(U_k)$ where $U_k$ is uniformly distributed from $1$ to $e$. Does $\{W_n\}_{n\geq 1}$ converge in distribution? I found ...
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20 views

approximate probability of geometric distribution using CLT

I have the following problem: For $i≥1$, let $X_i∼G_1/2$ be distributed Geometrically with parameter 1/2. Define $$Y_n=\frac{1}{\sqrt{n}}\sum_{i=1}^n (X_i-2)$$ Approximate $P(−1≤Y_n≤2)$ with large ...
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0answers
8 views

$X_i = i^{\delta}Z_i$, find the range of $\delta$ for which the law of large numbers and the central limit theorem are valid

$Z_1,Z_2,...$ are i.i.d., their expected value is zero, their variance $\sigma^2$, and $E[|Z_i^2|] = m_3 < \infty$. $X_i = i^{\delta}Z_i$, find the range of $\delta$ for which the law of large ...
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1answer
10 views

Confidence Interval of number of red marbles among 100 marbles where proportion of red marbles is uniformly distributed

A bag contains 100 marbles of colors red and black. The proportion of red color marbles is uniformly distributed between 0 and 1. How do I compute the confidence interval of the number of red marbles? ...
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2answers
29 views

What is the limit of the following expression?

I've been thinking about and trying to solve the following limit that I just feel lost by now. I always get an indeterminate form. I don't know what else to try. In the picture is just one way that I ...
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1answer
81 views

Proof of Convergence in Distribution for random variables with infinite variance

We are asked to prove that given $\{X_n\}$ being a sequence of iid r.v's with density $|x|^{-3}$ outside $(-1,1)$, the following is true: $$ \frac{X_1+X_2 + \dots +X_n}{\sqrt{n\log n}} \xrightarrow{\...
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1answer
28 views

Application of Central Limit Theorem to Sales

Consider the following problem and solution. (I am stuck at the modified problem.) Problem There are exactly two phone shops, $A$ and $B$, serving a town of 1000 people. Both shops sell an iPhoneX ...
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2answers
23 views

Approximating a random variable versus approximating probability statements about a random variable?

After formally stating the central limit theorem my statistics textbook says this: Interpretation: Probability statements about the sample mean $\overline{X}_n$ can be approximated using a Normal ...
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1answer
33 views

Bochner's theorem

I'm reading Bochner's theorem. Now I'm having problem with part on the third page: $\int_{s\in [0,T], s+u\in [0, T]} ds=1-\frac{|u|}{T}$? How to deduce it? Any help is welcome. Thanks in advance.
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2answers
87 views

Understanding the central limit theorem

I am an aspiring probabilist, and I definitely know the central limit theorem. However, I am trying to understand what idea it really embodies. I am aware that normal distribution arises as the limit ...
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21 views

Potentially new method for obtaining asymptotic distribution of M-estimators

Disclaimer. I'm not quite sure this is the best venue for this question, but I'll give it a try... So, in a comment to this MO post, it was said that one can use the comment right after Remark 7.3 in ...
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1answer
23 views

Central Limit Theorem and Hoeffding's bound yield different conclusions

Let $X_1,\ldots,X_n$ be i.i.d Bernoulli r.v with parameters $p$. For the sake of the example, say that $p=0.9$. I want to assess $P(\frac 1n \sum_{i=1}^nX_i\leq 0.1).$ By the CLT, $\frac 1n \sum_{i=1}...
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1answer
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Is my textbook definition of multidimensional CLT incorrect?

From All of Statistics by Wasserman, 2nd Edition: There are two ways I can interpret how $\mu$ is defined: The $\mu$ defined is the expected value of a single one of the $X_i$ vectors. But then $\...
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1answer
21 views

Convergence in distribution of the sum of two dependent random variables

I have the following question about the limiting distribution of the sum of two random variables say $Z_n = X_n+Y_n.$ I know the following: Conditioned on $X_n,$ $Y_n$ has a CLT i.e., $$\mathbb P (...
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2answers
54 views

Law of large numbers for sum

Liz is standing on the real number line at position 0. She rolls a die repeatedly. If the roll is 1 or 2, she takes one step to the right (in the positive direction). If the roll is 3, 4, 5 or 6, she ...
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1answer
27 views

Expectation of the mean of the sum of random variables [closed]

If $X_i$'s are independent and identified random variables, each with mean $\mu$ and variance $\sigma^2$. Let's say $S_m = \frac{1}{m} \sum_{i=1}^m X_i,~~ m = 1,2,\ldots,M.$ What are the values of $\...
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2answers
43 views

Why are these claims true?

I'm reading a textbook that makes these claims: (1) Given a random variable $X$ that is $1/N$ times the sum of $Np$ independent random numbers, each of which takes on the values $1$ or $-1$, $X$ has ...
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2answers
68 views

Central Limit Theorem for geometric mean

Suppose that $X_1,X_2,...$ be i.i.d. variable uniformly distributed on (0,1), and let $\tilde{X_n}$ denote the geometric average of $n$ of these variables, i.e.: $\tilde{X_n}=(X_1X_2\cdots X_n)^{1/n}$....
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1answer
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Convergence of average of i.i.d. Bernoulli to Normal will be slower for p closer to 0 or 1

this is true. If you fit a normal to the data you see, if you have small samples, the normal is spread out in shape. So the area cut out by the $x=0$ line or $x=1$ line is larger than the case when $p=...
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1answer
41 views

Question about the strong law of large numbers (to build understanding)

I'm learning about the LLN and CLT for the first time and I'm having some trouble. I've read through other posts but I have a quirky (likely dumb) question about denominators... The Strong LLN ...
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0answers
32 views

Choosing the samples that satisfies the Central Limit Theorem

My informal understanding of the CLT is that: if we draw a number of samples from a population, then the mean of all the samples will approach a normal distribution as the sample size (of each sample) ...
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1answer
138 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 ...
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0answers
58 views

Tail probability of sum of order statistics of distance from point to a set

Let $P$ be a distribution on a metric space $(\mathcal X, d)$. For a point $x \in \mathcal X$ and a Borel $B \subseteq \mathcal X$, let $d(x,B) := \inf_{y \in B}d(x,y)$ be the distance of $x$ from $B$....
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0answers
32 views

Mean for Lyapunov Condition on Probaiblity Function

Lyapunov condition $$\lim_{n\to\infty} \frac{E|V_n-E(V_n)|^{2+\Delta}}{n^{\Delta/2}\sigma^{2+\Delta}V_n}$$ is to be prove for $\Delta=0$, where $$V_n=\frac{1}{h}\exp\left\{-\left(\frac{j-x}{h} +\...
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1answer
17 views

Which distribution to use

After many generations, it is known that 75% of students pass certain subject. if a random sample of 40 students is such that it could be consider they have the same characteristics as the ones ...