# 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). The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.

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### Asymptotic distribution of repetitions in binary sequence

Let $N$ be a fixed positive integer. Let $b_1,\dots,b_M$ be a sequence of i.i.d. Bernoullay random variables with $p=\frac12$. So $P(b_i=0)=\frac12=P(b_i=1)$ for all $i$. Now let $X$ be the amount of ...
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### Central limit theorem of independent partial sums

Let $M,N\to\infty$ with $M/N\to 0$. Suppose there is a random sequence $X_{mi}$ with mean 0 and variance $\sigma_{m}^{2}$. $X_{mi}$ is independent of $X_{mj}$ for all $i\neq j$ but $X_{mi}$ is ...
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### Conditional central limit theorem

Let $\{S_m\}_{m\in\mathbb{N}}$ be a centered random walk with real values and i.i.d. increments $X_i$ with $\mathbb{E}(X_i) = 0 ,\mathbb{E}(X_i^2) = 1$. I'm having problems in studying limits of the ...
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### A question about the proof of Levy's continuity theorem

I'm reading the proof of Levy's continuity theorem by Christian Döbler but there is one part I don't understand. Namely, in the proof of Theorem 1.1, it is assumed that $(X_n), X, Z$ are independent ...
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### Existence of High Dimensional Central Limit Theorem

Given a sequence of $n$ i.i.d. random vector $\{\mathbf{X}_{i}\}_{1 \leq i \leq n}$ in $\mathbb{R}^{d}$, with covariance matrix $\mathbf{I}_{d \times d}$, and considering that the dimension $d$ is a ...
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### Central Limit Theorem for Bounded Random Vectors with Dependency Graphs

I am familiar with the following Central Limit Theorem (CLT) result for a family of bounded random variables with a dependency graph structure (Paper Link): Let $\{Y_{1}, \ldots, Y_{d}\}$ be a family ...
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1 vote
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### Law of Large Numbers for Changing Distributions

I thought of the following problem: Suppose there is a school with 1000 students. A random sample of 50 students is selected and each of these 50 students is asked to run 100 meters and the time is ...
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### Limit of integral of sum of cosine functions by CLT?

I want to show that $$\lim_{n\to \infty} (2\pi)^{-d}n^{d/2}d^{-2n}\int_{[-\pi, \pi]^d} (\cos(x_1)+\cdots +\cos(x_d))^{2n} dx_1\cdots dx_d =2(d/4\pi)^{d/2}$$ holds. How do I prove this? It seems that ...
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### Asymptotic Distribution and Describe Sources of Increasing Power in an hypothesis testing problem

I am currently dealing with the following problem in a past exam (with no solution): Suppose $S$ follows the Poisson distribution with mean $2\lambda>0$, here $\lambda$ is a parameter. Another two ...
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### Show that $S_n \xrightarrow{a.s.} S$ for some random variable $S.$

Let $\{X_n \}_{n \geq 1}$ be a sequence of independent random variables such that for some $\alpha \geq \frac {1} {2},$ \mathbb P \left (X_n= \pm n^{\alpha} \right ) = \frac {n^{1-2 \alpha}} {2} \...
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### Central limit theorem for two-sided Pareto distribution

I am trying to solve the following problem, which provides an example for a central limit theorem in spite of the fact that the variance is infinite. Consider the two-sided Pareto distribution with ...
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### Help me intuit Equal Probabilities in Poisson Distribution for $k = λ$ and $k = λ-1$

I was trying to understand Poisson distribution and I'm confused as to why the likelihood of $k=λ-1$ is equal to $k=λ$. Here is my understanding and where my confusion is: For a given time interval, ...
1 vote
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### Why is a factor of 24 used in this Central Limit Theorem Question?

I have a question from Example 12.3.1 from Marcel B. Finan book Lecture Notes in Actuarial Mathematics Question: Let $X_i$, $i = 1, 2, \dots, 48$ be independent random variables that are uniformly ...
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### Confidence interval for parameter of normal distribution $X_i\sim N(\theta,\theta^2)$ with equal mean and standard deviation

A sample $X_1,\dots,X_n$ is drawn from the normal distribution $N(\theta,\theta^2)$. I am asked to find a $90\%$ confidence interval for the population mean $\theta$. Let $X_i\sim N(\theta,\theta^2)$ ...
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### Infinite divisibility and Gaussian random variables

I was looking for a simple explanation of why the Gaussian random variable can be the only distribution appearing in the Central limit theorem. From the statement of the Central limit theorem, it is ...
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### Mean and variance when sampling temporal rates

I having some trouble understanding how mean and variance are be used in sampling rates and temporal rates (per unit of time). Example If I eg. wanted to evaluate car crashes per hours driven I would ...
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### Using CLT to define normal distribution

While reading the statement for the Central Limit Theorem on Wikipedia, I began wondering if the following "definition" for "normal distribution" makes sense in the context of real,...
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### Central Limit Theorem for Difference-in-Means Estimator

I am studying Lecture 1 of Stefan Wager's Causal Inference notes and come across a central limit theorem for the difference-in-means estimator, which I am unable to prove. The mathematical abstraction ...
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Question from an old test: The bakery produces doughnuts with weights $W_1 \text{g}, W_2 \text{g}, W_3 \text{g}, \ldots$, where $W_i$ for each doughnut is drawn from a distribution with a finite ...