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.
1,530
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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 ...
- 1,609
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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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2
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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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2
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Expectation of max of geometric r.v.
Let $(X_i)_{i \geq 1}$ be a sequence of i.i.d. geometric random variables such that $X_i \sim \text{Geom}(1/2)$ and $\mathbb{E}[X_i]=1$. I want to show that
$$\lim_{n \rightarrow \infty} \frac{1}{n^2}\...
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Is the Hadamard Product of two laplacian operators allowed to get some kind of biharmonic operator?
I'm currently working on my masters thesis in computer science and from this point I'm not that into this subject. Right know I try to understand the steps the authors of this paper did to get the ...
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Limit of expectation of ratio of sums of uniformly distributed variables
I saw this problem in past year exam of probability theory.
Let $U_1, \ldots, U_n, \ldots $ be independent standard uniformly distributed variables: $U_i \sim Uniform(0, 1)$ and $a>b>0$.
...
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Average of a sampled set
I have a set of natural numbers, denoted by A. The average of A is X. Then, I sample every item of A independently and add it to another set, denoted by A' with probability of p. My question is - what ...
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Rate of convergence (Berry-Esseen theorem) for the sum of an asymptotically normal random number of random variables?
Let $\{X_i, i \geq 1\}$ be independent, identically distributed random variables with finite mean and variance.
Let $M_t$ be a non-negative, integer-valued random variable that is independent from $\{...
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Writing a counting process $N(t)$ as $N(t) = \sum_{k=1}^{n(t)} Y_k$ : Existence and properties of $Y_k$?
Let $N(t)$ be the counting renewal process arising from interarrival times $\{X_k\}$ which are independent, identically-distributed, positive random variables, with mean $\mathbb{E}(X_k) = \mu$ and ...
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Confusion about variance statistic in OMC
Adding context, I'm studying Markov Chain Monte Carlo and I'm currently on Ordinary Monte Carlo or Independent Monte Carlo where I have confusion about what it states as estimator of "Variance in ...
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Is central limit theorem applicable to Poisson distributed samples multipled with different coefficients
I am developing a photodetector simulator. The electrons count produced by the incident monochromatic light is Poisson distributed. However, photodetectors respond to the incident light with a wide ...
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Finding $\lim_{\,n\to\infty} \, \sum_{k=n}^{3n} \binom{k-1}{n-1}\left(\frac{1}{3}\right)^n\left(\frac{2}{3}\right)^{k-n}$
The problem is to find the following limit:
$$\lim_{\,n\to\infty} \, \sum_{k=n}^{3n} \binom{k-1}{n-1}\left(\frac{1}{3}\right)^n\left(\frac{2}{3}\right)^{k-n}$$
I know it has something to do with CLT ...
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Show that $\frac{1}{\sqrt{n}} S_n$ does not converge in probability. [duplicate]
Let $(X_n)$ be a sequence of stochastically independent and identically distributed random variables with $\mathbb{E}X_1 = \mu < \infty$ and $\text{Var} \, X_1 = \sigma^2 < \infty$, where $\...
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Normal approximation to Poisson weighted sum
I am trying to come up with a way to approximate the sum given below by a normal distribution.
$$ P(F| \overline{n}) = \sum_{n=0}^{\infty}P(F|n)P(n|\overline{n}) $$
F is the sum of IID random ...
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Using CLT, Slutsky's theorem and delta method
Let $Y_n$ be a sequence of random variables with $\chi^2_n$ distribution. Using Slutsky' theorem or delta method prove that $$\sqrt{2Y_n}-\sqrt{2n-1}\stackrel{D}\to N(0,1)$$
In the first place I ...
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Difference between compensator of point process under real parameter an its MLE estimator
Suppose we have some point process $N=N_{\theta_0}$ on the real line, driven by a conditional intensity $\lambda_{\theta_0}$ dependent on some finite-dimensional parameter $\theta_0\in\Theta\subset\...
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CLT for exchangeable and symmetric dependent Bernoulli variables
Consider a finite set of $n$ dependent Bernoulli random variables $X_1$, $X_2$, ..., $X_n$ that are symmetric and therefore exchangeable. This means:
Each variable $X_i$ has the same probability of ...
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Approximating the poisson distribution using normal distribution
The number of calls $X$ to a telephone exchange during the busiest hour of the day follows a Poisson distribution $Po(λ)$. Over $8$ days, the following independent observations of $X$ have been ...
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CLT for an insurance company
An insurance company has 1,000 individuals of the same age insured. The probability of death in a given year for each insured individual is 0.01. The insured pay an annual premium of 1,200 eur, and in ...
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What is the sample mean in Central limit theorem?
Lindeberg–Lévy CLT
Suppose $X_1, X_2, X_3 … X_n$ is a sequence of i.i.d. random variables with $\mathbb{E}[X_i] = \mu$ and $\text{Var}[X_i] = \sigma^2 < \infty$.
Then, as $n$ approaches infinity, ...
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What probability distribution function is this?
This is sort of a followup to this question (I'll mention everything relevant in this post though so no need to click link).
Main Question:
I was trying to study a random variable $Y$. I will ...
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Asymptotic distribution of MLE of $\theta$ for $f(x) = (1-\theta)1_{(-1/2,0)}(x) + (1+\theta)1_{(0,1/2)}(x)$
I am trying to find the asymptotic distribution for $\theta$ given $f(x) = (1-\theta)1_{(-1/2,0)}(x) + (1+\theta)1_{(0,1/2)}(x)$. I've shown that $\hat{\theta} = \frac{T_2 - T_1}{n}$, where $T_1 = \...
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Equivalent condition for Lindeberg-Levy-Feller Central Limit Theorem.
Let $\left\{X_{n}\right\}_{n \geq 1}$ be a sequence of random variables. Let
$$
S_{n}=\sum_{j=1}^{n} X_{j}, \quad s_{n}^{2}=\sum_{j=1}^{n} \mathbb E\left(X_{j}^{2}\right)<\infty
$$
If $s_{n}^{2} \...
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MLE and limit distribution of ratio of parameters
I am solving an estimation problem and I can't make any progress. I have $ (X_{i1},X_{i2})^T $ iid from $\mathcal{N}_2 ( \mu , \Sigma)$. Define
$$
\lambda_j = \frac{\mu_j}{\sqrt{\sigma_{jj}^2}} , \...
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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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1
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Asymptotic distribution of MLE of $\sigma$ for $N(0,\sigma^2)$
I know that given $X_1,...,X_n \sim N(0,\sigma^2)$, the MLE for $\sigma$ is $\hat{\sigma} = \sqrt{\frac{1}{n}\sum_{i=1}^n (X_i - \bar{X})^2}$. I want to find the asymptotic distribution for $\hat{\...
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How to prove $p$ is in the confidence interval $[f-1/ \sqrt{n},f+1/ \sqrt{n}]$ where $f$ is one possible value of $\sum X_i /n$?
Let $X_1, ...,X_n$ be independent random variables such that $X_i \sim Bernoulli(p)$, $\forall i \leq n$
Let $F_n = \sum_{i=1}^n X_i /n$ and let $\omega_0$ be one event in the sample space of $F_n$ ...
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How to Calculate $\frac{1}{\sqrt{n}}x^{\top}y$ given random vectors $x$ and $y$?
Given random variable $X$ and Gaussian random variable $Y$ with zero mean and unit variance ($X$ is independent of $Y$), $\boldsymbol{x}\in \mathbb{R}^n$ is i.i.d. sampled from $X$ and $\boldsymbol{y}\...
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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, ...
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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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A multidimensional Central Limit Theorem for Pólya Urns
I know that if we fix $\alpha_1,\dots,\alpha_d\ge1$ and $S\ge1$ integers, and we define the $d$-dimensional process $(Z(n))_{n\ge0}$ as
follows: let $V$ be a Dirichlet random variable of parameter $\...
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How can an estimator be consistent and asymptotically normal at the same time?
I can't work out why the asymptotic distribution of an estimator matters if it is consistent?
My understanding is:
An estimator, $\hat{\theta}$, is consistent if it converges in probability to the ...
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1
answer
518
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Law of large number with subset of the variables
Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors with continuous joint density, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}\subset\mathbb{R}$ be decreasing ...
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Two ways of using the Central Limit Theorem on a basic example
The problem is as follows.
An insurance company sells 10,000 similar car insurance policies. They estimate that the amount paid out in claims on a typical policy has mean £240 and standard deviation £...
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What is the Asymptotic Order of the Sum of Random Variables with non-finite Moments?
Suppose $X_i$ is independently and identically distributed over $i$, and $E(|X_i|^d)$ with $d>0$ is not finite or undefined:
I am wondering whether $\sum_{i=1}^N |X_i|^d$ is of any asymptotic ...
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Higher version of Central Limit Theorem
I've been studying statistics for a while now, and honestly, I've been struggling to gain an intuition for both the Central Limit Theorem (CLT) and the Law of Large Numbers (LLN) up to this point. Now,...
- 1,560
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Multidimensional CLT: method of moments
Suppose that we have a sequence of random variable $X(n)$ and we know that each $X(n)$ has moments of all orders. The well-known `method of moments' tells us that if, for each $k \ge 1$, $\mathbb{E}(X(...
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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 ...
- 1,054
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0
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15
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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 ...
0
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0
answers
132
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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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39
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Convergence of the probability that the sum of the numbers drawn from a given distribution exceeds a given value.
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 ...
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