# Questions tagged [central-limit-theorem]

For questions about the central limit theorem, which states: "Given certain conditions, the mean of a sufficiently large number of iterates of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed." (Wikipedia)

604 questions
Filter by
Sorted by
Tagged with
13k views

### Are there any examples of where the central limit theorem does not hold?

Wikipedia says - In probability theory, the central limit theorem (CLT) establishes that, in most situations, when independent random variables are added, their properly normalized sum tends ...
160 views

### Central Limit Theorem - Vector of Random Variables - Imputation of Missing Values

I have a large dataset -- 300,000 records, each representing a customer-- and a variable holding their incomes. Since there were missing values, I used the Multiple Imputation Chained-Equations ...
12 views

### Implications of zero limiting variance

Assume that I have a sequence of random variables $X_1, X_2, \dots$ with means $\mu_1, \mu_2, \dots$ such that $\lim_{n \to \infty} \operatorname{Var}(X_n) = 0$. Can I claim that for large enough $n$ ...
28 views

### confidence interval for sample that is not normal distributed

I have one sample of only 86 values, which is not normally distributed according to the Shapiro-Wilk normality test. Can I still use this formula/code (sorry R code) to estimate the 95% confidence ...
48 views

### Confidence Interval for Estimator using Delta method

The statement I am given the following discrete distribution with $\theta>0$ $$p(x) = \left(\frac{\theta}{1+\theta}\right) ^{2-x}\left(\frac{1}{1+\theta}\right)^{x-1} \hspace{1cm} x=1,2$$ I need to ...
5k views

### Question about standard deviation and central limit theorem

I have a quick question about the central limit theorem. Lets say I measure some value that comes from an arbitrary distribution N times and I repeat this M times. I understand that if I calculcate ...
98 views

### theoretical confidence interval depending on sample size [closed]

I am using R and plain English to express my question. Let us say I have a "true"/made up population, which is normally distributed with a mean of 500000 and a standard deviation of 13000: <...
46 views

### Does Normality of a Time Series imply Stationarity and Viceversa?

I have a theory question which never became completely clear to me. Reading Hamilton (1995) I understod that the stationarity requirement for time series data stands as the normality requirement for ...
43 views

### possible use case of central limit theorem for analysts

This is a bit of a long shot but I would appreciate any help please. I have to do a basic stats course for our analysts, which I try to make as applicable and useful as possible using our data (e.g. ...
42 views

### Does the Central Limit Theorem imply that $(\hat{X}_n - \bar{x}) = o_p(1)$ at rate $O_p(1/\sqrt{n})$?

Let $\left\{\hat{X}_n\right\}$ be a sequence of estimators that converges in probability to the constant $\bar{x}$, i.e., $\left(\hat{X}_n - \bar{x}\right) = o_p(1)$. Then say that, by some applicable ...
29 views

### ICA: a question about the non-gaussian requirement

I'm new in the ICA processing and I'm trying to understand the non-gaussian requirement. I read that the problem is that, if the composed data is $\mathbf{x}=\mathbf{As}$ with $\mathbf{A}$ (unknown) ...
67 views

### Rate of convergence of $\hat Q_{xx}^{-1} = \left(\frac{\mathbf{X}^T \mathbf{X}}{n}\right)^{-1}$ to the probability limit?

Consider the simple linear regression model. $$y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \quad \quad \quad \quad i = 1,2,\dots,n.$$ Let $\mu_x$ and $\sigma_x^2$ represent the mean and variance of ...
27 views

### CLT-based confidence intervals not working in code

I wrote code to draw $n$ samples from a categorically distributed random variable $C$ with probabilities $p_i$ for each value $i$ and to use those samples to compute an approximation $q_i = n_i/n$ of ...
22 views

### Understanding rate of convergence for realized estimators

I'm a Econ student currently taking a small course on realized measures/estimators. I'm a bit confused about the meaning behind rate of convergence for each type of estimator. I'll give some ...
71 views

### Central Limit Theorem and Skewed Distribution

I'm looking for a simple answer to this question relating the central limit theorem and Gaussian and skewed distributions, if one exists. I used the binomial function to generate calculations of the ...
49 views

26 views

### Central Limit Theorem - does the number of times the samples are taken matter in terms of the CLT?

Let's say that in R, we generate $n$ random variables $Y_1, \dots, Y_n$ which all follow an exponential distribution. We then construct the mean $\bar{Y_n}$. The process is repeated $m$ times, so we ...
45 views

### Why is it that when you add normally distributed random variables the variance gets larger but in the Central Limit Theorem it gets smaller?

When you add two independent normal distributions the resulting distributions' variance is the sum of the variances i.e. it gets larger. However, the Central Limit Theorem states that when ...
46 views

66 views

### How is confidence interval related to central limit theorem?

I am currently looking into Confidence Interval and sees that Confidence Interval is possible based on Central Limit Theorem. So, I have been looking for how Central Limit Theorem is related to ...
64 views

### Interpretation of odd Central Limit Theorem (i.i.d) condition

My class was taught a third sufficient condition for the CLT to hold in the i.i.d. case that can replace the Lindeberg or Lyapunov conditions. I have never seen this condition before and am wondering ...
45 views

### Asymptotic Distribution Using CLT

I have random variables $X_1, X_2, ... , X_n \sim \text{IID } f_X$ using the density function: $$f_X(x) = \frac{2x}{\theta^2} \cdot \mathbb{I}(0 \leqslant x \leqslant \theta).$$ I have to use the ...
142 views

35 views

### CLT - Adding small samples to one big sample

According to CLT, the SE is the SD of the distribution of several samples means. This SE depends on each sample mean, the SD of each sample and N (the size of each sample which I test). Since there ...