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)

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Non-normal sample from a non-normal population (option returns) does the central limit theorem hold?

I'm testing a short call option strategy and found, as expected, non-normal return distributions. It is known that option returns are not normally distributed (i.e., also the population). I take the ...
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1answer
47 views

Proof Poisson converges to Normal [closed]

I am looking for a formal proof that, with the CLT transformation, a random variable $Y \sim POI(\lambda)$ converges to a normal distribution ($Z\sim N(0,1)$). I believe this can be formulated as: $$...
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8 views

Original vs Sample mean distribution in reconstructing time series data

Lets assume that the time series corresponds to some observable signal from a processor that computes a modulo 2 operation. The nature of the time series depends largely on the output of the modulo ...
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What is the meaning behind a sample distribution if you only have one sample?

I'm trying to understand the meaning behind the central limit theory and the importance of CLT for inferential statistics. The problem that I encountered has to do with sample distributions. ​ I do ...
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28 views

Sampling as adding random variables, especially binomial RVs

Is sampling equivalent to adding random variables? I'm a bit confused because as we can see that the binomial distribution becomes more and more shaped like a normal distribution as $n$ increases. We'...
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PDF of given complex exponential equation of random variables [closed]

Let $d_i$ and $d_k$ both are independent and identical random varibles with pdf $\frac{3d_i^2}{R_{max}^3-R_{min}^3}$ where $R_{max}$ and $R_{min}$ both are constants and $ R_{min}<d_i<=R_{max}$...
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CLT in Diebold & Mariano (1995)

The Diebold-Mariano (DM) statistic is derived as follows: Assuming the loss-differential between the two models $d_t$ is covariance stationary: $$\begin{cases} \mathbb{E}[d_t] = \mu> 0 & \...
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1answer
76 views

CLT so we have enough burgers!

An event has 32 people. Caterers need to make burgers for these people. They expect that a person at this event might need 0, 1 or 2 burgers with probabilities 0.2, 0.55, and 0.25 respectively. We ...
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Picking a specific estimated CDF from a set of CDFs provided by an ECDF

Let $F_X$ be a CDF of an unknown random variable $X$. If we have independent samples $x_1, x_2, \ldots, x_n$ of $X$ then we can estimate $F_X$ non-parametrically using an ECDF $\hat{F}_n$. By Central ...
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Convergence of variance of sample median, pt. 2

Follow on question to this, answered negatively by Thomas Lumley. We reprint it here for convenience. In this SE question, it is stated that there is a central limit theorem for the sample median, ...
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Cental limit theorem, Chebyshev's inequality, and convergence of distributions through rescaling

I've been thinking about this issue for a few days and although read some of relevant questions on this site, still couldn't get it off my mind. Suppose we have $n$ i.i.d random variables $X_i$ with ...
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Limit of $\mathbb{P}(\sqrt n - 1 \leq \sqrt{X_1^2 + \cdots X_n^2} \leq \sqrt{n} + 1)$ wrt $n$ for standard multivariate normal $X$

Suppose $X$ is a standard multivariate normal distribution. Then what is the limit of the $c_n$ where the $c_n$ are $\mathbb{P}(\sqrt n - 1\leq \sqrt{X_1^2 + \cdots X_n^2} \leq \sqrt n + 1)$? I want ...
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Sample and population: definition for workplace statistics?

This might be a bit of a dunce's question, but I was wondering about the difference between a sample and a population. Obviously, if you have data relating to 200,000 people but you only look at 200 ...
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1answer
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Convergence of variance of sample median

In this SE question, it is stated that there is a central limit theorem for the sample median, namely $$ \sqrt{n}(Y_n - m) \xrightarrow{d} N(0, [2f(m)]^{-2}), $$ as $n\to\infty$ where $Y_n$ is the ...
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Different regularity conditions for finite population CLT

I am having trouble understanding the different regularity conditions for different versions of the finite population central limit theorem. I would greatly appreciate any help or insight anyone has. ...
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1answer
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Does a Binomial converge to Poisson or Normal?

I have read the answer here. Here the distinction is that If $n\to\infty$ and $p\to0$ while $np$ approaches some positive number $\lambda,$ then the binomial distribution approaches a Poisson ...
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277 views

Plain English explanation of Ito's integral?

I'm looking for a plain English explanation of Ito's integral. I don't need an exhaustive proof, derivation, etc. Just a simple ~this is effectively what it does and why it's better than a Riemann sum ...
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55 views

Is bootstrapping redundant for computing confidence intervals for the sample mean in large samples?

If we are just interested in computing confidence intervals for the population mean $\mu$ using a sample $X_1,X_2,\dots,X_n$ of $n$ iid random variables is bootstrapping redundant if $n$ is large? I ...
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1answer
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CLT for non iid random variables

Assume $U_k$ are correlated standard normal random variables. Let $R_k := a_k U_k$. I'm looking for CLT of the sum $S_p := \sum_{k=1}^{p}\frac{R_k}{\sqrt{p}}$. Since $U_k$ are correlated, I'm looking ...
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Differences Between the Central Limit Theorem and Consistency

I have recently finished studying the central limit theorem and the idea of consistency. I am still a little fuzzy about them, so I was wondering what are some key similarities and differences of the ...
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Approximating $E[g(\overline X_n)]$ and want to bound the remainder using some form of CLT or Berry-Essen Theorem

If we have a set $X_1,\dots,X_n$ of iid random variables with finite mean $\mu$ and variance $\sigma$, the CLT says that $\sqrt{n}(\overline X_n - \mu) \stackrel{d}{\to} \mathcal{N}(0,\sigma^2)$. If ...
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1answer
76 views

Limiting distribution of $\sum_{j=1}^{p}\lambda_j U_j$

Assume $U_j$ are $\chi^2(1)$ random variables and $\lambda_1, \ldots, \lambda_p$ are the eigenvalues of a covariance matrix $\Sigma = (r^{|i-j|})_{ij}$ with a Toeplitz-type structure (for some fixed $|...
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Asymptotic properties of between-group estimates?

Suppose my data consists of the mean of several separate groups (sizes may vary) from an iid sample of an outcome variable Y and an independent variable X. The data generating process is $Y = \beta X +...
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75 views

what is the standard deviation of the geometric mean sample distribution?

I wrote a python script to take a population distribution of a random variable in the interval (0,1) to be uniform and make 2 sample distribution: The fist is the distribution of the arithmetic mean ...
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1answer
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confidence interval of $\beta$, where $X$'s are from exponential distribution

Suppose $X_i\overset{ind}{\sim}\mathcal{E}(\lambda_i)$, where $\lambda_i=(t_i\beta)^{-1}$, where $t_i$'s are positive known values and $\beta$ is positive unknown parameter. Here $i=1,\dots,n$. It can ...
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How to think about confidence intervals in the context of linear regression?

I think I understand confidence intervals for sampling distributions, but am trying to connect this understanding to the confidence bands I see around linear regression lines. According to the ...
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Limit distribution for a linear discrete-time stochastic process: limit of the sum of linearly transformed uniform distributions

I have posted this in math stack exchange, but I figured maybe this is a better forum for this kind of question. Suppose we have a stochastic linear process: $$x_{k+1} = Ax_{k} + Bw_{k} \qquad \text{...
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1answer
114 views

Confidence interval for the mean of the uniform distribution

I can take samples of a random variable $X \sim U(a, b)$, where the length of $(a, b)$ is known. I am interested in its mean $E[X]$, estimated with $\hat{X_n} = \frac{1}{n} \sum_{i=1}X_n$, but I need ...
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OLS Regression: How can the Central Limit Thorem justify the assumption of normality of the error term?

Relative noob here. My understanding is that the CLT is used to derive the sampling distribution of the sample means. So from a population distribution, whether normal or not, if you take a sample and ...
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1answer
57 views

Central limit theorem for the function of an iid random variable

Given an iid random variable $X$, instead of the distribution $\sqrt{n}(n^{-1}\sum{X_{i}}-E[X])$ which is the result that the central limit theorem provides , I am interested in the distribution of $\...
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1answer
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Mean and variance of the Gaussian resulting from Central Limit Theorem

Let $\{x_i\}$ be a set of iid random variables (not necessarily Gaussian distributed). The CLT states that $\frac{1}{n}\sum_{i=1}^n x_i$ is asymptotically normal. What do we know about the mean and ...
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Asymptotics of the maximum of k subsample means

Suppose we have $n$ i.i.d samples $X_1, X_2, \cdots, X_n$ from some real-valued distribution $P$. Let $\alpha \in [0,1]$ be a fixed constant. Select an uniformly random subset $S_1 \subset \{1,2,\...
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1answer
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How many times do we need to sample to get the sampling distribution of means?

I was looking into the central limit theorem and noticed that every article talks about how big the sample size should be but they never indicate how many times should we be sampling to get the ...
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1answer
120 views

Does central limit theorem help in making inference only about the population mean and not other parameters?

As per the weak law of large numbers, if your sample size is large, your mean of the sample is likely to be closer to the population mean than in a smaller sample. Additionally the CLT tells us the ...
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2answers
288 views

Central Limit Theorem: Sample Size or Number of Samples?

The central limit theorem states that if we take a take a large enough sum of random variables, the sum will approach a normal distribution. I am confused about why we focus only on the sample size ...
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2answers
109 views

OLS assumption normallity of error term really needed?

As the title explains I was wondering whether the additional OLS assumption of having a normally distributed error term isn't redundant if the sample is large enough. I understand that we want the ...
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Finding cross-correlation p-value for non-normal time series with CLT

I have two non-normally distributed time series with 77 data points (each) which I am cross-correlating. The time series were pre-whitened to remove autocorrelation. Since I am not trying to use ...
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2answers
718 views

Why a sample of skewed normal distribution is not normal?

I was under the impression that if I randomly sample from a skewed normal distribution, the distribution of my sample would be normal based on central limit theorem, but the graph clearly shows that ...
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Stochastic Convergence - Radford Neal's Prior

There's question 1.3 in this set of questions already solved I found about stochastic convergence. I cannot understand where the expression $$\sum_{i} \phi_{i}^{2}(x)$$ came from and why is the last ...
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1answer
92 views

OLS vs LAD in cases with large n

Thanks to lots of helpful answers in the community, I figured that Least Absolute Deviations regression can give better estimations when the normality of residuals is violated (e.g. residuals ...
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1answer
71 views

Can this problem really be solved using central limit theorem?

My friend had this question on a test: Let $\{X_n\}_{n \in N}$ be a sequence of independent random variables with the same normal distribution $N(0, 2n)$. Check for the convergence of a sequence $\{...
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26 views

Network version of central limit theorem?

Is there a version of the central limit theorem for random variables that are connected through a network? More precisely, imagine I have $N$ vertices in a graph, and a random variable $X_{i}$ for ...
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52 views

Convergence of sum of (1) random variable that converges in distribution to Normal and (2) degenerate random variable that diverges to infinity?

Say that we have $\sqrt{n}(\hat{\mu} - \mu_0)$, which we can equivalently write as $\sqrt{n}(\hat{\mu} - \mu) + \sqrt{n}(\mu - \mu_0)$, where $\mu$ is the population mean, $\hat{\mu}$ is the sample ...
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1answer
115 views

Multinomial & Covariances

Assume we have $\mathbf{X} = (X_1,\ldots,X_k) \sim Multinomial(n,\mathbf{p}=(p_1,\ldots, p_k)).$ How can we find $\operatorname{Cov}(e^{X_i},e^{X_j})$? Tried to do CLT but it's led me nowhere.
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1answer
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How to establish asymptotic normality of unbiased estimator and find asympotic variance?

Given a linear regression model with deterministic regressors $$y_{i} = x'_{i}\beta + \epsilon_i, \quad \epsilon \sim (0,\sigma^2) i.i.d., \quad i = 1, ...,n, \quad with \quad \mu_4 := \quad \mathbb{E}...
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41 views

Probability of averages

I have a random variable $Y$ and I am taking an independent sample of $n$ from this RV. I'll refer to this sample as $Y_n$, and I define the average of this sample as $\bar{Y}_n$. The maximum of this ...
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17 views

Which distributions originates from sampling near boundaries [closed]

I have a certain non-deterministic process which receives a parameter k that belongs to a fixed domain [a, b]. I am able to generate samples from the output of the process in the domain of k and by ...
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1answer
46 views

Central Limit Theorem Approximation and Relation to Law of Large Numbers

Assume the Linberg-Levy CLT to where we know $$\sqrt{n}\frac{\bar{X}_n-\mu}{\sigma}\xrightarrow{d}N(0,1).$$ I feel like I commonly see then that $$\bar{X}_n\approx N(\mu,\frac{\sigma^2}{n}),$$ but ...
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80 views

Why does bootstraping not seem to produce a normal distribution for this data?

I am trying to calculate the 95% confidence interval of the mean value of the population. I have this data: ...
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0answers
32 views

Sample mean of a geometric distribution

Let $D$ be a distribution with finite mean $\mu$ and finite variance $\sigma^2$. Consider the distribution $S_n$ of the sample mean of $n$ i.i.d. values from $D$. I understand that the Central Limit ...

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