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Central Limit Theorem for Lévy Process

Answer 1 is a bit of an overkill. The question seems to be about the case where the second moment is finite and not something as general as Doney and Maller. Below are two different simple proofs. ...
user165536's user avatar
7 votes
Accepted

Using CLT, Slutsky's theorem and delta method

From the CLT, $$\sqrt{n}\left(\frac{Y_n}{n} - 1\right) \overset{d}{\to} N(0,2).$$ With $g(u) = \sqrt{u}$, the delta method implies $$\sqrt{n}\left(g\left(\frac{Y_n}{n}\right) - g(1)\right) \overset{d}{...
angryavian's user avatar
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4 votes

Using CLT, Slutsky's theorem and delta method

As the statement is about distributions, we assume that $Y_{n}=\sum_{k=1}^{n}Z_{k}^{2}$ for some iid $N(0,1)$ variates defined on some fixed probability space. $$\frac{2Y_{n}-(2n-1)}{\sqrt{2Y_{n}}+\...
Mr.Gandalf Sauron's user avatar
2 votes
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Approximating the poisson distribution using normal distribution

I think the point of the question is to appeal to the central limit theorem (CLT), which states that for i.i.d. observations $X_{i}$ with $E[X_{i}^{2}] < \infty$ (so that the variance exists), we ...
minginator's user avatar
0 votes
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CLT for an insurance company

The model you are using for the future lifetime random variable is incorrect. You seem to be using a Bernoulli distribution, but this does not correctly take into account the number of years a given ...
heropup's user avatar
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1 vote

What probability distribution function is this?

This is just my (not very rigorous) summary of the arguments in the comments (community wiki). Let $R$ be the number of runs ($R=3$ in your example). In the original setting we have $L$ as a fixed ...
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What is the sample mean in Central limit theorem?

You wrote a scalar version central limit theorem. Since the distribution converges to a normal distribution with mean $\mu$ and covariance $\sigma^2/n$, each sample is a scalar. Of course, you could ...
mjw's user avatar
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