# Stable Distributions And Limit Theorems

• mikeyork
In summary: The convergence I was intending to refer to (sorry I didn't make this clear) was not to the distribution of the sum which comes from a convolution of the distribution, but to the density of the mean which comes from the convolution of the...
mikeyork
I'm an oldie and not well-versed in the modern formalism used in stochastic calculus, so please bear with me. I'm aware of Levy's characteristic function for stable distributions, though not well-versed in its practicalities.

I have read that for alpha=2 the stable distribution is Gaussian and also that the Gaussian is the only stable function with finite variance. However, I think I have found a pdf that is a rational function with finite variance and power law tails |x|^-4 and symmetric in x, for which an infinite recurrence of convolutions with itself appears to converge to another rational function with |x|^-4 tails. If so, then does this not imply convergence to a non-Gaussian stable distribution? And since the cumulative distribution will behave as |x|^-3 this implies alpha = 2 does it not?

On a similar note, I have read that the Central Limit Theorem implies that any distribution that is bounded converges to a Gaussian. Does bounded mean that all moments are finite? If we are always dealing with sampled data then of course all moments will be finite. But I have also read statements to the effect that any distribution with finite variance will converge to a Gaussian. But if we have a continuous distribution and make convolutions, as in my case, then it would seem reasonable that the limiting distribution need not be Gaussian if not all moments are finite!

Can anyone clear these issues up for me? Are cases like mine well known and understood? Is there any literature on them?

mikeyork said:
On a similar note, I have read that the Central Limit Theorem implies that any distribution that is bounded converges to a Gaussian

It says something about the distribution of the mean of independent samples from a distribution F converging, not that the distribution F itself converges to some other distribution.

Stephen Tashi said:
It says something about the distribution of the mean of independent samples from a distribution F converging, not that the distribution F itself converges to some other distribution.

You refer to samples. Any finite number of samples will have all finite moments -- even if F does not. Now suppose we do repeated convolutions of F instead of taking samples. That's the situation I am asking about.

mikeyork said:
I'm an oldie and not well-versed in the modern formalism used in stochastic calculus, so please bear with me. I'm aware of Levy's characteristic function for stable distributions, though not well-versed in its practicalities.

I have read that for alpha=2 the stable distribution is Gaussian and also that the Gaussian is the only stable function with finite variance. However, I think I have found a pdf that is a rational function with finite variance and power law tails |x|^-4 and symmetric in x, for which an infinite recurrence of convolutions with itself appears to converge to another rational function with |x|^-4 tails. If so, then does this not imply convergence to a non-Gaussian stable distribution? And since the cumulative distribution will behave as |x|^-3 this implies alpha = 2 does it not?

On a similar note, I have read that the Central Limit Theorem implies that any distribution that is bounded converges to a Gaussian. Does bounded mean that all moments are finite? If we are always dealing with sampled data then of course all moments will be finite. But I have also read statements to the effect that any distribution with finite variance will converge to a Gaussian. But if we have a continuous distribution and make convolutions, as in my case, then it would seem reasonable that the limiting distribution need not be Gaussian if not all moments are finite!

Can anyone clear these issues up for me? Are cases like mine well known and understood? Is there any literature on them?

You've encountered an example of a two-sided sub-exponential distribution. The one-sided variants are very commonly used in extreme value theory. You may well find a factor of n somewhere in the asymptotic nth convolution (though the two-sidedness makes this more complicated) so the "raw" limit won't exist but with appropriate shifting and scaling the limiting distribution will indeed be a stable Gaussian distribution.

bpet said:
You've encountered an example of a two-sided sub-exponential distribution. The one-sided variants are very commonly used in extreme value theory. You may well find a factor of n somewhere in the asymptotic nth convolution (though the two-sidedness makes this more complicated) so the "raw" limit won't exist but with appropriate shifting and scaling the limiting distribution will indeed be a stable Gaussian distribution.

Thanks a lot for that. I had not encountered the term "sub-exponential" before and I definitely see the relevance.

The convergence I was intending to refer to (sorry I didn't make this clear) was not to the distribution of the sum which comes from a convolution of the distribution, but to the density of the mean which comes from the convolution of the density divided by n. (Since the convolution of the density with itself gives the density of the sum.) The density of the mean has the same variance as the original density (which, unlike the density of the sum, does not have the factor n you referred to). It is this density of the mean that seems to converge to a limiting density (without any further shifting or scaling) and every n I have computed behaves as |x|^-4, the same as the original density. I don't see how, then, it can converge to a Gaussian.

Since I could then, in principle, take that asymptotic density (if it converges) and apply the same convolution, I should surely obtain the same density, because of the convergence, should I not? Is that not the condition that implies a stable distribution?

Perhaps I am confused about the terminology "stable" and "limiting density". What is the proper term to use for the asymptotic (n-->infinity) density of the mean? Is it correct to call the result of asymptotic application of convolution a "stable distribution" if it converges? Can you help out again here with some more explanation of what is going on and/or clarification of terms?

mikeyork said:
...

The convergence I was intending to refer to (sorry I didn't make this clear) was not to the distribution of the sum which comes from a convolution of the distribution, but to the density of the mean which comes from the convolution of the density divided by n. (Since the convolution of the density with itself gives the density of the sum.) The density of the mean has the same variance as the original density (which, unlike the density of the sum, does not have the factor n you referred to). It is this density of the mean that seems to converge to a limiting density (without any further shifting or scaling) and every n I have computed behaves as |x|^-4, the same as the original density. I don't see how, then, it can converge to a Gaussian.

...

It won't (Law of Large Numbers applies). CLT includes a factor of sqrt(n).

HTH

bpet said:
It won't (Law of Large Numbers applies). CLT includes a factor of sqrt(n).

HTH

So am I correct that the density of the mean converges to a stable distribution? It seems to me it must since repeated convolution of asymptotic density divided by n will reproduce itself, won't it?

mikeyork said:
So am I correct that the density of the mean converges to a stable distribution? It seems to me it must since repeated convolution of asymptotic density divided by n will reproduce itself, won't it?

No - remember LLN says the sample mean converges to a constant, so the limiting density doesn't exist. Also check the tail cdf, it should look something like n/|nx|^3.

1 person

## 1. What is a stable distribution?

A stable distribution is a probability distribution that remains unchanged after undergoing certain mathematical operations, such as addition or multiplication. This property makes stable distributions useful in modeling phenomena that exhibit long-term stability or persistence.

## 2. What are the characteristics of a stable distribution?

There are four main characteristics of a stable distribution: stability, symmetry, heavy tails, and infinite divisibility. Stability means that the distribution remains unchanged after being scaled by a constant factor. Symmetry means that the distribution is symmetric around its mean. Heavy tails refer to the tendency of the distribution to have a higher frequency of extreme events than a normal distribution. Infinite divisibility means that the distribution can be decomposed into smaller independent distributions.

## 3. What is the central limit theorem and how is it related to stable distributions?

The central limit theorem states that the sum of a large number of independent and identically distributed random variables will tend towards a normal distribution, regardless of the underlying distribution of the individual variables. Stable distributions are a generalization of the central limit theorem, where the sum of a large number of random variables will tend towards a stable distribution, instead of a normal distribution. In other words, stable distributions are a more flexible and robust version of the central limit theorem.

## 4. What are some real-world applications of stable distributions?

Stable distributions have various applications in finance, economics, and physics. In finance, they are used to model stock market returns, currency exchange rates, and other financial variables that exhibit long-term stability. In economics, stable distributions are used to model income and wealth distributions. In physics, they are used to model turbulence in fluids and the distribution of energy in particle collisions.

## 5. How are stable distributions calculated and estimated?

Stable distributions can be calculated analytically using mathematical formulas, but these can be complex and difficult to work with. They can also be estimated using statistical methods, such as maximum likelihood estimation, which involves finding the parameters that best fit the data to a stable distribution. There are also computer software programs and libraries available that can estimate stable distributions from data, making it more accessible for researchers and practitioners.

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