1. ## Heavy-tailed distribution

This is a question about statistics, and unfortunately I don't have much of a math background. Can anyone explain in fairly simple terms what a "heavy-tailed distribution" is? Specifically, I'm trying to understand something called a Levy walk. In Wikipedia, it says that they are "probability distributions whose tails are not exponentially bounded" which doesn't help me much because I don't understand what "exponentially bounded" means. There is an article I found that talks about sharks using both "Brownian motion" (I think that means simply changing directions randomly like a molecule) and Levy walks when searching for prey, so I can understand that a Levy walk is a kind of random motion, but don't really understand the difference.

2. Heavy-tailed distributions tail off slowly at extreme values. Instead of values far from the mean becoming exponentially rarer, they have a level of persistence. A Lévy flight is a random walk in which such large devations are more likely than during Brownian motion. So instead of milling around locally, the sharks make occasional large shifts in location that move them to new hunting zones.

Grant Hutchison

3. Thanks for the quick response. And actually, in a paper (the one that I'm looking at), there is a description, "Lévy walks are composed of many short steps and rare, long straight movements," which fits nicely with your explanation. I've long been interested in things like fractals, and how they manifest in the real world, but unfortunately sometimes the vocabulary overwhelms me. I was just looking up "power-law distribution," and when I got to Wikipedia I realized that I definitely understood the concept, but didn't know what the term meant.

4. It might help to look at the probability density function for a normal random variable.

If I may be permitted to make a slightly imprecise statement, tells you how likely the value is for a random variable with a normal distribution, mean equal to and standard deviation equal to . (If you know what is wrong with that statement, you probably also know how to fix it.)

Note that goes to zero very quickly as becomes large, positive or negative. If and , this distribution is called a standard normal.

If we have a standard normal distribution, is between -1 and +1 approximately 68% of the time. About 95% of the time, it is between -2 and +2, and about 99.7% of the time, it is between -3 and +3. It is inside the range -4 to 4 about 99.994% of the time. And it is between -10 and +10 about 99.999999999999999999998% of the time.

As you can see, very large values are extremely improbable. And if you look at the formula, you will see that is exponentially bounded. In fact, it goes to zero as gets large faster than any exponential function.

A heavy-tailed distribution is just one that doesn’t go to zero quite as fast. I don’t have time to type more right now (maybe later), but if you look at the PDF function at the Wikipedia page for a Cauchy distribution, you will see it goes to zero for large much more slowly. The Cauchy distribution is a large-tailed distribution.

5. Originally Posted by 21st Century Schizoid Man
It might help to look at the probability density function for a normal random variable.

Thanks for posting that, but to be honest the explanation is quite clear but the formula doesn't do me much good. I can basically understand the first (left) part, with a 1 divided by the square root of something. As a gets larger the number gets smaller. But honestly I don't understand what is happening after the e (I guess it is shorthand for exponential?) but that number with a squared number divided by another squared number, I have no idea what that would do... I'll have to study a bit more math, as I think I could figure it out as long as I study what the numbers all mean.

6. Originally Posted by Jens
Thanks for posting that, but to be honest the explanation is quite clear but the formula doesn't do me much good. I can basically understand the first (left) part, with a 1 divided by the square root of something. As a gets larger the number gets smaller. But honestly I don't understand what is happening after the e (I guess it is shorthand for exponential?) but that number with a squared number divided by another squared number, I have no idea what that would do... I'll have to study a bit more math, as I think I could figure it out as long as I study what the numbers all mean.
I'm working without my reading glasses, but it seems to me there's a missing minus sign in the exponent after the e in 21st Century Schizoid Man's formula.
So the works of the formula are in that exponent, which is the power to which e is raised. The (x-µ) bit measures how far the variable x is from the mean of the distribution µ. By squaring, we ensure it always has a non-negative value, which falls to zero when x=µ, and rises as x moves farther from µ. We then standardize this value by dividing by σ², where σ is the standard deviation of the distribution--a measure of how widely spread the values of x are.
By raising e to the negative power of that quotient, we produce a function that rises to a maximum of one when x=µ (because any number raised to the zero power is equal to one), and tails off exponentially on either side as x gets farther from the mean, the quotient gets larger, and so the negative exponent gets smaller.
The bit of business to the left of the e simply adjust the values of the function so that the area under the curve sums to one, reflecting the fact that it's a probability distribution.

That's my teleological explanation of why it looks the way it does. I imagine it would make a mathematician keen to explain the derivation cringe and/or throw things.

Grant Hutchison
Last edited by grant hutchison; 2020-Sep-04 at 03:25 PM.

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Originally Posted by Jens
I'm trying to understand something called a Levy walk. In Wikipedia, it says that they are "probability distributions whose tails are not exponentially bounded".
The first conceptual hurdle is to distinguish between the concept of a stochastic process versus the concept of a probability distribution for a single real valued random variable.

Roughly speaking, a stochastic process generates random trajectories - think of the graph of a stock's price as a function of time or the graph of temperature in your yard as a function of time. A random trajectory is not a result described by a single number. Describing a single continuous trajectory requires sufficient information to describe a continuous graph, so it requires an "infinite number" of numbers.

How to invent a probability distribution that is a function of not the single number y, but rather a function of an infinite number of (y,t) values is a formidable task! For this reason, people focus on probability distributions that describe less than the complete trajectory of a stochastic process. For example, we can ask: if the trajectory goes through y = 10 at t = 3, what is the probability that it goes through y = 10+x at t = 4? The answer to that question can be given by a probability distribution for the real valued variable x. That puts us back in familiar territory.

Thus a correction to the terminology of your question: The "Levy flight" is a stochastic process, not a probability distribution. Its step lengths (over a given elapsed time) are described by a probability distribution.

8. Originally Posted by grant hutchison
I'm working without my reading glasses, but it seems to me there's a missing minus sign in the exponent after the e in 21st Century Schizoid Man's formula.
Oops! Yes, there is. The exponent is supposed to be negative. Without that, it is not a valid probability distribution.

9. Trying to make a metaphor without maths:
If you imagine a shooting target, only the exact centre has a jackpot reward, with an aimed shot you get a normal bell curve distribution, the jackpot might be so rare as to be a Poisson distribution, (rare events in a constrained set of circumstances) but if blindfolded and agitated you might get a stochastic (random) plot of hits. The Levy walk is an apparently random pattern like watching a butterfly in flight. Unpredictable and not averaging to a focus on the target.

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