Heavy-tailed distribution

In probability theory, a heavy- tailed distribution ( also: heavy -tail distribution ) or fat-tailed distribution is a probability distribution with an infinite variance. Clearly stating the notion that on the "tail" or the " tails " of the distribution is more mass than for example with the normal distribution. The distribution of a random variable is called depending on the condition of short-, medium - or heavy- tailed. If we define the conditional mean excess function

So true

Are independent and identically distributed random variables, is valid under the assumption that they are heavy- tailed that the distribution of the sum of which is asymptotically determined by the distribution of the maximum.

Interpretation

To illustrate the random variable model a waiting period. Follows a short- tailed distribution, then: The longer one has been waiting for, the shorter the expected remaining waiting time. Follows a medium -tailed distribution, the previous waiting time has no effect on the still expected remaining waiting time. The other hand, follows a long-tailed distribution, then: The longer one has been waiting for, the longer it is still expected remaining waiting time.

Application

In the actuarial used heavy -tail (or heavy- tailed ) distributions for modeling of large losses and extreme events. Liability lines is called because of your long run- as so-called long-tail lines of business. In contrast, lines of business are as comprehensive insurance, which household or glass insurance so-called short -tail lines of business. The settlement of the damage in these short -tail lines of business is generally short. In the long-tail lines of business execution durations over 40 years are not uncommon.