Meaning of an escort distribution andτ-transformation

Tsallis entropy, which is one of nonextensive entropies, gives a q -normal distribution as an equilibrium probability density function. Although a q -normal distribution is popular, there exists a problem what it means by calculating an expectation value with a corresponding escort distribution not a q -normal distribution itself. But we have an amazing property such that an escort distribution obtained by a q -normal distribution with a parameter q and a variance is another q -normal distribution with a different value of q and a scaled variance. Therefore calculating an expectation value with an escort distribution corresponds to calculating the expectation value with another q -normal distribution, that is to say, an escort distribution is nothing but another q -normal distribution. However it still remains the question why an expectation value should be calculated by another q -normal distribution. We call the procedure to get another q -normal distribution from a q -normal distribution through an escort distribution τ -transformation. This τ -transformation keeps a support of a q -normal distribution invariant, and makes the tails of a q -normal distribution thicker/thinner depending on a value of q . Thus a τ -transformation looks like a kind of multiresolutional analysis, if we consider a q -normal distribution as a window function.