Limitation of Log-Normal Theory of Turbulence

The probability distribution of the sum of n independent random variables with finite mean and finite variance does not generally approach the normal distribution with variance proportional to n , contrary to the log-normal theory of turbulence in which the central limit theorem is applied in an easy manner. Although the expectations of all the moments of the sum agree with those of the limiting normal distribution in the leading order for n ≫1, the moments of the exponential of the sum do not. Thus the log-normal theory is not so robust as expected from the generality of the central limit theorem.