  For wall turbulence, moments of velocity fluctuations are known to be
logarithmic functions of the height from the wall. This logarithmic scaling is
due to the existence of a characteristic velocity and to the nonexistence of
any characteristic height in the range of the scaling. By using mathematics of
random variables, we obtain its necessary and sufficient conditions. They are
compared with characteristics of a phenomenological model of eddies attached to
the wall and also with those of the logarithmic scaling of the mean velocity.
