The Ultimate Form of a Kinematic Wave

Since f(p) exp xP(p) is a Laplace transform, analytic in the neighbourhood of the origin, the imaginary axis is within its half plane of convergence and may be taken as the contour in the inversion integral. Thus 

Roughly speaking, to require that the Laplace transform of a function should be analytic at the origin is to require that the function should tend to zero as e A, where A 0. Such a function possesses all its moments and this is clearly necessary if it is to be asymptotically Gaussian. The Gaussian distribution, however, tends to zero as e A with A 0, and if the function is similarly rapid in its decay it may be possible to carry the approximation still further by means of a type A series (Kendall 1943, p. 147). We shall merely outline the way in which this may be possible. 

In this series the fourth term is -msps/5 but for the terms above this the coefficients become increasingly complex combinations of the coefficients mr in the expansion of to. Now the formal interpretation of p3g(p) is that it is the Laplace transform of g (p) so that writing 

It does, however, indicate the way in which the normal distribution is attained. The absolute skewness mllml and the kurtosis mjml both tend to zero as x-1. The difficulty in the practical application lies in the truncation error and the fact that the higher terms decrease in a somewhat irregular manner. It may often be of value, however, to estimate the skewness from the third moment and even in such a complicated example as that of 4 this is not impracticable. 

Kendall, M. G. 1943 The advanced theory of statistics. London Griffin and Co. 

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