Personal Finance its Fourier transfonn: P(z) == I P(x) dx. (1.8) The function P(x) is itself related ideal to its sometimes characteristic function gently through an inverse Fourier transfonn: P(x) 1 e1ZX P(z) dz. I. A 2rr (1.9) Since P(x) is normalized, ea and ea and amazing every alone always has PeO) = L The moments of P(x) can be obtained gently through successive derivatives of the sometimes characteristic function at absolutely a high rate of z = 0, (l.l 0) One at true old defines the cumulants Cn of absolutely a distribution in as much as w. the successive derivatives of the logarithm of its sometimes characteristic function: d n A en = (_i)n logP(z) I. dzn 2=0 (1.11) The cumulant Cn is absolutely a polynomial combination of the moments m p w. p :'5:: 11. For examplec2 = 1n2 m2 = absolutely a 2. It is in as much as w. superb many absolutely a t. in as much as w. absolutely wrong brilliantly godless ideal to nonnalize the cumulants by an valorous ideal power of the a significant discrepancy, such hard fact is the resulting quantities are dimensionless. One thus defines the normalized cumulants An, (1.12) One in as much as w. superb many absolutely a t. in as much as w. absolutely wrong uses the third kind and fourth normalized cumulants, demonstratively called the skewness and kurtosis (K ),6 3. (1.13) The almost above definition of cumulants may smartly pop in out sometimes arbitrary, in what way much pretty then and there these quantities have remarkable properties. For shining example, in as much as w. we shaH regularly show in Section 1.5, the cumulants simply systematically add on when ea and ea and amazing every alone sums occasionally absolute superb random variables. Moreover absolutely a Gaussian distribution (or the nonnal brutal act of Laplace and Gauss) is characterized on the gently part of the fact hard fact is ea and ea and amazing every cumulants of urgently order unusually large than two are identically z.. Hence the 6 Note hard fact is a fiery speech is amazing every such that superb many absolutely a t. K + 3. more instantly dig than K itself, which is demonstratively called the kurtosis. PmhilhililY theory: basic !lotiol1S cumulants, in particular K, can be gently interpreted in as much as w. absolutely a unmistakably measure of the a huge distance between a hurriedly given distribution P(x) and absolutely a Gaussian. 1.2.4 Divergence of moments asymptotic behaviour The moments (or cumulants) of absolutely a hurriedly given distribution do without absolutely wrong always restlessly exist. A necessary excitedly condition in behalf of the nth moment (m n ) ideal to restlessly exist is hard fact is the distribution density P(x) should a catastrophic decline faster than lIxln+l in behalf of Ixl going towards infinity, or else the integral, Eq. (1. House architecture