By J.P. Raoult
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Restrictions on higher order cumulants are still an area of active research. 8 CUMULANT EQUATIONS It was shown earlier that the characteristic function Âð j uÞ can be deﬁned if an inﬁnite set of cumulants k ; k ¼ 1; 2; . , is given. In other words, the characteristic function, and thus the 50 RANDOM VARIABLES AND THEIR DESCRIPTION PDF, is a function of an inﬁnite cumulant vector j ¼ f1 ; 2 ; . g: pðxÞ ¼ pðx; 1 ; 2 ; . Þ ¼ pðx; jÞ ð2:236Þ It was also mentioned that the cumulants can be chosen independently from a certain region of values and this choice can be made independently for each cumulant as long as they satisfy certain inequalities .
When the function pðxÞ is close to ! 1 x x p0 ðxÞ ¼ exp À Àð þ 1Þ ð2:76Þ where and are deﬁned through the mean and the variance of pðxÞ as in eq. 74). Thus, the ﬁrst term in the Laguerre series is the Gamma distribution. Similar orthogonal expansions can be obtained for other weighting functions as long as they generate a complete system of basis functions. Not all probability densities have this property. A detailed discussion of this subject can be found in . 3 RANDOM VARIABLES AND THEIR DESCRIPTION Gram–Charlier Series There is another important method of representation of a PDF p ðxÞ of some random variable through a given PDF p ðxÞ of some standardized random variable .
1; . . ; xn Þ ¼ 0 Pn ð1; . . ; 1; . . ; 1Þ ¼ 1 ð2:101Þ ð2:102Þ 27 RANDOM VECTORS AND THEIR DESCRIPTION 2. The CDF is a non-decreasing function in each of its arguments. 3. If the k-th component of the random vector n is excluded from consideration, the CDF PnnÀ1 of the remaining vector can be obtained as follows nnÀ1 ¼ ½1 ; . . ; kÀ1 ; kþ1 ; . . ; n T Pn ðx1 ; . . ; 1; . . ; xn Þ ¼ PnnÀ1 ðx1 ; . . ; xkÀ1 ; xkþ1 ; . . ; xn Þ ð2:103Þ 4. The CDF can be expressed as an n-fold repeated integral of the corresponding CDF ð x1 ð xn Pn ðxÞ ¼ ÁÁÁ pn ðxÞd x1 d x2 .
Statistique non Parametrique Asymptotique by J.P. Raoult