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This case is obtained when√the shape parameters are such that λ > 0 and α ¯ = 0, that is, δ = 0 and α = c2 λ . 31) for x ∈ . The variance gamma density is a three parameter density. The parameter λ is the shape parameter with smaller λ implying larger tail heaviness. Furthermore, when λ → ∞ the variance gamma density asymptotically approaches the Gaussian density. 10 plots the logarithm of the variance gamma density. The densities of the Student t, normal inverse Gaussian, hyperbolic and variance gamma distribution look very similar when plotted directly.

Thus a gamma distributed random variable is leptokurtic. 21) we have for a Student t distributed random variable X ∼ t(n) the kurtosis ∞ κX = 0 ⎛ ⎝ ⎞4 x n n−2 ⎠ Γ ( n+1 2 ) √ Γ ( n2 ) π n 1+ x2 n − n+1 2 dx = 3 n−2 n−4 . 37) This is finite only for n > 4 degrees of freedom. This type of random variable is also leptokurtic. The Student t density approaches asymptotically a Gaussian density as n → ∞. This is also reflected in its limiting kurtosis of three as n → ∞. In Fig. 24), in dependence on the two shape parameters λ and α.

Xn ) = P (Xi ≤ xi , i ∈ {1, 2, . . , n}) . 1) Its properties can be illustrated by considering the case of two random variables X1 and X2 . 2) xi →−∞ for i = 1 and fixed x2 ∈ condition or i = 2 and fixed x1 ∈ lim x1 ,x2 →∞ , and also the limit FX1 ,X2 (x1 , x2 ) = 1. 3) Furthermore, FX1 ,X2 (x1 , x2 ) is nondecreasing and continuous from the right in x1 and x2 . 4) . The marginal distribution FX1 satisfies FX1 (x1 ) = lim FX1 ,X2 (x1 , x2 ). 5) for x1 ∈ . For continuous random variables the joint distribution function is often differentiable, except possibly at some isolated or boundary points.