Key Takeaway: This paper presents a probabilistic approach to the tails of infinitely divisible real random variables, enabling the study of their characteristic functions and their deterministic properties.

Abstract

Consider an arbitrary non-degenerate infinitely divisible real random variable X. Usually this is described by the Levy formula for its characteristic function. For any t in the real line ℝ, we have
$$ \begin{array}{*{20}c} {C(t)} \hfill & { = \,E\,\exp (iXt)} \hfill \\ {\,} & {= \,\exp \,\{ i\theta t\, - \,\frac{{\sigma ^2 }}{2}t^2} \hfill \\ {\,} & {+ \,\int\limits_{ - \infty }^0 {\left( {e^{ixt} \, - \,1\, - \,\frac{{ixt}} {{1\, + \,x^2 }}} \right)\,dL\left( x \right)\, + \,\int\limits_0^\infty {\left( {e^{ixt} \, - \,1\, - \,\frac{{ixt}} {{1\, + \,x^2 }}} \right)\,dR\left( x \right)} \} ,}} \\ \end{array}$$
where θ є ℝ and σ ≥ 0 are uiquely determined constants and L and R are uniquely determined left-continuous and right-continuous functions (the so-called Levy measures) on (−∞,0) and (0, ∞), respectively, that is L(·) and R(·) are non- decreasing functions such that L(−·) = 0 = R(·) and
$$ \int_{ - \in }^0 {x^2 dL\left( x \right)\, + \,\int_0^ \in {x^2 } \,dR\left( x \right)\, \,0.}$$
(1.1)