| Abstract |
Let an infinitely divisible law for a one-dimensional random variable be given by the logarithm of its characteristic function (ch. f.), $\varphi (t)$: \[ \log \varphi (t) = i\beta t - \gamma t^2 + \int_{ - \infty }^0 {\left( {e^{itu} - 1 - \frac{{itu}} {{1 + u^2 }}} \right)dG_ - (u)} + \int_0^\infty {\left( {e^{itu} - 1 - \frac{{itu}} {{1 + u^2 }}} \right)dG_ + (u)} , \]where $\beta $ and $\gamma \geqq 0$ are real constants; ${dG_ - (u)}$ and ${dG_ + (u)}$ are non-decreasing functions such that \[ G_ - ( - \infty ) = G_ + (\infty ) = 0;\quad \int_{ - a}^0 {u^2 dG_ - (u)} + \int_{ - 0}^a {u^2 dG_ + (u)} < \infty \] for any finite $a > 0$, and $\int_{ - \varepsilon }^0 {u^2 dG_ - } + \int_0^\varepsilon {u^2 dG_ + } \to 0$ as $\varepsilon \to 0$. We call ${G_ - (u)}$ and ${G_ + (u)}$ the negative and positive part of the Poisson spectrum of the corresponding law and $\gamma \geqq 0$ its Gaussian component.We introduce also the concepts of the bounded Poisson spectrum, continuous Poisson spectrum, countable or finite Poisson spectrum. In particular, in case of the countable or finite Poisson spectrum, we may write \[ \log \varphi (t) = i\beta t - \gamma t^2 + \sum\limits_{m = 1}^\infty {\lambda _m \left( {e^{it\mu _m } - 1 - \frac{{it\mu _m }} {{1 + \mu _m^2 }}} \right)} + \sum\limits_{n = 1}^\infty {\lambda _n \left( {e^{it\nu _n } - 1 - \frac{{it\nu _n }} {{1 + \nu _n^2 }}} \right)} , \] where ${\mu _m }$, ${\nu _n }$ are Poisson frequencies; $\lambda _m \geqq 0,\gamma _{ - n} \geqq 0$ are energy parameters.Theorem 1. For an infinitely divisible law with a Gaussian component$\gamma > 0$to decompose only into infinitely divisible components it is necessary that its Poisson spectrum be finite or countable. Moreover, the corresponding Poisson frequencies${\mu _m }$and${\nu _n }$must coincide with the following series of numbers:For ${\mu _m }$\[ \cdots k_{ - 1} k_{ - 2} \mu ,k_{ - 1} \mu ,\mu ,\frac{\mu } {{k_1 }},\frac{\mu } {{k_1 k_2 }}, \cdots ,\frac{\mu } {{k_1 k_2 \cdots k_s }}, \cdots . \]For ${\nu _n }$\[ \cdots l_{ - 1} l_{ - 2} \nu ,l_{ - 1} \nu ,\nu ,\frac{\nu } {{l_1 }},\frac{\nu } {{l_1 l_2 }}, \cdots ,\frac{\nu } {{l_1 l_2 \cdots l_s }}, \cdots , \]where$k_{ - 2} ,k_{ - 1} ,k_1 ,k_2 $and$l_{ - 2} ,l_{ - 1} ,l_1 ,l_2 $are arbitrary sets of natural numbers$ > 1$ (repetitions being permitted). If the Poisson spectrum is bounded, this necessary condition is also sufficient.The presence of the Gaussian component is essential. If the Poisson spectrum is not bounded, it is not known whether the necessary condition is also sufficient. It is proved only that it is sufficient if the high frequency energy is sufficiently small (Theorem 4), it is sufficient if \[ \log \log \frac{1} {{\lambda _m }} > c\mu _m^{1 + \alpha } ,\,\log \log \frac{1} {{\lambda _{ - n} }} > c\nu _n^{1 + \alpha } \] for some $c > 0$, $\alpha > 0$ and sufficiently large $\mu _m $ and $\nu _n $.Theorem 2. If the Poisson spectrum of the law is bounded so that$dG_ - (u) = 0(u \leqq - b),dG_ + (u) = 0(u > a)$, then all its components have ch. f. of the form\[ \varphi _j (t) = \exp \left\{ {P_{3j} (it) + t^4 \int_{ - b}^a {e^{itu} \Phi _j (u)du} } \right\}, \]where$P_{3j} (it)$is a polynomial of degree$\leqq 3$and$\Phi _j (u) \in L^2 ( - b,a)$is a Lebesgue square-summable function.Theorem 3. LetFbe an infinitely divisible law with a bounded Poisson spectrum which is rational to the right of the point$\alpha \geqq 0$. Then all its possible components have ch. f. of the form\[ \log \varphi (t) = P_3 (it) + t^4 \int_{ - b}^\alpha {e^{eitu} \Phi (u)du} + \sum\limits_{n = 1}^p {\left( {\alpha _n + \beta _n it} \right)} \left( {\exp \left( {it\frac{n} {q}\mu } \right) - 1} \right). \]where$\alpha _n ,\beta _n $are real numbers (not necessarily positive), $\Phi (u) \in L^2 ( - b,a);P_3 (it)$is a polynomial of degree$\leqq 3$.A stability theorem (Theorem 5) is also proved: Let$L_1 $be the set of all infinitely divisible laws with bounded Poisson spectra satisfying the necessary condition of Theorem 1 (the Gaussian component may be absent). Let$F \in I_1 $and$K_F $be the set of all infinitely divisible laws with the Poisson spectra contained in the spectrum ofF. Let$\{ {F_j } \}$be the sequence of laws such that\[ \mathop {\sup }\limits_x \left| {F_j (x) - F(x)} \right| = \varepsilon _j \to 0{\textit{ as }}j \to 1 \]and$F_j = F_{1j} * F_{2j} $(composition).Then$\mathop {\inf }\limits_{{\bf F}^{(1)} \in {\bf K}_{\bf F} } \mathop {\sup }\limits_x \left| {F_{1j} (x) - F^{(1)} (x)} \right|\delta _j \to 0.$ |
