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Reports of the Academy of Sciences of the USSR
- Vol. 161, No. 1
MATHEMATICS
I. V. OSTROVSKII
ON DECOMPOSITIONS OF INFINITELY DIVISIBLE LAWS WITHOUT A GAUSSIAN COMPONENT
(Presented by Academician Yu. V. Linnik, 30 IX 1964)
1°. Introduction. We shall use the terminology adopted in the monograph [1]. In particular, by the Poisson spectrum of an infinitely divisible (i.d.) law \(F\) we shall mean the set of growth points of the functions \(M_1(x)\) and \(M_2(x)\) occurring in the representation of its characteristic function (c.f.) \(\varphi(t)\) by Lévy’s formula
\[ \varphi(t)=\exp\left\{ i\alpha t-\gamma t^2+ \int_{-\infty}^{0}\left(e^{itx}-1-\frac{itx}{1+x^2}\right)dM_1(x)+ \right. \]
\[ \left. +\int_{0}^{\infty}\left(e^{itx}-1-\frac{itx}{1+x^2}\right)dM_2(x)\right\}. \]
By \(I_0\) we shall denote the class of i.d. laws having only i.d. components.
Yu. V. Linnik proved ([1], Ch. VIII) that, for an i.d. law with a Gaussian component to belong to \(I_0\), it is necessary that the Poisson spectrum of this law be finite or countable and, in addition, satisfy a very stringent condition of arithmetic character. We shall consider i.d. laws without a Gaussian component. From the theorems established by P. Lévy [2] and D. A. Raikov [3] it follows that, for such laws, the condition of arithmetic character appearing in Yu. V. Linnik’s result is not necessary for membership in \(I_0\). However, it was not known ([1], p. 257) whether it is necessary that the Poisson spectrum be finite or countable.
We give a negative answer to this question, indicating two classes of i.d. laws lying in \(I_0\) such that among the components of each of them there are laws with continuous Poisson spectrum.
2°. Formulation of the results.
Theorem 1. Let \(F\) be an i.d. law without a Gaussian component, whose Poisson spectrum lies on the segment \([a,b]\), and suppose that the condition \(0<a<b\le 2a<\infty\) is fulfilled. Then \(F\in I_0\).
Following [4], we shall call a set with linearly independent points a set on the real axis such that every finite system of its points is linearly independent over the field of rational numbers. It is known ([4], p. 209) that every perfect set contains a perfect subset with linearly independent points.
Theorem 2. Let \(F\) be an i.d. law without a Gaussian component, whose Poisson spectrum is positive and is a closed bounded set with linearly independent points. Then \(F\in I_0\).
Theorems 1 and 2 describe the two classes of i.d. laws lying in \(I_0\) which were mentioned in the introduction. We derive Theorems 1 and 2 from a single theorem, possibly of independent interest, on the form of the c.f. of components of an i.d. law without a Gaussian component and with positive closed bounded Poisson spectrum. To formulate it, we introduce the following notation: \(V^{(b)}\) is the collection of all funct-
tions having bounded variation on the axis \((-\infty,\infty)\), continuous from the left and equal to zero at \(-\infty\); \(S(\sigma)\), where \(\sigma \in V^{(b)}\), is the smallest closed set whose complement consists of points of constancy of \(\sigma(x)\); \((n)A\), where \(n=1,2,3,\ldots\), is the set defined by the conditions: \(A=A\), \((n)A=(n-1)A+A\) (\(A+B\) is the set consisting of numbers of the form \(\alpha+\beta\), where \(\alpha \in A,\ \beta \in B\)).
Theorem 3. Let \(F\) be an i.d. law without a Gaussian component, with Poisson spectrum \(A\), and let \(0<a<b<\infty\), where \(a=\inf_{x\in A}x,\ b=\sup_{x\in A}x\). Then the ch.f. of any component of the law \(F\) has the form
\[ \exp\left\{ i\alpha t+\int_{-\infty}^{\infty}(e^{ixt}-1)\,d\sigma(x)\right\}, \tag{1} \]
where \(\alpha\) is real, and the function \(\sigma(x)\in V^{(b)}\) is nondecreasing on \([a,2a]\) and
\[ S(\sigma)\subset [a,b]\cap \bigcup_{n=1}^{\infty}(n)A. \]
Theorems 1, 2, and 3 for the case when the Poisson spectrum of the law \(F\) consists of a finite number of points were proved by D. A. Raikov \((^3)\). The method that we use in this article is a certain development of D. A. Raikov’s method.
3°. Results used.
(a) If \(\sigma_1(x),\sigma_2(x)\in V^{(b)}\), then \(S(\sigma_1*\sigma_2)=\overline{S(\sigma_1)+S(\sigma_2)}\); moreover, if \(\sigma_1(x)\) and \(\sigma_2(x)\) are nondecreasing functions, then \(S(\sigma_1*\sigma_2)=S(\sigma_1)+S(\sigma_2)\).
(b) (A special case of Theorem 9.02 from \((^1)\), p. 184). If \(F\) is an i.d. law whose Poisson spectrum lies in \([-b,b]\), \(b>0\), then the ch.f. of any component of the law \(F\) has the form \(\exp g(t)\), where \(g(t)\) is an entire function of exponential type not exceeding \(b\).
(c) (Corollary of the Paley—Wiener theorem). If an entire function \(g(t)\) of exponential type not exceeding \(b\) is representable for real \(t\) in the form
\[ g(t)=\int_{-\infty}^{\infty} e^{ixt}\,d\sigma(x), \tag{2} \]
where \(\sigma(x)\in V^{(b)}\), then \(S(\sigma)\subset[-b,b]\), and representation (2) holds in the whole complex \(t\)-plane.
4°. Proof of Theorem 3. Without loss of generality, one may assume that the ch.f. of the law \(F\) has the form
\[ \varphi_0(t)=\exp\left\{\int_{-\infty}^{\infty}(e^{ixt}-1)\,d\sigma_0(x)\right\}, \]
where \(\sigma_0(x)\in V^{(b)}\) is a nondecreasing function and \(S(\sigma_0)=A\). Then we have
\[ \varphi_0(t)=c_0\sum_{n=0}^{\infty}\frac{1}{n!}\left\{\int_{-\infty}^{\infty} e^{ixt}\,d\sigma_0(x)\right\}^n, \]
where \(c_0=\exp\{\sigma_0(-\infty)-\sigma_0(\infty)\}\). Hence the relation* follows:
\[ F(x)=c_0\left\{\varepsilon(x)+\sum_{n=1}^{\infty}\frac{1}{n!}\sigma_0^{n*}(x)\right\}, \tag{3} \]
where \(\varepsilon(x)=0\) for \(x\le 0\), \(\varepsilon(x)=1\) for \(x>0\). By virtue of (a) we have \(S(\sigma_0^{n*})=(n)S(\sigma_0)=(n)A\). Since \((n)A\subset[na,nb]\), \(a>0\), for each
* If \(\sigma\in V^{(b)}\), then \(\sigma^{n*}\) is defined by the conditions \(\sigma^{1*}=\sigma,\ \sigma^{n*}=\sigma*\sigma^{(n-1)*}\).
can intersect a finite interval for only a finite number of sets \((n)A\). Therefore the set \(\displaystyle \bigcup_{n=1}^{\infty} (n)A\) is closed, and from (3) it follows that
\[ S(F)=\{0\}\cup \bigcup_{n=1}^{\infty} (n)A. \]
Now let \(F=F_1*F_2\), where \(F_1\) and \(F_2\) are some probability laws. Since (by (a)) \(S(F)=\overline{S(F_1)+S(F_2)}\), and \(S(F)\subset[0,\infty]\), the sets \(S(F_1)\) and \(S(F_2)\) are bounded on the left. Without loss of generality, we may assume that the infimum for \(S(F_1)\) is \(0\) (this can be achieved by replacing \(F_1(x)\) by \(F_1(x+\delta)\), and \(F_2(x)\) by \(F_2(x-\delta)\), where \(\delta\) is the infimum of the set \(S(F_1)\)). But then \(0\in S(F_1)\), \(0\in S(F_2)\), and, consequently, \(S(F_1)\cup S(F_2)\subset S(F)\).
Since \(0\) is an isolated point for \(S(F)\), \(0\) is also an isolated point for \(S(F_1)\). Therefore the law \(F_1\) has a jump at the point \(0\). Denoting the size of the jump by \(c_1\) \((c_1>0)\), we may write the relation: \(F_1(x)=c_1\varepsilon(x)+G(x)\), in which \(G(x)\in V^{(b)}\) is nondecreasing and
\[ S(G)\subset \bigcup_{n=1}^{\infty} (n)A\subset [a,\infty). \]
For the c.f. \(\varphi_1(t)\) of the law \(F_1\), we obtain the expression
\[ \varphi_1(t)=c_1+\int_a^\infty e^{ixt}\,dG(x). \tag{4} \]
By virtue of (b) we have \(\varphi_1(t)=\exp g(t)\), where \(g(t)\) is an entire function of exponential type not exceeding \(b\), and, consequently, the function \(\varphi_1(t)\) is entire. Then relation (4) holds in the whole complex \(t\)-plane.
Choose \(\eta>0\) so large that
\[ \int_a^\infty e^{-x\eta}\,dG(x)<c_1. \]
Put
\[ G_\eta(x)=\int_{-\infty}^{x} e^{-y\eta}\,dG(y). \]
Noting that for real \(\xi\)
\[ \left|\int_a^\infty e^{ix\xi}\,dG_\eta(x)\right|\le \int_a^\infty dG_\eta(x)<c_1, \tag{5} \]
we obtain the relation
\[ g(\xi+i\eta)=\ln\varphi_1(\xi+i\eta) =\ln\left\{c_1+\int_a^\infty e^{ix\xi}\,dG_\eta(x)\right\} \]
\[ =\ln c_1+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n c_1^n} \left\{\int_a^\infty e^{ix\xi}\,dG_\eta(x)\right\}^{n} \tag{6} \]
(\(\xi\) real). Consider now the series
\[ \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n c_1^n}\,G_\eta^{n*}(x). \]
From the right-hand inequality in (5) it follows easily that this series converges on \((-\infty,\infty)\) and that its sum belongs to \(V^{(b)}\). Denote this sum by \(\sigma_\eta(x)\). Since \(S(G_\eta)=S(G)\), we have
\[ S(\sigma_\eta)\subset \bigcup_{n=1}^{\infty} (n)S(G) \subset \bigcup_{n=1}^{\infty} (n)\left\{\bigcup_{k=1}^{\infty} (k)A\right\} = \bigcup_{n=1}^{\infty} (n)A. \]
Further, from (6) it follows that for real \(\xi\)
\[ g(\xi+i\eta)=\ln c_1+\int_{-\infty}^{\infty} e^{ix\xi}\,d\sigma_\eta(x). \tag{7} \]
With the aid of (b) we conclude that \(S(\sigma_\eta)\subset[-b,b]\) and that relation (7) holds for all complex \(\xi\). Setting in (7) \(\xi=t-i\eta\) and taking
\[ \sigma(x)=\int_{-\infty}^{x} e^{y\eta}\,d\sigma_\eta(y), \]
we shall have
\[ g(t)=\ln c_1+\int_{-\infty}^{\infty} e^{ixt}\,d\sigma(x), \tag{8} \]
where
\[ S(\sigma)=S(\sigma_n)\subset[-b,b]\cap\bigcup_{n=1}^{\infty}(n)A =[a,b]\cap\bigcup_{n=1}^{\infty}(n)A. \]
Now we have
\[ \varphi_1(t)=\exp g(t)=c_1\sum_{n=0}^{\infty}\frac1{n!} \left\{\int_{-\infty}^{\infty} e^{ixt}\,d\sigma(x)\right\}^{n}. \]
Hence we obtain
\[ F_1(x)=c_1\left\{\varepsilon(x)+\sum_{n=1}^{\infty}\frac1{n!}\sigma^{n*}(x)\right\}. \tag{9} \]
Since
\[ S\left(\sum_{n=2}^{\infty}\frac1{n!}\sigma^{n*}\right) \subset\bigcup_{n=2}^{\infty}(n)S(\sigma)\subset[2a,\infty), \]
it follows from (9), for \(a\le x_1<x_2\le 2a\), that
\[
F_1(x_2)-F_1(x_1)=c_1\{\sigma(x_2)-\sigma(x_1)\};
\]
therefore the function \(\sigma(x)\) does not decrease on \([a,2a]\). Noting that \(g(0)=\ln\varphi_1(0)=0\), from (8) we obtain
\[
\ln c_1=\sigma(-\infty)-\sigma(\infty),
\]
and consequently the function \(\varphi_1(t)\) has the form (1).
5°. Because of lack of space, we omit the proofs of Theorems 1 and 2. We note that when \(b<2a\), Theorem 1 is a special case of Theorem 3.
6°. Remarks on Theorem 1.
1) Theorem 1 may be supplemented by the following assertion. If the Poisson spectrum of an infinitely divisible law \(F\) is concentrated on \([-b,-a]\) and \(0<a<b\le 2a<\infty\), then \(F\in I_0\).
2) From Theorem 1 and the preceding remark it follows that every infinitely divisible law \(F\) can be represented in the form
\[
F=F_1*F_2*F_3*\cdots,
\]
where \(F_k\in I_0\) \((k=1,2,3,\ldots)\). Indeed, if \(\varphi(t)\) is the characteristic function of the law \(F\), then
\[ \varphi(t)=e^{i\alpha t-\gamma t^2}\prod_{k=-\infty}^{\infty}\varphi_k^{(1)}(t)\varphi_k^{(2)}(t), \]
where \(\alpha\) and \(\gamma\) are the quantities appearing in the representation of \(\varphi(t)\) by Lévy’s formula, and \(\varphi_k^{(1)}(t)\) \((\varphi_k^{(2)}(t))\), \(k=0,\pm1,\pm2,\ldots\), are the characteristic functions of an infinitely divisible law with Poisson spectrum contained in \([-2^{k+1},-2^k]\) (\([2^k,2^{k+1}]\)); moreover, the corresponding function \(M_1(x)\) \((M_2(x))\) in Lévy’s formula for \(\varphi_k^{(1)}(t)\) \((\varphi_k^{(2)}(t))\), for \(x\in[-2^{k+1},-2^k]\) (\([2^k,2^{k+1}]\)), is taken equal to the function \(M_1(x)\) \((M_2(x))\) corresponding to \(\varphi(t)\).
3) The condition \(b\le 2a\) in Theorem 1 is essential. Using the method of G. Cramér’s paper \((^5)\), it is not difficult to show that, for \(0\le a\le 2a<b<\infty\), an infinitely divisible law for which
\[
M_2'(x)\ge \mathrm{const}>0\quad\text{for }a\le x\le b
\]
does not belong to \(I_0\).
I express my deep gratitude to B. Ya. Levin for his attention to the work and for valuable comments.
Kharkov State University
named after A. M. Gorky
Received
26 IX 1964
REFERENCES
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- D. A. Raikov, Izv. Acad. Sci. USSR, Ser. Math., 2, 91 (1938).
- I. M. Gelfand, D. A. Raikov, G. E. Shilov, Commutative Normed Rings, M., 1960.
- H. Cramér, Ark. Mat., 1, No. 7, 61 (1949).