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This note studies decompositions of infinitely divisible probability laws into independent components, focusing on laws with positive finite or countable Poisson spectrum and a Gaussian component. It characterizes, in the rational bounded-spectrum case, when such a law belongs to the class of laws whose every component is again infinitely divisible: the Poisson frequencies must form a descending divisibility chain with zero as the only possible accumulation point. The paper also gives a necessary condition for the unbounded rational case, describes the analytic form of components for semipositive bounded spectra and for spectra rational beyond a point, and derives these results using Paley-Wiener theory, special auxiliary functions, and a saddle-point lemma.
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MATHEMATICS
Corresponding Member of the Academy of Sciences of the USSR Yu. V. Linnik
SOME THEOREMS ON THE DECOMPOSITION OF INFINITELY DIVISIBLE LAWS
Let \(X\) be a random variable with an infinitely divisible (i.d.) law \(F(x)\). Its characteristic function (c.f.) \(\varphi(t)\) is uniquely represented by the formula
\[ \ln \varphi(t)=\beta it-\gamma t^{2} +\int_{-\infty}^{0}\left(e^{itx}-1-\frac{itx}{1+x^{2}}\right)dM(x) +\int_{0}^{\infty}\left(e^{itx}-1-\frac{itx}{1+x^{2}}\right)dN(x), \tag{1} \]
where \(\beta\) is a real number; \(\gamma\geq 0\); \(M(x)\) and \(N(x)\) are nondecreasing; \(M(-\infty)=N(\infty)=0\);
\[ \int_{-a}^{0} x^{2}\,dM(x)+\int_{0}^{a} x^{2}\,dN(x)<\infty,\quad a>0. \]
If \(X_{1}\) and \(X_{2}\) are independent random variables with distribution laws \(F_{1}\) and \(F_{2}\), and \(X=X_{1}+X_{2}\), \(F=F_{1}*F_{2}\) (composition), then we shall speak of a decomposition of the law \(F\) into components. We shall be interested in possible decompositions of i.d. laws into components. Denote the class of i.d. laws by \(I\). With respect to the functions \(M(x)\) and \(N(x)\) we shall say that they characterize the Poisson spectrum of the random variable \(X\); the values \(x>0\) will give the positive, and \(x<0\) the negative, Poisson frequencies. In this note, for the time being, only i.d. laws with positive Poisson spectrum will be considered, i.e. \(dM(x)=0\). If \(dN(x)=0\) for \(x\geq X_{0}>0\), then we shall speak of a bounded Poisson spectrum; the case of a finite or countable Poisson spectrum will be given by a c.f. under the condition
\[ \ln \varphi(t)=\beta_{1}it-\gamma t^{2}+\sum_{m=1}^{N}\lambda_{m}\left(e^{it\mu_{m}}-1\right), \tag{2} \]
where \(N<\infty\) or \(N=\infty\); \(\lambda_{m}>0\); \(\mu_{m}>0\); the series \(\sum \lambda_{m}\) converges. The separate Poisson component \(Y_{m}\) has c.f. \(\exp\{\lambda_{m}(e^{it\mu_{m}}-1)\}\), and here \(\mu_{m}=D(Y_{m})/E(Y_{m})\).
We shall be interested in the class \(I_{0}\subset I\) of i.d. laws that can be decomposed only into i.d. components. It is easy to see that laws from \(I_{0}\) with a finite or countable Poisson spectrum (c.f. of type (2)) have the same components, i.e. components expressible by a formula of type (2) with the same spectrum (allowing some \(\lambda_{m}\) to be equal to zero). In simplified terms, the components of laws from \(I_{0}\) have the same form as these laws themselves. In the work of H. Cramér [1] it was proved that the normal law belongs to \(I_{0}\) (i.e., it can have only normal components); in the works of D. A. Raikov [2, 3] it was proved that the Poisson law belongs to \(I_{0}\) (has only Poisson
components); in the author’s papers \((^{4,5})\) it was proved that the composition of Gaussian and Poisson laws belongs to \(I_0\). The method of \((^{4,5})\) can be carried over to a more general case, and the theorems stated below are obtained.
For laws with a positive finite or denumerable Poisson spectrum we introduce the concept of rationality of the spectrum. A Poisson spectrum will be called rational if \(\mu_m/\mu_l=r_{lm}\) is rational for any \(l\) and \(m\).
If a finite or denumerable Poisson spectrum is not rational, and \(\gamma>0\) (a Gaussian component is present), then the membership of the law in \(I_0\), apparently, is a quite exceptional phenomenon, which will be considered in another paper.
In the present paper we restrict ourselves only to the case of a rational spectrum. For it the following theorems hold.
Theorem 1. Let \(F\) be an infinitely divisible law with a bounded positive rational finite or denumerable Poisson spectrum and with \(\gamma>0\) (having a Gaussian component). In order that \(F\in I_0\) (have only infinitely divisible components), it is necessary and sufficient that the Poisson frequencies \(\mu_m\) in representation (2) coincide with a decreasing sequence of numbers
\[ \mu,\quad \frac{\mu}{k_1},\quad \frac{\mu}{k_1k_2},\quad \frac{\mu}{k_1k_2k_3},\ldots,\quad \frac{\mu}{k_1k_2\ldots k_s},\ldots, \tag{3} \]
where \(\mu>0;\ k_1,k_2,k_3,\ldots\) is some set of natural numbers (allowing repetitions).
Thus, if a law \(F\in I_0\) has a denumerable spectrum, then it can have only one limiting point of accumulation of frequencies—zero.
Theorem 2. If in the conditions of Theorem 1 the requirement of boundedness of the spectrum is omitted, then, for \(F\) to belong to the class \(I_0\), it is necessary that the Poisson frequencies \(\mu_m\) in representation (2) coincide with a sequence of numbers
\[ \ldots,\ k_{-3}k_{-2}k_{-1}\mu,\quad k_{-2}k_{-1}\mu,\quad k_{-1}\mu,\quad \mu,\quad \frac{\mu}{k_1},\quad \frac{\mu}{k_1k_2},\quad \frac{\mu}{k_1k_2k_3},\ldots, \tag{4} \]
where \(\ldots,k_{-3},k_{-2},k_{-1},k_1,k_2,k_3,\ldots\) is some set of natural numbers.
Whether this condition is sufficient has not yet been clarified.
Theorems 1 and 2 are connected with Theorem 3.
Theorem 3. Let \(F\) be an infinitely divisible law with a semipositive bounded Poisson spectrum, so that \(dN(x)=0\) for \(x>a\) in formula (2). Then all its components have characteristic functions of the form
\[ \varphi_1(t)=\exp\left(P_3(it)+t^4\int_0^a e^{itu}\phi(u)\,du\right), \tag{5} \]
where \(P_3(it)\) is a polynomial of degree not higher than the third; \(\phi(u)\) is a real function summable with its square on the segment \([0,a]\).
Theorem 4. Let \(F\) be an infinitely divisible law with a semipositive bounded Poisson spectrum which is rational to the right of the point \(b\), so that the characteristic function has the form
\[ \varphi(t)=\exp\left(\beta it-\gamma t^2+\int_0^b (e^{itx}-1)\,dN(x) +\sum_{j=1}^{m}\lambda_j\left(\exp\left(it\frac{a_j}{q}\mu\right)-1\right)\right), \tag{6} \]
where \(\lambda_j>0;\ a_j,q\) are integers; \(a_1<a_2<\cdots<a_m;\ \dfrac{a_1}{q}\mu<b\). Let \(\gamma>0\). Then
all its components have characteristic functions of the form
\[ \varphi_1(t)=\exp\left(P_3(it)+t^4\int_0^b e^{itu}\phi(u)\,du +\sum_{n=1}^{q}(\alpha_n+\beta_n it)\exp\left(\left(it\frac{n}{q}\mu\right)-1\right)\right), \tag{7} \]
where \(\alpha_n,\beta_n\) are real numbers, not necessarily positive; \(\phi(u)\) is a function summable with its square on \([0,b]\); \(P_3(it)\) is a cubic polynomial.
We note that from Theorem 4 it is easy to derive the main result of papers \((^{4,5})\): a composition of Gaussian and Poisson laws belongs to \(I_0\), i.e., can be decomposed only into compositions of the same kind. For this it suffices in (6) to take \(b=0;\ q=m=a_j=1\). One obtains the equality
\[ \varphi_1(t)=\exp\bigl(P_3(it)+a_1(\exp(i\mu t)-1)\bigr). \]
It is readily established that \(\alpha_1>0;\ P_3(it)=\beta it-\gamma t^2;\ \gamma\geqslant 0\). Theorems 1 and 2 follow from Theorem 4 with the aid of a lemma that is also of independent interest.
Lemma. Let \(\mu>0,\ \gamma>0;\ 0<m<M;\ m\) and \(M\) are integers, and \(m\) does not divide \(M\). Then, for every sufficiently small \(\nu>0\), the function
\[ \psi(t)=\exp\left(-\gamma t^2+\lambda_1(e^{M\mu it}-1)+\lambda_2(e^{m\mu it}-1)-\nu(e^{\mu d it}-1)\right), \tag{8} \]
where \(d\) is the greatest common divisor of the numbers \(m\) and \(M\); \(\lambda_1,\lambda_2>0\) are given numbers, will be the characteristic function of some random variable.
This lemma is proved by means of the saddle-point method and the transformation formula for \(\vartheta\)-functions; in this way one also obtains an asymptotic expression for the probability density corresponding to the characteristic function (7).
The main tools for proving Theorems 3 and 4 are the same as in \((^{4,5})\): the Paley—Wiener theorem on the representation of entire functions of exponential type belonging to \(L_2\) on some axis, and the use of special functions—the “little cups of I. M. Vinogradov.”
The indicated tools, in combination with the saddle-point method, make it possible to study, with respect to possible membership in the class \(I_0\), laws with a bounded spectrum of all the remaining types: a continuous spectrum, a finite or countable spectrum that is not rational, and a spectrum containing negative frequencies. This, however, requires a considerable technical complication of the available apparatus and will be done subsequently. The case of an unbounded spectrum so far only partially admits the indicated treatment.
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
12 IV 1957
REFERENCES
- H. Cramér, Math. Zs., 41, 405 (1936).
- D. A. Raikov, DAN, 14, No. 1, 9 (1937).
- D. A. Raikov, Izv. AN SSSR, Ser. Mat., No. 1, 91 (1938).
- Yu. V. Linnik, DAN, 114, No. 1 (1957).
- Yu. V. Linnik, Theory of Probability and Its Applications, No. 1 (1957).