V. M. ZOLOTAREV

GENERAL THEORY OF THE MULTIPLICATION OF INDEPENDENT RANDOM VARIABLES

(Presented by Academician A. N. Kolmogorov on 18 XI 1961)

1. The question of constructing a general theory of multiplication of independent random variables was posed by Paul Lévy \((^{1,2})\) as an independent problem, in connection with the fact that the scheme of multiplication of random variables (the \(M\)-scheme), being an essential extension of the scheme of summation (the \(A\)-scheme), cannot in general be reduced to the latter.

In the \(M\)-scheme a new operation on distribution functions (d.f.’s) appears, which we shall call \(M\)-composition and denote by \(\circ\). Namely, if \(F_\xi, F_\eta\) are the d.f.’s of independent random variables \(\xi, \eta\), and \(\zeta=\xi\eta\), then \(F_\zeta(x)=F_\xi(x)\circ F_\eta(x)\). This operation is commutative, associative, and distributive with respect to ordinary addition.

In view of the obvious analogy between the \(M\)- and \(A\)-schemes, it is natural to extend the terminology of the \(A\)-scheme to the \(M\)-scheme, accompanying terms and concepts with the prefix \(M\) for the \(M\)-scheme and the prefix \(A\) for the \(A\)-scheme, and to begin the construction of a general theory in the following direction.

1) Construction of an analytic apparatus playing in the \(M\)-scheme the same role as characteristic functions (c.f.’s) in the \(A\)-scheme.

2) Description in the \(M\)-scheme of the class \(\mathfrak M\) of infinitely divisible laws (\(M\)-i.d.l.). By definition, a d.f. \(F(x)\in\mathfrak M\) if there exists for it such a sequence of integers \(0<n_1<n_2<\cdots\) that \(F\) is representable in the form of an \(n_i\)-fold \(M\)-composition of some d.f. \(F_{n_i}\) (i.e. \(F=F_{n_i}\circ F_{n_i}\circ\cdots\circ F_{n_i}\)). If, moreover, the sequence \(\{n_i\}\) contains an infinite subsequence of even numbers, then we assign the d.f. \(F\) to the subclass \(\mathfrak M'\) of the class \(\mathfrak M\) (unlike the \(A\)-scheme, \(\mathfrak M'\) does not coincide with \(\mathfrak M\)).

3) Construction in the \(M\)-scheme of general limit theorems.

4) Clarification of the arithmetic properties of distributions with respect to the operation of \(M\)-composition.

This is the program to which we shall mainly adhere in what follows.

1. We associate with the random variable \(\xi\) the following pair of functions, putting here \(0^{it}=0\) (\(t\) takes real values):

\[ w_k(t)=M|\xi|^{it}\operatorname{sgn}^k\xi,\qquad k=0,1. \]

Definition. The characteristic transform (c.t.) \(W_\xi(t)\) of the random variable \(\xi\) will mean the diagonal matrix of second order with elements \(\{w_0(t),w_1(t)\}\) on the main diagonal.

For matrices with diagonal elements \(\{w_0(t),-w_1(t)\}\) and \(\{\overline{w_0(t)},\overline{w_1(t)}\}\) (complex conjugates) we shall adopt respectively the notations \(W_\xi^*(t)\) and \(\overline{W}_\xi(t)\). Let us note the most important properties of c.t.’s.

I. If the random variables \(\xi,\eta\) are independent, then

\[ W_{\xi\cdot\eta}(t)=W_\xi(t)\cdot W_\eta(t). \]

II.

\[ W_{-\xi}(t)=W_\xi^*(t);\qquad W_{1/\xi}(t)=W_\xi(-t)=\overline{W_\xi(t)}. \]

III.

\[ c^+=P\{\xi>0\}=\tfrac12\operatorname{sp} W_\xi(0),\qquad c^-=P\{\xi<0\}=\tfrac12\operatorname{sp} W_\xi^*(0). \]

IV. The functions \(w_k(t)\) admit a unique representation of the form

\[ w_k(t)=c^+ f^+(t)+(-1)^k c^- f^-(t), \]

where \(f^+, f^-\) are certain ch.f.’s defined by the equalities

\[ c^+ f^+(t)=\frac12 \operatorname{sp} W_\xi(t), \qquad c^- f^-(t)=\frac12 \operatorname{sp} W_\xi^*(t). \]

Conversely, any pair of ch.f.’s \(f^+, f^-\), together with a pair of nonnegative numbers \(c^+, c^-\) whose sum does not exceed 1, determine some ch.p. \(W_\xi(t)\). Let us note that applying Pólya’s sufficient criterion to the construction of the ch.f.’s \(f^+\) and \(f^-\) gives us a sufficient condition for a ch.p.

V. For all positive points of continuity of the d.f. \(F_\xi(x)\), the following equality is valid (the case of negative values of \(x\) is reduced to this by means of property II):

\[ F_\xi(t)=1-\frac12 c^+-\frac{1}{2\pi}\int_0^\infty \operatorname{Im}\{x^{-it}\operatorname{sp} W_\xi(t)\}\,\frac{dt}{t}. \]

VI. The functions \(w_k(t)\) determine some ch.p. \(W(t)\) if and only if the functions \(u(t)=\frac12\operatorname{sp} W(t)\) and \(v(t)=\frac12\operatorname{sp} W^*(t)\): a) are continuous, \(u(0)\geq 0\), \(v(0)\geq 0\), \(u(0)+v(0)\leq 1\); b) are positive definite.

VII. If a sequence of ch.p.’s \(W_n(t)\) converges elementwise to a matrix \(W(t)\) with continuous elements, then \(W(t)\) is a ch.p. and the d.f.’s \(F_n(x)\), determined by the ch.p.’s \(W_n(t)\), converge weakly \((\Rightarrow)\) to the d.f. \(F(x)\) determined by the ch.p. \(W(t)\). Conversely, if \(W_n(t), W(t)\) are ch.p.’s for the d.f.’s \(F_n(x)\) and \(F(x)\), then from \(F_n(x)\Rightarrow F(x)\) as \(n\to\infty\) there follows uniform, in every finite interval, elementwise convergence of \(W_n(t)\) to \(W(t)\).

VIII. The role of moments with respect to a random variable \(\xi\) in the \(M\)-scheme is played by the quantities
\(m_k(n)=M(\log^n|\xi|\,\operatorname{sgn}^k\xi\mid \xi\ne0)\), \(n\geq0;\ k=0,1\).
The formal expansion of the ch.p. \(W_\xi(t)\) has the form

\[ W_\xi(t)=(c^+ + c^-)\sum_{n=0}^{\infty}\frac{(it)^n}{n!}M_n,\quad \text{where }\ M_n= \begin{pmatrix} m_0(n) & 0\\ 0 & m_1(n) \end{pmatrix}. \]

IX. Let \(a,b\) be arbitrary positive numbers and \(\eta=a|\xi|^b\operatorname{sgn}\xi\). Then
\(\xi=|\eta/a|^{1/b}\operatorname{sgn}\eta\),
\(F_\eta(x)=F_\xi(|x/a|^{1/b}\operatorname{sgn}x)\), and
\(W_\eta(t)=a^{it}W_\xi(bt)\).

1. A description of the class \(\mathfrak M\) (\(M\)-i.d. d.l.) was partially carried out by Lévy \((^1)\) in terms of ch.f.’s of logarithms of random variables. A complete description of the class \(\mathfrak M\) is given by the following theorem.

Theorem 1. The d.f. \(F\) belongs to \(\mathfrak M\) if and only if the ch.p. corresponding to it has the form

\[ w_0(t)=\alpha_0 f_1(t)f_2(t), \qquad w_1(t)=\alpha_1 f_1(t)/f_2(t), \]

where: a) \(f_1(t)\) is the ch.f. of some \(A\)-i.d. d.l., \(f_2(t)\) is the ch.f. of an \(A\)-i.d. d.l. of the form

\[ \log f_2(t)=\int (e^{itu}-1)\,dH(u); \]

b) \(0\leq \alpha_0\leq 1\);
\(|\alpha_1|\leq \alpha_0 \lim_{\varepsilon\to0}\exp\{2[H(\varepsilon)-H(-\varepsilon)]\}\).
Membership in \(\mathfrak M'\) is ensured by the additional (necessary and sufficient) condition \(\alpha_1\geq0\).

We shall call a d.f. \(F(x)\) \(M\)-stable if for any \(a_i>0\), \(b_i>0\), \(i=1,2,3\), there exist such \(a>0\), \(b>0\) that

\[ F(a_1|x|^{b_1}\operatorname{sgn}x)\circ F(a_2|x|^{b_2}\operatorname{sgn}x)\circ F(a_3|x|^{b_3}\operatorname{sgn}x) = F(a|x|^b\operatorname{sgn}x). \]

Between \(M\)-stable (class \(\mathfrak M\)) and \(A\)-stable laws there exists a direct relationship, so that every \(A\)-stable law with density \(g_A(x)\) determines in the following way exactly three \(M\)-stable laws with densities
\(g_M^{(1)}(x)\), \(g_M^{(2)}(x)\), and \(g_M^{(3)}(x)\):

\[ g_M^{(1)}(x)= \begin{cases} 0, & x<0,\\ g_A(\log x), & x\geq 0; \end{cases} \]

\[ g_M^{(2)}(x)=\frac12 g_A(\log|x|); \qquad g_M^{(3)}(x)=g_M^{(1)}(-x). \]

If in the definition of the class \(\mathfrak{N}\) of \(M\)-stable laws \((\mathfrak{N}\subset \mathfrak{M})\) one takes not three but two components in the left-hand side, then, in contrast to the \(A\)-scheme, we obtain a narrower class \(\mathfrak{N}'\) \((\mathfrak{N}'\subset \mathfrak{N})\), consisting of laws of the form \(g_M^{(1)}, g_M^{(2)}\). Let us note that an even number of components in the left-hand side of the definition leads to \(\mathfrak{N}'\), and an odd number to \(\mathfrak{N}\).

Analogues of the \(L\)-laws of \(A\)-schemes are distinguished in the class \(\mathfrak{M}\), if in Theorem 1, as defining characteristic functions \(f_1, f_2\), one chooses the characteristic functions of the \(A\)-infinitely divisible laws of the class \(L\). The role of the Poisson law in the \(M\)-scheme is played by the distribution

\[ P\{\xi=(-1)^r h^n\}=\frac{e^{-\lambda}}{2n!}\left[\alpha_0 e^{-\mu}(\lambda+\mu)^n+(-1)^r\alpha_1 e^\mu(\lambda-\mu)^n\right] \]

\[ P\{\xi=0\}=1-\alpha_0,\qquad r=0,1;\qquad n=0,1,2,\ldots \]

where \(0\leq \alpha_0\leq 1,\ |\alpha_1|\leq \alpha_0 e^{-2\mu};\ h,\lambda,\mu>0\).

1. The formulation of limit theorems in the \(M\)-scheme is the same as in the \(A\)-scheme. A sequence of series of independent, within the series, random variables \(\xi_{n1},\ldots,\xi_{nk_n}\), \(n=1,2,\ldots\), and the products

\[ \zeta_n=a_n\xi_{n1}\xi_{n2}\cdots \xi_{nk_n}, \tag{1} \]

are considered, where \(a_n\) are some positive constants. It is required to give a description of the class of possible limiting distributions and convergence conditions for the products (1) under the assumption of “limit negligibility” of the random factors (\(M\)-negligibility): for every \(\varepsilon>0\)

\[ \lim_{n\to\infty}\max_k P\{(|\xi_{nk}|-1)^2>\varepsilon\}=0. \tag{2} \]

In the scheme of increasing products \(\zeta_n=\xi_1\xi_2\cdots\xi_n\), such \(M\)-negligibility is attained by means of a power transformation of the factors: \(\bar{\zeta}_n=a_n|\zeta_n|^{b_n}\operatorname{sgn}\zeta_n\), where \(a_n\) and \(b_n\) are positive constants. Such a transformation is convenient and natural within the framework of the \(M\)-scheme by virtue of property IX.

Theorem 2. The class of limiting distribution laws for products of the form (1), (2) coincides with the class \(\mathfrak{M}\).

Let \(F_{nk}(x)\) denote the distribution functions of the random variables \(\xi_{nk}\),

\[ c_{nk}^{+}=1-F_{nk}(+0),\qquad c_{nk}^{-}=F_{nk}(0); \]

\[ \widetilde F_{nk}^{(1)}(x)=1-\frac{1}{c_{nk}^{+}}\left[1-F_{nk}(e^x)\right],\qquad \widetilde F_{nk}^{(2)}(x)=1-\frac{1}{c_{nk}^{-}}F_{nk}(-e^x), \]

and let \(\widetilde{\xi}_{nk}^{(\nu)}\), \(\nu=1,2\), \(k=1,\ldots,k_n\), be a sequence of series of mutually independent random variables with distribution functions \(\widetilde F_{nk}^{(\nu)}(x)\).

Theorem 3. For convergence of the distribution functions of the products (1), (2), under some choice of constants \(a_n>0\), to a law of the class \(\mathfrak{M}\), determined by \(\alpha_0,\alpha_1,f_1(t)\) and \(f_2(t)\), it is necessary and sufficient that the following conditions be fulfilled:

In the case \(\alpha_1\neq 0\),

a)

\[ \prod_{k=1}^{k_n}\left(c_{nk}^{+}+(-1)^r c_{nk}^{-}\right)\to \alpha_r \quad \text{as } n\to\infty,\quad r=0,1. \]

b) The sums \(\widetilde{\xi}_{n1}^{(\nu)}+\cdots+\widetilde{\xi}_{nk_n}^{(\nu)}+\widetilde a_n^{(\nu)}\), for some choice of constants \(\widetilde a_n^{(\nu)}\), have limiting laws with characteristic functions \(f_\nu(t)\), \(\nu=1,2\), and as the constants \(a_n\) one may take the quantities \(\exp(\widetilde a_n^{(1)}+\widetilde a_n^{(2)})\).

In the case \(\alpha_1=0\), condition a) remains, and b) is replaced by:

b\(_1\)) The sums \(\widetilde{\xi}_{n1}^{(1)}+\widetilde{\xi}_{n1}^{(2)}+\cdots+\widetilde{\xi}_{nk_n}^{(1)}+\widetilde{\xi}_{nk_n}^{(2)}+\widetilde a_n\), for some choice of constants \(\widetilde a_n\), have a limiting distribution law with characteristic function \(f_1(t)f_2(t)\) (here one may take \(\exp(\widetilde a_n)\) as \(a_n\)).

1. The arithmetic properties of distributions with respect to the operation of \(M\)-composition are basically preserved, although peculiarities also arise.

Theorem 4. A ch. f. having no \(M\)-indecomposable components is \(M\)-infinitely divisible.

Theorem 5. If one of the components of the ch. f. \(W(t)\) vanishes somewhere on the real axis, without being identically zero, then the ch. f. corresponding to this ch. f. has an \(M\)-indecomposable component. If, however, \(w_1(t)\equiv 0^*\) (i.e., the distribution is symmetric), then the ch. f. admits a decomposition containing an \(M\)-indecomposable component.

Theorem 6. Every ch. f. is representable in the form of an \(M\)-composition of \(M\)-indecomposable ch. f.’s in a finite or countable number, and of an \(M\)-i. d. ch. f.

The peculiarities of the arithmetic of ch. f.’s in the \(M\)-scheme may be seen in the following assertions.

A nonsymmetric \(M\)-Poisson distribution law decomposes into an \(M\)-composition only of \(M\)-Poisson laws, whereas a symmetric one can also decompose into laws that are not \(M\)-Poisson. An identical assertion holds for \(M\)-normal laws.

1. It is of interest to describe and study distributions of the class \(\mathfrak{F}=\mathfrak{G}\cap\mathfrak{M}\) (the \(\mathfrak{G}\)-class of \(A\)-i. d. laws). The fact that such a class is essentially nonempty is shown by the following theorem.

Theorem 7. All \(A\)-stable laws of the class \(W^{**}\) belong to \(\mathfrak{M}\); moreover, the subclass \(\mathfrak{M}'\) contains the laws having a nonnegative median.

1. Let \(\xi,\eta\) be independent random variables, \(\zeta=\xi\eta\), and let \({}_{\xi}w_k(t)\), \({}_{\eta}w_k(t)\) be elements of the ch. f.’s corresponding to them. Then the condition \({}_{\xi}w_1(t)\cdot{}_{\eta}w_1(t)\equiv 0\) is necessary and sufficient for the symmetry of the ch. f. \(F_\zeta(x)\). This is the answer to the question posed by Lévy in \((^1)\). Property IV makes it possible rather simply to construct examples of nonsymmetric ch. f.’s whose \(M\)-composition is symmetric.

Consider one-vertex (with vertex at zero) and two-vertex (with extrema at the points \(-1,0,1\)) distributions.

Theorem 8. A ch. f. \(F_\xi(x)\) is one-vertex if and only if the matrix \((1+it)W_\xi(t)\) is some ch. f.

As a consequence, from this we obtain that the \(M\)-composition of two ch. f.’s, one of which is one-vertex (with vertex at zero), is again one-vertex.

Theorem 9. A ch. f. \(F_\xi(x)\) is two-vertex if and only if the corresponding ch. f. \(W_\xi(t)\) is representable in the form

\[ W_\xi(t)=\frac{1}{t}\int_0^t W(u)\,du, \]

where \(W(u)\) is some ch. f.

It follows from Theorem 9 that the \(M\)-composition of two-vertex ch. f.’s with real ch. f.’s is again two-vertex.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
16 XI 1961

CITED LITERATURE

\(^1\) P. Lévy, Ann. Sci. de l’Ecole Norm., 76, No. 1, 59 (1959).
\(^2\) P. Lévy, C. R., 248, No. 13, 1920 (1959).
\(^3\) V. M. Zolotarev, Theory of Probability and Its Applications, 2, issue 4, 444 (1957).

* The case \(w_0(t)\equiv 0\) corresponds to an improper distribution.
** The definition of the class \(W\) is contained in \((^3)\), p. 453.