A. A. ZINGER

ON A PROBLEM OF B. V. GNEDENKO

(Presented by Academician Yu. V. Linnik on 24 XII 1964)

1. Let (\xi_1,\xi_2,\ldots) be a sequence of mutually independent random variables and
[
\zeta_n=\frac{1}{B_n}(\xi_1+\xi_2+\cdots+\xi_n)-A_n,\quad n=1,2,\ldots,
]
be a sequence of normalized sums which, for a suitable choice of the normalizing constants ((B_n\to\infty)), has a proper limiting distribution (G(x)).

Several years ago B. V. Gnedenko posed the problem of characterizing the class of limiting distributions (G(x)) in the case where, among the distributions of the random variables (\xi_i) ((i=1,2,\ldots)), only (r) different ones occur. Denote this class by (\mathfrak{P}_r). Some special cases of the problem under consideration were studied in ((^2,^3)). In particular, in the note of V. M. Zolotarev and V. S. Korolyuk ((^2)) the case (r=2) was investigated. It turned out there that the class (\mathfrak{P}_2) consists of compositions of two stable laws. The case (r>2) remained open. The following example shows that for (r>2) the class (\mathfrak{P}_r) is broader than the class of compositions of a finite number of stable distributions.

Example. Take (r=3). Put (\lambda=\sqrt{2}) and construct three symmetric distribution functions (F_j(x)) ((j=1,2,3)) as follows:
[
F_1(x)=1-F_1(-x)=
\begin{cases}
\dfrac12, & 0\le x\le 1,\[4pt]
1-\dfrac13 x^{-\lambda}\left[1+\dfrac{1}{\sqrt3}\cos\left(\log x-\dfrac{\pi}{6}\right)\right], & x\ge 1;
\end{cases}
]
[
F_2(x)=1-F_2(-x)=
\begin{cases}
\dfrac12, & 0\le x\le 1,\[4pt]
1-\dfrac12 x^{-\lambda}\left[1-\dfrac{1}{\sqrt3}\sin(\log x)\right], & x\ge 1;
\end{cases}
\tag{1}
]
[
F_3(x)=1-F_3(-x)=
\begin{cases}
\dfrac12, & 0\le x\le 1,\[4pt]
1-x^{-\lambda}\left[1-\dfrac{1}{\sqrt3}\cos\left(\log x+\dfrac{\pi}{6}\right)\right], & x\ge 1.
\end{cases}
]

Construct the sequence ({T_n;\ n=1,2,\ldots}); (T_n) is defined as the positive solution of the equation
[
b(T_n)=n,
]
where
[
b(T)=e^{\lambda T}\frac{1}{\sqrt3}\left[2\sqrt3+\cos\left(T-\frac{\pi}{3}\right)+\frac23\cos\left(T+\frac{\pi}{3}\right)-\frac13\cos T\right].
]

Next put
[
B_n=\exp T_n,
\tag{2}
]
[
n_1(n)=\mathrm{E}\left{B_n^\lambda\left(1+\frac{1}{\sqrt3}\cos\left(T_n-\frac{\pi}{3}\right)\right)\right}^{*},
]
[
n_2(n)=\mathrm{E}\left{\frac23 B_n^\lambda\left[1+\frac{1}{\sqrt3}\cos\left(T_n+\frac{\pi}{3}\right)\right]\right},\qquad
n_3(n)=n-n_1(n)-n_2(n).
]

* Here the symbol (\mathrm{E}{\ }) denotes the integer part of the number enclosed in braces.

Now, in constructing the normalized sums $\zeta_n$, we shall be guided by the following rules: in the sum $\zeta_n$, the random summands $\xi_j$ with distribution $F_j(x)$ are taken with $j=1,2,3$, and as normalizing constants we take $B_n$ from (2).

Using the general theorem on convergence of sums of mutually independent random variables (see (1), § 25), it is easy to verify the existence and nondegeneracy of the limiting distribution in the example under consideration and to compute the spectral function of the limiting distribution

[
H_n(x)=\sum_{j=1}^{3} n_j(n)\,[F_j(B_n x)-E(x)]\to
]

[
\to -|x|^{-\lambda}\left[1+\frac16\cos\left(\log |x|-\frac{\pi}{6}\right)\right]\operatorname{sign}x.
\tag{3}
]

Here $E(x)$ is the improper distribution.

[
D_n(\varepsilon)=2\varepsilon^2 H_n(\varepsilon)-2\int_{|x|<\varepsilon} xH_n(x)\,dx
=O(\varepsilon^{2-\lambda}).
]

It is immediately clear from (3) that the limiting distribution is not a composition of stable distributions.

1. The limiting distribution $G(x)$, as is known from the general theory (see, for example, (1), § 24), is infinitely divisible. Let $H(x)$ be the spectral function corresponding to it in the representation of the logarithm of the characteristic function by P. Lévy’s formula.

Take some $q_0>0$ and construct the square matrix of order $r$ $\mathcal H(x)$ as follows:

[
\mathcal H(x)=
\left|
\begin{array}{cccc}
H(x) & H(q_0x) & \ldots & H(q_0^{r-1}x)\
H(q_0x) & H(q_0^2x) & \ldots & H(q_0^r x)\
\ldots & \ldots & \ldots & \ldots\
H(q_0^{r-1}x) & H(q_0^r x) & \ldots & H(q_0^{2r-2}x)
\end{array}
\right|.
\tag{4}
]

We shall call the limiting distribution $G(x)$ nondegenerate if, for at least one pair of positive numbers $x_0,q_0$, the matrices $\mathcal H(\pm x_0)$ are nonsingular.*

This note contains results characterizing the class $\mathfrak P_r^1$ of limiting distributions for B. V. Gnedenko’s problem in the nondegenerate case. In particular, conditions are established under which the limiting distribution is a composition of stable distributions. Namely, in the nondegenerate case the following theorem holds:

Theorem 1. If the limiting distribution $G(x)\in\mathfrak P_r^1$, then:

$1^\circ$. For any $q>0$, on each of the half-axes of definition $(-\infty,0)$, $(0,\infty)$, $\mathcal H(x)$ satisfies the equation

[
\mathcal H(qx)=\mathcal H(x)h^{\log q},
\tag{5}
]

where the matrix $h$ does not depend on $x$.

$2^\circ$. For $x>0$ (respectively, for $x<0$) there may be specified constant matrices $A$ and $B$ ($AB=\mathcal H(x_0)$) such that $\mathcal H(x)B^{-1}$ and $A^{-1}\mathcal H(x)$ preserve sign and are monotone.

* In note (2) it is shown that the spectral function $H(x)$ must satisfy the equation $\beta_0H(x)+\ldots+\beta_rH(q^r x)=0$. The definition of nondegeneracy corresponds to the condition $\beta_0\ne0$, $\beta_r\ne0$.

3°. All characteristic numbers of the matrix (h) are simple (they do not depend on (q)). They may be written in the form

[
\exp(-\lambda_1);\quad \exp(-\lambda_1\pm i\nu_{11});\ \ldots\ \exp(-\lambda_1\pm i\nu_{1k_1}),
]

[
\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots
]

[
\exp(-\lambda_\rho);\quad \exp(-\lambda_\rho\pm i\nu_{\rho 1});\ \ldots\ \exp(-\lambda_\rho\pm i\nu_{\rho k_\rho});
]

here

[
0<\lambda_1<\lambda_2<\cdots<\lambda_\rho<2;\quad
0<\nu_{\alpha\beta}\leq \pi;\quad
\sum_{l=1}^{\rho}(1+2k_l)=r.
]

The theorem formulated above makes it possible to describe the class (\mathscr{P}_r^1) in the following way.

Let us denote by (\mathscr{G}_l) the totality of those distribution laws from Khintchine’s class (L) (see ((^1)), § 29) whose spectral functions on each of the half-axes ((-\infty,0)), ((0,\infty)) are represented in the form

[
H(x)=|x|^{-\lambda_l}\left{a_l+\sum_{j=1}^{k_l}\left[a_{lj}\cos(\nu_{lj}\log|x|)+b_{lj}\sin(\nu_{lj}\log|x|)\right]\right},
]

where (a_l\ne 0); (a_{lj}^2+b_{lj}^2\ne 0); (j=1,2,\ldots,k_l).

For (G(x)) there is the representation

[
G(x)=G_1G_2\ldots*G_\rho(x),
\tag{6}
]

where (G_l(x)\in\mathscr{G}_l); (l=1,2,\ldots,\rho).

It should be noted that laws of the form (6) belong to a family of laws of a more general form, studied for the first time by Yu. V. Linnik in work ((^4)) in connection with the investigation of identically distributed linear statistics in repeated homogeneous samples. For the case where the limiting distribution must be a composition of stable ones, the following theorem holds:

Theorem 2. In order that the limiting distribution (G(x)\in\mathscr{P}_r^1) be a composition of stable distributions, it is necessary and sufficient that all characteristic numbers of the matrix (h) be real and positive.

In this case the eigenvalues of the matrix (h) may be rewritten in the form (\exp(-\lambda_1)); (\exp(-\lambda_2)); (\ldots); (\exp(-\lambda_r)), where (0<\lambda_1<\cdots<\lambda_r<2), and (\lambda_l) ((l=1,\ldots,r)) are the exponents of stable laws, whose composition forms the limiting law (G(x)).

Received
11 XII 1964

REFERENCES

(^1) B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Moscow–Leningrad, 1949.
(^2) V. M. Zolotarev, V. S. Korolyuk, Probability Theory and Its Applications, 6, No. 4, 469 (1961).
(^3) E. K. Lebedintseva, Dokl. AN SSSR, No. 1, 12 (1955).
(^4) Yu. V. Linnik, Ukr. Mat. Zh., 5, No. 2, 3, 207 (1953).