UDC 519.21

MATHEMATICS

I. N. KOVALENKO

ON THE STATISTICAL CHARACTERIZATION OF SYMMETRIC STABLE DISTRIBUTION LAWS

(Presented by Academician Yu. V. Linnik on 11 XII 1965)

Let \(\xi_1, \xi_2, \ldots, \xi_n, \ldots\) be jointly independent random variables having one and the same distribution \(F(x)\). A linear type of distributions is the set \(\tilde F\) of distributions of the form \(\{F(ax+b), a>0, -\infty<b<\infty\}\), where \(F(x)\) is the fixed distribution function generating the given type. We introduce classes \(\mathfrak A_n,\ n\geqslant 1\), of types of distributions, defining them by the following conditions:

A. \(\mathfrak A_m\cap \mathfrak A_n=\varnothing,\ m<n.\)

B. \(\tilde F\in \bigcup_{i=1}^{n}\mathfrak A_i\) if and only if, for every \(\tilde G\ne \tilde F\), there exists a continuous bounded function \(f(x_1,x_2,\ldots,x_n)\) with the properties
\[ f(ax_1+b, ax_2+b,\ldots, ax_n+b)=f(x_1,x_2,\ldots,x_n),\quad a>0,\ -\infty<b<\infty; \]
\[ M_{\tilde F}f(\xi_1,\xi_2,\ldots,\xi_n)\ne M_{\tilde G}f(\xi_1,\xi_2,\ldots,\xi_n). \]

In the author’s paper \((^1)\) it was shown that, for any \(n\geqslant 1\), the set \(\mathfrak A_n\cup\mathfrak A_{n+1}\cup\ldots\) is nonempty. Moreover, an analogous assertion is valid for additive types of distributions. In the literature there are a number of results that can be interpreted as assigning concrete types of distributions to certain classes. Thus, the well-known result of A. A. Zinger \((^2)\) consists in the fact that the normal type of distributions belongs to \(\mathfrak A_4\cup\mathfrak A_5\cup\mathfrak A_6\). In a recent article by Yu. V. Prokhorov \((^3)\) this result is extended to other symmetric laws.

In the present note the following assertion is proved.

Theorem. The linear type generated by a symmetric stable distribution law belongs to \(\mathfrak A_3\cup\mathfrak A_4\cup\mathfrak A_5\).

In particular, A. A. Zinger’s theorem is refined as follows: the normal law is characterized by the totality of the distributions of the random variables
\[ \eta_\alpha=\frac{\alpha\xi_1+(1-\alpha)\xi_2-\xi_3}{|\xi_4-\xi_5|},\quad 0\leqslant \alpha\leqslant 1. \tag{1} \]

Proof of the theorem. Let \(f(t)=Me^{it\xi_1}\). It is known \((^4)\) that
\[ f(t)=\exp\{-|t|^\gamma\}, \tag{2} \]
where \(0<\gamma\leqslant 2\). (Since only statistics invariant with respect to shift and stretching will be considered, one may choose any distribution belonging to the given type.)

For brevity of notation, introduce the notation \(\mathcal L[\varphi]\) for the distribution of any vector statistic \(\varphi\), and the symbol \(\Rightarrow\), understanding by the expression \(\mathcal L[\varphi]\Rightarrow \mathcal L[\psi]\) that from the distribution of the statistic \(\varphi\) the distribution of \(\psi\) is uniquely recovered.

First of all,
\[ \{\mathcal L[\operatorname{sign}\eta_\alpha,\ |\eta_\alpha|]\}_{0\leqslant\alpha\leqslant 1}\Longleftrightarrow \{\mathcal L[\eta_\alpha]\}_{0\leqslant\alpha\leqslant 1}. \tag{3} \]
Since in the present case the distribution of \(\eta_\alpha\) is symmetric,
\[ \mathcal L[|\eta_\alpha|]\Rightarrow \mathcal L[\operatorname{sign}\eta_\alpha,\ |\eta_\alpha|], \tag{4} \]

and therefore,

\[ \{\mathcal L[|\eta_\alpha|]\}_{0\leq\alpha\leq1}\Rightarrow \{\mathcal L[\eta_\alpha]\}_{0\leq\alpha\leq1}. \tag{5} \]

Under condition (2), for \(t=\operatorname{Re} t\)

\[ M\exp\{it[\alpha\xi_1+(1-\alpha)\xi_2-\xi_3]\} =f(\alpha t)f((1-\alpha)t)f(-t)= \]

\[ =\exp\{-[\alpha^\gamma+(1-\alpha)^\gamma+1]|t|^\gamma\}. \tag{6} \]

Introduce the random variable

\[ \zeta_\alpha= \left|\alpha\xi_1+(1-\alpha)\xi_1-\xi_3\right|/ [\alpha^\gamma+(1-\alpha)^\gamma+1]^{1/\gamma}. \tag{7} \]

Then, by virtue of equality (6), the distribution of the random variables
\(\mu_\alpha=\zeta_\alpha|\xi_4-\xi_5|^{-1}\) does not depend on \(\alpha\) \((0\leq\alpha\leq1)\). On the other hand,

\[ \{\mathcal L[\eta_\alpha]\}_{0\leq\alpha\leq1}\Longleftrightarrow \{\mathcal L[\mu_\alpha]\}_{0\leq\alpha\leq1}. \tag{8} \]

Further, we have

\[ \ln\mu_\alpha=\ln\zeta_\alpha-\ln|\xi_4-\xi_5|. \tag{9} \]

Let us prove that if \(\xi\) is a random variable with a symmetric stable distribution, then \(\ln|\xi|\) has a characteristic function analytic in some strip containing \(\{\operatorname{Im} z=0\}\). Indeed, when \(\operatorname{Im}x=\operatorname{Im}y=0\),

\[ \left|M\exp\{i(x+iy)\ln|\xi|\}\right|\leq M|\xi|^{-y}<\infty \qquad(-y<\gamma). \tag{10} \]

Consequently, the characteristic function of the random variable \(\ln\mu_\alpha\) is not equal to 0 in any interval. Then from formula (9) the following conclusion follows. If the \(\mu_\alpha\) have distributions compatible with the hypothesis of stability of \(\xi_1\), i.e., distributions not depending on \(\alpha\), then \(\zeta_\alpha\) will also have a distribution not depending on \(\alpha\) \((0\leq\alpha\leq1)\). Owing to symmetry, the distribution of the random variable
\([\alpha\xi_1+(1-\alpha)\xi_2-\xi_3][\alpha^\gamma+(1-\alpha)^\gamma+1]^{-1/\gamma}\) also does not depend on \(\alpha\) and, in particular, is equal to the distribution corresponding to the case \(\alpha=1\). Thus we arrive at the functional equation

\[ f(\alpha t)f((1-\alpha)t)\overline{f(t)} = \left| f\left(\left[\frac{\alpha^\gamma+(1-\alpha)^\gamma+1}{2}\right]^{1/\gamma}t\right) \right|^2. \tag{11} \]

Denote

\[ f(t)=e^{a(t)+ib(t)}, \tag{12} \]

where the newly introduced functions are real for \(\operatorname{Im}t=0\). Formula (11) is equivalent to two equalities

\[ a(\alpha t)+a((1-\alpha)t)+a(t) = 2a\left(\left[\frac{\alpha^\gamma+(1-\alpha)^\gamma+1}{2}\right]^{1/\gamma}t\right), \qquad a(0)=0; \tag{13} \]

\[ b(\alpha t)+b((1-\alpha)t)-b(t)=0, \qquad b(0)=0, \tag{14} \]

which hold in the interval \(|t|<t_0\leq\infty\), where \(f(t)\) does not vanish.

From formula (13) it follows that the function \(a(t)\) has growth of at most a power order. (For example, when

\[ \left[\frac{\alpha^\gamma+(1-\alpha)^\gamma+1}{2}\right]^{1/\gamma} =\rho_\alpha<1, \]

taking into account that \(a(t)\leq0\), we have the inequality
\(|a(t)|\leq 2\max_{0<u<\rho_\alpha t}|a(u)|\).) Hence it follows that equations (13) and (14) are satisfied for all \(t<0\) and that the Laplace integral

\[ \varphi(s)=\int_0^\infty e^{-st}a(t)\,dt \tag{15} \]

is analytic in the domain \(\operatorname{Re}s>0\). Equality (13) leads to the formula

\[ \frac{1}{\alpha}\varphi\left(\frac{s}{\alpha}\right) + \frac{1}{1-\alpha}\varphi\left(\frac{s}{1-\alpha}\right) + \varphi(s) = \frac{2}{\rho_\alpha}\varphi\left(\frac{s}{\rho_\alpha}\right). \tag{16} \]

Expanding the terms of formula (16), except the first, in a Maclaurin series in \(\alpha\) and making the obvious cancellations, we arrive at the conclusion that there exists a finite limit of the expression \(\alpha^{-\gamma-1}\varphi(s/\alpha)\) as \(\alpha\to0\), and the differential equation is satisfied

\[ s\varphi'(s)+\varphi(s)-cs^{\gamma-1}=0. \tag{17} \]

Equations of this kind admit only solutions of the form

\[ \varphi(s)=c_1s^{-\gamma-1}+c_2s^{-1}. \tag{18} \]

Substituting (18) into equation (16), for example for \(\alpha=1/2\), we find that \(c_2=0\), and, therefore, by virtue of (15), \(a(t)=ct^\gamma,\ t>0\). In a similar way, from the functional equation (14) we find that \(b(t)=t\cdot\mathrm{const}\). The theorem is proved.

Remark. In the proof, the distribution of the statistic \(\eta_\alpha\) was used not for all, but only for several values of \(\alpha\), namely: for \(\alpha=1\), at an infinitely close point \(\alpha\to1\), \(\alpha=1/2\), and also (for proving symmetry) at the point \(\alpha=0\) and at a point infinitely close to it. Therefore, from general analytic considerations there follows the possibility of generalizing our result in the direction indicated by Yu. V. Linnik and A. A. Zinger \({}^{5}\); however, the author has not carried out the corresponding calculations.

Received
5 XII 1965

REFERENCES

\({}^{1}\) I. N. Kovalenko, Proceedings of the All-Union Conference on Probability Theory and Mathematical Statistics, 1958, Yerevan, 1960.
\({}^{2}\) A. A. Zinger, Vestnik Leningrad University, 1 (1956).
\({}^{3}\) Yu. V. Prokhorov, Theory of Probability and Its Applications, 10, No. 3 (1965).
\({}^{4}\) M. Loève, Probability Theory, IL, 1962.
\({}^{5}\) A. A. Zinger, Yu. V. Linnik, Theory of Probability and Its Applications, 9, No. 4 (1964).