# Application of a theorem of mamay's to a "denumerable decomposition" of the Poisson law

B. Ramachandran

► To cite this version:

B. Ramachandran. Application of a theorem of mamay's to a "denumerable decomposition" of the Poisson law. Annales de l'ISUP, 1964, XII (1), pp.13-20. hal-04085118

HAL Id: hal-04085118

https://hal.science/hal-04085118

Submitted on 28 Apr 2023

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
APPLICATION OF A THEOREM OF MAMAY'S
TO A "DENUMERABLE DECOMPOSITION"
OF THE POISSON LAW (1)

B. RAMACHANDRAN

The subject of α-decomposition of random variables (strictly speaking, of their characteristic functions) has its origin in an analytical generalization by Linnik and Zinger [4] of the Lévy-Cramer Theorem on the Normal Law. This extension seems to have been motivated by the search for a proof of the celebrated Darmois-Skitovich Theorem (on the independence of linear forms in a finite number of independent random variables), using the technique of obtaining a differential equation involving the characteristic functions concerned. Allied results were obtained by Linnik in [5] and in [6]. An account of these labours may be found in [7]. Results of a similar character, but using a different approach, were independently obtained by Dugué [1] ; in particular, he obtained an analytical generalization of Raikov's theorem on the Poisson law, similar to the Linnik-Zinger generalization of the Lévy-Cramer Theorem. The result of Linnik-Zinger was used by Laha [3] to characterize the Normal distribution by means of properties of certain linear statistics. A common feature of all these works was that they concerned themselves only with a finite number of characteristic functions entering as factors.

In his paper [5], and, later, in his book [7], Linnik posed the problem of extending the α-decomposition theorem on the Normal law to the case of a demumerable number of factors ; from such an extension, one could then easily derive an extension of the Darmois-Skitovich Theorem to the case of independent linear forms in a

(1) Research supported by the Council of Scientific and Industrial Research, Government of India, under their "Scientific Pool" Scheme.

The reader is also referred to a Research Announcement by R.G. Laha of a proof of essentially the same result as here (Bull. Amer. Math. Soc., Vol. 67, 1961, pp. 148-150) ; however, according to this Announcement, the proof will be based on properties of functions having non-negative derivatives of all orders at the origin, instead of directly on Mamay's theorem as the present paper is. The author understands that the details of the proof by R.G. Laha are in the process of being published.
denumerable number of independent random variables. This problem was solved (under the restriction that the $\alpha$'s be bounded below by a positive quantity) by Mamay, [9] and [10], whose main result may be stated as follows:

**THEOREM 1:** Let $f(t)$ be a function of the complex variable $t$, defined, analytic, and non-vanishing in a strip $|Im t| < \delta$, $\delta > 0$. Let, for $j = 1, 2, \dots$ ad inf., $\alpha_j \ge \alpha_0 > 0$, and the characteristic functions $f_j(t)$ satisfy the relation:

$$ \prod_{j=1}^{\infty} \{f_j(t)\}^{\alpha_j} = f(t) \quad (1) $$

in some real neighbourhood of $t=0$. Then, every $f_j(t)$ is defined, analytic and non-vanishing in (at least) the same strip $|Im t| < \delta$, and relation (1) makes sense and is valid throughout this strip.

Theorem 1 almost immediately yields the desired $\alpha$-decomposition theorem for the Normal law (in the case where the $\alpha_j$ are bounded below by a positive quantity). We deal here with an application of the theorem to the establishing of a similar result for the Poisson law, thus generalizing the results of Raikov and Dugué. The author's indebtedness to Dugué's stimulating paper [1], cited above, should be too obvious to need mention.

Our result may be stated as follows:

**THEOREM 2:** Let $\alpha_j \ge \alpha_0 > 0$ for every $j$ ($j = 1, 2, \dots$ ad inf.), and let $\{f_j(t)\}$ be a sequence of characteristic functions such that, in some real neighbourhood of $t=0$, we have

$$ \prod_{j=1}^{\infty} \{f_j(t)\}^{\alpha_j} = \exp\{\lambda(e^{it} - 1)\}, \quad \lambda > 0. \quad (2) $$

Then, every $f_j(t)$ is of the form: $\exp[i\xi_j t + \lambda_j(e^{it} - 1)]$, where every $\xi_j$ is real, every $\lambda_j$ is real and non-negative, $\sum \alpha_j \xi_j = 0$, and $\sum \alpha_j \lambda_j = \lambda$.

*Proof:* Since $\exp[\lambda(e^{it} - 1)]$ is a non-vanishing entire function of the complex variable $t$, and the conditions of Theorem 1 are otherwise satisfied, it follows that every $f_j(t)$ is also a non-vanishing entire function of $t$, and that (2) is valid for all complex $t$. The integral representation: $f_j(t) = \int_{-\infty}^{\infty} e^{itx} dF_j(x)$ is valid for all complex $t$, since $f_j(t)$ is entire, where $F_j(x)$ is the distribution function corresponding to $f_j(t)$ (cf. [8], Theorem 7.1.1).
Setting $t = 2\pi$ in (2), we have

$$ \prod_{j=1}^{\infty} |f_j(2\pi)|^{a_j} = 1 $$

Since every $a_j > 0$ and $|f_j(t)| < 1$ for all real $t$, it follows that we must have $|f_j(2\pi)| = 1$ for every $j$. This implies (see, for instance, [2], p.59) that every $F_j(x)$ is a "lattice distribution" with unit "span"; that is, that every $f_j(t)$ has the form :

$$ f_j(t) = e^{it\xi_j} \left( \sum_{n=-\infty}^{\infty} p_{jn} e^{itn} \right) \quad (3) $$

where $\xi_j$ is a real number, every $p_{jn} \ge 0$, and $\sum_{-\infty}^{\infty} p_{jn} = 1$. We shall - as we certainly may - take $\xi_j$ to be a point of increase of $F_j$, so that $p_{jo} > 0$.

Now, choosing for every characteristic function that branch of its logarithm which vanishes at $t=0$, we have for all complex values of $t$,

$$ \sum \alpha_j \ln f_j(t) = \lambda (e^{it} - 1) \quad (4) $$

Setting $t = iy$ in (4), with $y > 0$, and dividing through by $y$, we have for all such $y$,

$$ \sum \alpha_j [\ln f_j(iy)/y] = \lambda(e^{-y} - 1)/y \quad (5) $$

For convenience, we shall denote $[\ln f_j(iy)/y]$ by $\psi_j(y)$ and $[\lambda(e^{-y} - 1)/y]$ by $\psi(y)$ in what follows.

We proceed to show that $\lim_{y \to \infty} \psi_j(y)$ exists finitely for all $j$. It is known that $\ln f_j(iy)$ is a convex function of the real variable $y$. (The derivatives of $f_j(iy)$ can be obtained by differentiating under the integral sign, as often as necessary, in the relation : $f_j(iy) = \int_{-\infty}^{\infty} e^{-yx} dF_j(x)$; the non-negativeness of the second derivative of $\ln f_j(iy)$ is then an immediate consequence of Schwarz's inequality). We observe further that, if $\Phi(y)$ is a convex function of $y$ in $y \ge 0$ such that $\Phi(0) = 0$, then $[\Phi(y)/y]$ is a non-decreasing function of $y$ in $y > 0$; this follows at once from the fundamental convexity relation that, for $0 < y_1 < y_2$, the determinant

$$ \begin{vmatrix} 0 & \Phi(0) & 1 \\ y_1 & \Phi(y_1) & 1 \\ y_2 & \Phi(y_2) & 1 \end{vmatrix} \ge 0 $$
Hence, $\psi_j(y)$ is a non-decreasing function of y in $y > 0$; in particular, therefore, $\lim_{y \to \infty} \psi_j(y)$ exists, whether finitely or not. We shall now show that it exists finitely for every j.

Given any $\varepsilon > 0$, choose and fix $Y > 0$ such that $\lambda/Y < \varepsilon/2$. Then, for any $y > Y$, $\sum_{j} \alpha_j [\psi_j(y) - \psi_j(Y)]$ is a series with non-negative terms (since $\psi_j(y)$ is a non-decreasing function of y) whose sum is $\psi(y) - \psi(Y)$, which is (of course, positive and) less than $\lambda/Y < \varepsilon/2$. Hence, for all $y > Y$, we have:

$$0 < \psi_j(y) - \psi_j(Y) < \varepsilon/2\alpha_j \quad \text{for every } j \text{; and} \quad (6.1)$$

$$0 \le \sum_{n+1}^{\infty} \alpha_j [\psi_j(y) - \psi_j(Y)] < \varepsilon/2 \quad \text{for every } N. \quad (6.2)$$

From (6.1), on letting $y \to \infty$, it follows that $\lim_{y \to \infty} \psi_j(y)$ is finite for every $j$; let us denote this limit by $-\theta_j$. It then follows (cf. [8], Theorem 7.2.2) that $F_j(x)$ is "bounded below" and that $\theta_j$ is its left-most point of increase ("left extremity"). (In this connection, see also the "Additional Remarks" at the end of this paper). We may identify $\theta_j$ with $\xi_j$ in representation (3), so that every $\xi_j(t)$ has the form

$$f_j(t) = e^{i\xi_j t} \left( \sum_{n=0}^{\infty} p_{jn} e^{itn} \right) = e^{i\xi_j t} g_j(t), \quad \text{say,} \quad (7)$$

where $\xi_j$ is the left extremity of $F_j$, $p_{j0} > 0$, every $p_{jn} \ge 0$, $\sum_{n=0}^{\infty} p_{jn} = 1$.

We shall now show that $\sum \alpha_j \xi_j = 0$. Recalling our choice of $Y$ above, since $\sum \alpha_j \psi_j(Y)$ converges, there exists $N_0$ such that for all $N \ge N_0$, $\left| \sum_{n+1}^{\infty} \alpha_j \psi_j(Y) \right| < \varepsilon/2$. Hence, for all $N \ge N_0$ and all $y > Y$, we have from (6.2)

$$\left| \sum_{n+1}^{\infty} \alpha_j \psi_j(y) \right| \le \sum_{n+1}^{\infty} \alpha_j [\psi_j(y) - \psi_j(Y)] + \left| \sum_{n+1}^{\infty} \alpha_j \psi_j(Y) \right| < \varepsilon,$$

$$\text{i.e.,} \qquad \left| \psi_j(y) - \sum_{n=1}^{N} \alpha_j \psi_j(y) \right| < \varepsilon.$$

Letting $y \to \infty$ in the above relation, we have, since the limits of all the functions concerned exist, that, for all $N \ge N_0$,
$$ \left| \sum_{j=1}^{n} \alpha_j \xi_j \right| \le \varepsilon. $$

Since $\varepsilon > 0$ is arbitrary, the desired relation follows.

Henceforward, the proof is simple if we follow the lines of [1], as we shall. Every $z \ne 0$ has a representation as $e^{it}$ for a suitable complex $t$; the power-series $\sum p_j z^n$ in $z$ then represents a non-vanishing entire function $G_j(z)$ of the complex variable $z$ (note that $p_{j0} > 0$). Let $H_j(z) = \ln G_j(z)$, where $H_j(1) = 0$. Then, $H_j(z)$ is also an entire function of $z$. For all $z \ne 0$, we have from (4) - in view of the fact that $\sum \alpha_j \xi_j = 0$ - that

$$ \sum \alpha_j H_j(z) \text{ exists and is } = \lambda (z - 1) $$

The maximum modulus of $G_j(z)$ in $|z| \le r$ is $G_j(r)$, for every $r > 0$, since $G_j(z)$ is represented by a power-series with non-negative coefficients. For the same reason, $G_j(r)$ is a non-decreasing function of $r$ in $r > 0$, and therefore, so is $H_j(r)$, which is, besides, the maximum value of the real part of $H_j(z)$ in $|z| \le r$. But, we also have from (8) that, for $r \ge 1$,

$$ H_j(r) = H_j(r) - H_j(1) \le (\lambda/\alpha_j) (r - 1), $$

since the $\alpha_j$ are all positive, and every $H_j(r)$ is non-decreasing. Therefore, by the Liouville-Hadamard theorem on the maximum of the real part of an entire function (see, for instance, [11], p.87), every $H_j(z)$ is a linear function of $z$. The facts that, in representation (7), $g_j(t)$ is a characteristic function, and, in particular, that $g_j(0) = 1$, then show that

$$ g_j(t) = \exp [\lambda_j(e^{it} - 1)] $$

where $\lambda_j \ge 0$. Thus, our theorem stands completely proved.

*Additional Remarks:* In the light of our above discussion of the behaviour of $[\ln f_j(iy)/y]$, we may enunciate the following sharper form of a known result on the "extremities" of "one-sided" distribution functions which have analytic characteristic functions (cf. [8], Theorem 7.2.2, Corollary 1):

**THEOREM:** If $F(x)$ be a distribution function having an analytic characteristic function $f(t)$, and is

a) bounded below, then its left extremity is given by

$$ \text{let } F = - \lim_{y \to \infty} [\ln f(iy)/y] ; $$

<figure><img src="image_2.png" /></figure>
b) bounded above, then its right extremity is given by

$$ \operatorname{rext} F = \lim_{y \to \infty} [\ln f(-iy)/y] . $$

The existence of the limits above is part of the assertion of the theorem.

In particular, if $F(x)$ be a "finite" distribution function, i.e., is bounded both below and above, then its two extremities are given by the above relations. (Its characteristic function is known to be an entire function of order one and of exponential type, unless it is identically equal to one : cf. [8], Theorem 7.2.3).

REFERENCES

[1] D. DUGUE - Sur le théorème de Lévy-Cramér, *Publ. Inst. Stat. Univ. Paris*, **6**, (1958), 213-225.

[2] B.V. GNEDENKO and A.N. KOLMOGOROV - *Limit Distributions for Sums of Independent Random Variables* (Transl. K. L. Chung), Addison-Wesley, (1954).

[3] R.G. LAHA - On a characterization of the Normal distribution from properties of suitable linear statistics, *Ann. Math. Stat.*, **28**, N° 1, (1957), 126-139.

[4] Yu.V. LINNIK and A.A. ZINGER - On an analytical extension of a theorem of Cramér's and its application, *Vestnik Leningrad Univ.*, **10**, (1955), 51-56.

[5] Yu.V. LINNIK - A problem concerning the characteristic functions of probability distributions, *Uspekhi Matem. Nauk*, **10**, (1955), 137-138.

[6] Yu.V. LINNIK - On $\alpha$-decomposition of infinitely divisible probability laws, *Vestnik Leningrad Univ.*, **14**, (1959), 14-23.

[7] Yu.V. LINNIK - *Factorization of Probability Laws*, Leningrad Univ., (1960).

[8] E. LUKACS - *Characteristic Functions*, Charles Griffin, London, (1960).

[9] L.V. MAMAY - On some theorems in the theory of positive-definite functions, *Dokl. Akad. Nauk SSSR*, **126**, (1959), 271-273.

[10] L.V. MAMAY - On the theory of characteristic functions, *Vestnik Leningrad Univ.*, **15**, (1960), 85-99.
[11] E.C. TITCHMARSH - *The Theory of Functions*, Second Edition, Oxford University Press, (1939).

Institute of Math. Sciences
Madras, Inde

Reçu le 1er mai 1963
APPLICATION OF THE METHOD TO THE THEORY OF LINEAR DIFFERENTIAL EQUATIONS

(1) B.G. BORISOVSKII and the theory of linear differential equations, (1988) (in Russian)

Oxford University Press

EQUATIONS IN LINEAR DIFFERENTIAL EQUATIONS

APPLICATION OF THE METHOD TO THE THEORY OF LINEAR DIFFERENTIAL EQUATIONS

The problem of determining the general solution of the
independent equations is here in part of the subject of the
講義.

EQUATIONS IN LINEAR DIFFERENTIAL EQUATIONS

In particular, if $F(x)$ is a "true" characteristic function, i.e.,
is independent of $\sin\omega t$ and $\cos\omega t$, then the two equations are given
by the above relations. The characteristic function is known to be
an analytic function of $\omega$ and of exponential type, where it
is identically zero at $\omega = 0$ (Theorem 1.2.1).

CERTIFIED COPY

(1) B.G. BORISOVSKII, On the theory of Lyapunov exponents, Mat. Fiz. Sbornik, V. Series, 5 (1984), 88-96.

(2) B.V. GREBINSKI and B.V. KOLMOGOROV - An initial approximation for the theory of Lyapunov exponents in Banach spaces, S. L. Sobolev, Moscow, (1984).

(3) B.G. LADKIN - On a characterization of the spectral distribution from properties of finite linear equations, in: A. D. OFMANN, S. L. Sobolev, (1981), 1-10.

(4) V.V. LILYANIK and A.A. GUMMEL - On an analytical solution of a system of equations of motion and its application, Soviet J. Math., 18 (1967), 51-60.

(5) V.V. LILYANIK - A problem concerning the characteristic function of a positive distribution, Soviet J. Math., 18 (1967), 107-116.

(6) V.V. LILYANIK - On a decomposition of inhomogeneous products, Amer. J. Math., 15 (1943), 26-38.

(7) V.V. LILYANIK - Characterization of positive exponents, Soviet J. Math., 15 (1943), 26-38.

(8) B. LITOVSKY - Characteristic functions, Charles Griffiths, London, (1966).

(9) L.Y. MORAT - On some properties in the theory of positive definite functions, J. Math. Anal. and Prob. 15 (1969), 271-278.

(10) L.Y. MORAT - On the theory of characteristic functions, Soviet J. Math., 15 (1969), 271-278.