Abstract
Full Text
UDC 519.24+517.535.4
MATHEMATICS
V. N. LOGVINENKO, I. V. OSTROVSKII, L. I. RONKIN
ON ANALYTIC TRANSFORMATIONS OF A NORMAL VECTOR
(Presented by Academician Yu. V. Linnik on 15 VI 1970)
Let (X=(x_1,\ldots,x_n)) be a random vector whose components (x_j), (1\le j\le n), are independent random variables distributed according to the normal law (N(0,1)). Yu. V. Linnik posed the problem of describing the class (\Xi) of entire functions (\varphi(Z)), (Z=(z_1,\ldots,z_n)\in C^n), real for (Z=X\in R^n), and such that the random variable (y=\varphi(X)) is distributed according to the law (N(0,1)).
In measure-theoretic terms, the class (\Xi) can be defined as the class of entire functions (\varphi(Z)), (Z\in C^n), real for (Z=X\in R^n), and such that for every Borel set (B\subset R^1) the equality
[
\frac{1}{\sqrt{2\pi}}\int_B e^{-y^2/2}\,dy
=
\left(\frac{1}{\sqrt{2\pi}}\right)^n
\int_{\varphi^{-1}(B)} e^{-|X|^2/2}\,dX,
\tag{1}
]
holds, where (\varphi^{-1}(B)={X:X\in R^n,\ \varphi(X)\in B}) is the full preimage of the set (B) in (R^n), and (dX) is the volume element in (R^n).
As is known, the class (\Xi) is nonempty; the general form of a linear function (\varphi(Z)) of class (\Xi) is as follows:
[
\varphi(Z)=\sum_{k=1}^{n} a_k z_k,\quad (a_1,\ldots,a_n)\in R^n,\quad
\sum_{k=1}^{n} a_k^2=1.
\tag{2}
]
However, examples can also be given of nonlinear functions of the class (\Xi) (Yu. V. Linnik):
[
\varphi(z_1,z_2)=z_1\cos[\eta(z_1^2+z_2^2)]+z_2\sin[\eta(z_1^2+z_2^2)],
\tag{3}
]
[
\varphi(z_1,z_2,z_3)=z_1\cos[\eta(z_3)]+z_2\sin[\eta(z_3)],
]
where (\eta(z)) is an arbitrary nonconstant entire function, real for (z\in R^1).
For every entire function (\varphi(Z)), (Z\in C^n), put
[
M(r,\varphi)=\max_{Z\in C^n,\ |Z|\le r}|\varphi(Z)|,\quad
\left(|Z|^2=\sum_{k=1}^{n}|z_k|^2\right).
]
Yu. V. Linnik and V. L. Eidlin ((^1)) showed that if (\varphi(Z)\in\Xi) and (\ln M(r,\varphi)=O(\ln^2 r)), then the function (\varphi(Z)) has the form (2). In the present paper we strengthen this result.
Theorem 1. If (\varphi(Z)\in\Xi), then either (\varphi(Z)) is a linear function of the form (2), or
[
\liminf_{r\to\infty} r^{-1}\ln M(r,\varphi)>0.
\tag{4}
]
For the function (3) with (\eta(z)=kz) one has
[
\lim_{r\to\infty} r^{-1}\ln M(r,\varphi)=|k|,
]
and therefore the assertion of Theorem 1 is, in a certain sense, unimprovable.
Theorem 1 follows directly from the two following lemmas.
Lemma 1. If (\varphi(Z) \in \Xi), then
[
\int_{|X|>1} |\varphi(X)|\, |X|^{-n-2}\,dX < \infty
\qquad (X \in R^n).
\tag{5}
]
Lemma 2. If an entire function (\varphi(Z)) does not satisfy condition (4), but does satisfy condition (5), then it is linear.
For the proof of Lemma 1, denote by (E_{k,N}) the set
[
E_{k,N}={X:X\in R^n,\ |X|<N,\ \sqrt{k}N\le |\varphi(X)|<\sqrt{k+1}N},
]
[
(N>1,\ k=0,1,2,\ldots).
]
Let (\operatorname{mes}n E) be the Lebesgue measure of this set in (R^n). Obviously,
(\operatorname{mes}n E<(2N)^n). We shall show that for (k\ge 1) the following sharper estimate is valid:
[
\operatorname{mes}n E\le
2(\sqrt{2\pi})^{\,n-1}(\sqrt{k}N)^{-1}
\exp{-(k-1)N^2/2}.
\tag{6}
]
For this, put in (1)
[
B={y:y\in R^1,\ |y|>\sqrt{k}N}.
]
We have
[
\frac{1}{\sqrt{2\pi}}\int_B e^{-y^2/2}\,dy
=
\frac{2}{\sqrt{2\pi}}\int_{\sqrt{k}N}^{\infty} e^{-y^2/2}\,dy
<
\frac{2}{\sqrt{2\pi}}\frac{1}{\sqrt{k}N}e^{-kN^2/2},
\tag{7}
]
and, on the other hand,
[
\left(\frac{1}{\sqrt{2\pi}}\right)^n
\int_{\varphi^{-1}(B)} e^{-|X|^2/2}\,dX
=
\left(\frac{1}{\sqrt{2\pi}}\right)^n
\int_{|\varphi(X)|>\sqrt{k}N} e^{-|X|^2/2}\,dX
\ge
]
[
\ge
\left(\frac{1}{\sqrt{2\pi}}\right)^n
\int_{E_{k,N}} e^{-|X|^2/2}\,dX
\ge
\left(\frac{1}{\sqrt{2\pi}}\right)^n
e^{-N^2/2}\operatorname{mes}n E.
\tag{8}
]
From (7), (8), and (9) we obtain (6).
Using (6), we obtain
[
\int_{|X|<N} |\varphi(X)|\,dX
=
\sum_{k=0}^{\infty}\int_{E_{k,N}} |\varphi(X)|\,dx
\le
\sum_{k=0}^{\infty}\sqrt{k+1}\,N\,\operatorname{mes}n E
\le
]
[
\le
N(2N)^n+
\sum_{k=1}^{\infty}
\sqrt{\frac{k+1}{k}}\,
2(\sqrt{2\pi})^{\,n-1}
\exp{-(k-1)N^2/2}
=
O(N^{n+1}).
]
Hence (5) follows easily.
We shall first prove Lemma 2 in the one-dimensional case. Let (\varphi(z)), (z\in C^1), be an entire function satisfying the hypotheses of the lemma, i.e.
[
\liminf_{r\to\infty} r^{-1}\ln M(r,\varphi)=0,
\qquad
\int_{|x|>1} |\varphi(x)|\,|x|^{-3}\,dx<\infty.
\tag{9}
]
It follows from the second condition that if (\varphi(z)) is a polynomial, then (\varphi(z)) is a linear function. We shall suppose that the function (\varphi(z)) is transcendental.
First we show that the set of zeros of the function (\varphi(z)) is infinite. Indeed, otherwise one could specify a polynomial (p(z)) such that the function
(\varphi_1(z)=\varphi(z)/p(z)) has no zeros at all. The function (\varphi_1(z)), obviously, satisfies the condition
[
\liminf_{r\to\infty} r^{-1}\ln M(r,\varphi_1)=0.
]
It is known (\bigl((^2),\ \text{p. }48,\ (^3),\ \text{p. }51\bigr)) that every entire function (\varphi_1(z)) satisfying such a condition and having no zeros is constant. We arrive at the conclusion that the function (\varphi(z)) is a polynomial, which contradicts the assumption of its transcendentality.
Now let (a_1,a_2,a_3) be arbitrary three zeros of the function (\varphi(z)). Consider the entire function
[
\psi(z)=\int_0^z \psi_1(\xi)\,d\xi,
\qquad
\psi_1(\xi)=\varphi(\xi)/[(\xi-a_1)(\xi-a_2)(\xi-a_3)].
]
It follows from (9) that the function (\psi(z)) satisfies the conditions
[
\liminf_{r\to\infty} r^{-1}\ln M(r,\psi)=0,\qquad
\sup_{-\infty<x<\infty}|\psi(x)|<\infty .
\tag{10}
]
By the Phragmén–Lindelöf theorem (((^{4}), p. 211), every entire function satisfying the conditions (10) is bounded in each of the half-planes (\operatorname{Im}z\geqslant 0) and (\operatorname{Im}z\leqslant 0) and, consequently, by Liouville’s theorem is constant. Since (\varphi(z)=\psi(z)(z-a_1)(z-a_2)(z-a_3)), we arrive at the conclusion that (\varphi(z)\equiv 0), which contradicts the transcendence of the function (\varphi(z)).
We now prove Lemma 2 in the case of dimension (n\geqslant 2). Denote by (S^n) the unit sphere in (R^n) and put
[
\varphi_\Theta(z)=\varphi(z\Theta),\qquad z\in C^1,\quad \Theta\in S^n .
]
Denoting by (d\Theta) the element of surface area of the sphere (S^n) and taking into account that (\varphi_{-\Theta}(z)=\varphi_\Theta(-z)), we have
[
\int_{|X|>1}|\varphi(X)|\cdot |X|^{-n-2}\,dX
=
\int_{S^n}d\Theta\int_1^\infty |\varphi_\Theta(x)|\,x^{-3}\,dx
=
]
[
=\frac12\int_{S^n}d\Theta\int_{|x|>1}|\varphi_\Theta(x)|\cdot |x|^{-3}\,dx .
]
Since, by the condition of the lemma, the integral (5) converges, it follows that for all (\Theta\in S^n), except, at most, for a set of measure 0 on (S^n), the integral
[
\int_{|x|>1}|\varphi_\Theta(x)|\cdot |x|^{-3}\,dx
]
will converge. Observing that (\liminf_{r\to\infty}r^{-1}\ln M(r,\varphi_\Theta)=0), and applying the one-dimensional case of Lemma 2, we conclude that the function (\varphi_\Theta(z)) is linear for all (\Theta\in S^n), except, at most, for a set of measure 0.
Let
[
\varphi(Z)=\sum_{k=0}^{\infty} p_k(Z)
]
be the expansion of the function (\varphi(Z)) into a series of homogeneous polynomials. Then
[
\varphi(z\Theta)=\sum_{k=0}^{\infty} p_k(\Theta)z^k .
]
Consequently, (p_k(\Theta)=0) almost everywhere on (S^n) for (k=2,3,\ldots). Therefore (p_k(Z)\equiv 0), (k=2,3,\ldots), and the function (\varphi(Z)) is linear. Lemmas 1 and 2, and together with them Theorem 1, are proved.
The class (\Xi) admits a generalization. Let (X) be the random vector discussed at the beginning of the article. Denote by (\Xi_m), (1\leqslant m\leqslant n), the class of vector-functions (W=\Phi(Z)=(\varphi_1(Z),\ldots,\varphi_m(Z))), where (Z\in C^n), and the (\varphi_j(Z)) are entire functions, real for (Z=X\in R^n), satisfying the following condition: the components of the random vector (Y=\Phi(X)) are independent and distributed according to the law (N(0,1)). Obviously, (\Xi_1=\Xi), and the class (\Xi_m) can be characterized by the condition
[
\left(\frac{1}{\sqrt{2\pi}}\right)^m
\int_B e^{-|Y|^2/2}\,dY
=
\left(\frac{1}{\sqrt{2\pi}}\right)^n
\int_{\Phi^{-1}(B)} e^{-|X|^2/2}\,dX,
\tag{11}
]
where (B) is any Borel set in (R^m), and (\Phi^{-1}(B)) is its full preimage in (R^n). Putting in (11) (B={Y:Y=(y_1,\ldots,y_m)\in R^m,\ y_j\in A}), where (A) is any Borel set in (R^1), and observing that then (\Phi^{-1}(B)=\varphi_j^{-1}(A)), we are convinced of the validity of the following assertion (Yu. V. Linnik): if (\Phi(Z)=(\varphi_1(Z),\ldots,\varphi_m(Z))\in\Xi_m), then (\varphi_j(Z)\in\Xi), (j=1,\ldots,m). With the aid of this assertion, from Theorem 1 we obtain a more general fact.
Theorem 2. Let (\Phi(Z)=(\varphi_1(Z),\ldots,\varphi_m(Z))\in\Xi_m). Then about each of the functions (\varphi_j(Z)), (1\leqslant j\leqslant m), one may assert the following: either
(\varphi_j(Z)) is a linear function of the form (2), or else (\varphi_j(Z)) satisfies the condition
[
\liminf_{r\to\infty} r^{-1}\ln M(r,\varphi_j)>0 .
]
We express our deep gratitude to V. S. Azarin, A. A. Gol’dberg, and B. Ya. Levin for valuable comments.
Kharkov State University
named after A. M. Gorky
Physico-Technical Institute of Low Temperatures
of the Academy of Sciences of the Ukrainian SSR
Kharkov
Received
18 V 1970
References
- Yu. V. Linnik, V. L. Eidlin, Theory of Probability and Its Applications, 13, no. 4, 751 (1968).
- U. Hayman, Meromorphic Functions, Moscow, 1966.
- A. A. Gol’dberg, I. V. Ostrovskii, Distribution of Values of Meromorphic Functions, Moscow, 1970.
- A. I. Markushevich, Theory of Analytic Functions, 2, Moscow, 1968.