UDC 519.21

MATHEMATICS

O. V. SARMANOV

GENERALIZED NORMAL CORRELATION

AND TWO-DIMENSIONAL FRÉCHET CLASSES

(Presented by Academician S. N. Bernstein on 17 VIII 1965)

1. Consider the class of functions \(N\{f(x,y)\}\) representable by bilinear expansions in Hermite polynomials

\[ f(x,y)=\frac{\exp\left[-\frac{x^2+y^2}{2}\right]}{2\pi} \left[1+\sum_{k=1}^{\infty} c_k H_k(x)H_k(y)\right]. \tag{1} \]

The series (1) is assumed to converge in the mean, which is equivalent to the condition

\[ \sum_{k=1}^{\infty} c_k^2<\infty, \tag{2} \]

where \(H_k(x)\) are orthogonal and normal with weight \(e^{-x^2/2}/\sqrt{2\pi}\) on the whole real axis, i.e.

\[ H_k(x)=\frac{(-1)^k}{\sqrt{k!}}\,e^{x^2/2}\frac{d^k}{dx^k}\left(e^{-x^2/2}\right) =\frac{1}{\sqrt{k!}}\left[x^k-\frac{k(k-1)}{2}x^{k-2}+\cdots\right]. \tag{3} \]

It follows from (1) that

\[ \int_{-\infty}^{\infty} f(x,y)\,dx=\frac{e^{-y^2/2}}{\sqrt{2\pi}}; \qquad \int_{-\infty}^{\infty} f(x,y)\,dy=\frac{e^{-x^2/2}}{\sqrt{2\pi}}. \tag{4} \]

Definition. When the sum of the series (1) is nonnegative for all real \(x\) and \(y\), we shall say that there is a generalized normal correlation between \(x\) and \(y\).

2. Theorem. In order that the sum of the series (1) be the density of a generalized normal correlation, it is necessary and sufficient that the coefficients \(c_k\) be the moments of some probability distribution concentrated inside the interval \([-1,1]\).

Necessity. Let the sum of the series (1) be nonnegative for all \(x\) and \(y\). Find the conditional mathematical expectation of the quantity \((y/x)^k\) for fixed \(x\ne 0\). Express \(y^k\) through the polynomials (3):

\[ y^k=\sqrt{k!}H_k(y)+a_{k-2}H_{k-2}(y)+a_{k-4}H_{k-4}(y)+\cdots, \]

where \(a_{k-2}, a_{k-4},\ldots\) are fully determined constants.

\[ M_x y^k=\sqrt{k!}c_k H_k(x)+a_{k-2}c_{k-2}H_{k-2}(x)+\cdots \]

\[ = c_k x^k+b_{k-2}x^{k-2}+b_{k-4}x^{k-4}+\cdots, \]

where \(b_{k-2}, b_{k-4},\ldots\) are also certain constants. Thus,

\[ M_x (y/x)^k=c_k+b_{k-2}/x^2+b_{k-4}/x^4+\cdots . \tag{5} \]

and for sufficiently large \(x\), \(c_k\) differs arbitrarily little from the \(k\)-th moment of the variable \(y/x\); moreover, by virtue of (2), \(c_k\) cannot be a moment of a variable that, with positive probability, goes outside the interval \([-1,1]\) or assumes the values \(\pm 1\).

Sufficiency. Let \(\{c_k\}\) form the sequence of moments of some variable \(\xi\), concentrated inside the interval \([-1,1]\) and given by the distribution function \(G(\xi)\); then for each fixed \(|\xi|<1\) consider the well-known expansion for the density of normal correlation with correlation coefficient \(\xi\)

\[ \frac{\exp\left[-\frac{x^2+y^2-2\xi xy}{2(1-\xi^2)}\right]}{2\pi\sqrt{1-\xi^2}} = \frac{\exp\left[-\frac{x^2+y^2}{2}\right]}{2\pi} \left[ 1+\sum_{k=1}^{\infty}\xi^k H_k(x)H_k(y) \right], \tag{6} \]

where, by assumption,

\[ \int_{-1}^{1}\xi^k\,dG(\xi)=c_k,\qquad k=1,2,\ldots \tag{7} \]

From (6) and (7) it follows that

\[ \frac{\exp\left[-\frac{x^2+y^2}{2}\right]}{2\pi} \left[ 1+\sum_{k=1}^{\infty}c_k H_k(x)H_k(y) \right] = \]

\[ = \frac{1}{2\pi}\int_{-1}^{1} \frac{\exp\left[-\frac{x^2+y^2-2\xi xy}{2(1-\xi^2)}\right]}{\sqrt{1-\xi^2}}\,dG(\xi)>0, \tag{8} \]

which completes the proof.

Remark. In normal correlation, \(\xi=R\) with probability 1, and \(c_k=R^k\). If \(\xi\) assumes the two values \(\pm R\) with probabilities \(1/2\), so that \(c_{2k}=R^{2k}\), \(c_{2k-1}=0\), \(k=1,2,\ldots\), the corresponding density was considered in (¹) and was given the name pseudonormal correlation; finally, if all \(c_k=0\), then \(x\) and \(y\) are independent.

Corollary 1. If the sum (1) has constant sign and the variables \(x\) and \(y\) are dependent, then \(c_{2k}>0\) for all \(k=1,2,\ldots\).

Corollary 2. It is easy to verify that the characteristic function of the density (8) has the form

\[ \varphi(t_1,t_2) = \exp\left[-\frac{t_1^2+t_2^2}{2}\right] \int_{-1}^{1}\exp[-Rt_1t_2]\,dG(R). \tag{9} \]

In particular, if it is known that the correlation between \(x\) and \(y\) is normal, but the correlation coefficient is not known exactly and is uniformly distributed on the interval \([R-\varepsilon, R+\varepsilon]\), \(R-\varepsilon>-1\), \(R+\varepsilon<1\), then the characteristic function of such a generalized normal correlation is expressed with the aid of (9) as follows:

\[ \varphi(t_1,t_2,\varepsilon) = \exp\left[-\frac{t_1^2+t_2^2+2Rt_1t_2}{2}\right] \frac{\operatorname{sh}(\varepsilon t_1t_2)}{\varepsilon t_1t_2}. \tag{10} \]

2. M. Fréchet in (², ³) posed the following general problem: to find the class of two-dimensional distribution functions \(\{P(x,y)\}\) having given partial (extreme or marginal) one-dimensional distributions \(F(x)\) and \(F_1(y)\). In the preceding section an important part of the normal Fréchet class has been found.

Morgenstern and Gumbel (⁴) considered the following example of functions belonging to the Fréchet class \(K\{F(x),F_1(y)\}\) for arbitrary given \(F(x)\) and \(F_1(y)\):

\[ P(x,y)=F(x)F_1(y)\left[1+\lambda(1-F(x))(1-F_1(y))\right], \qquad -1<\lambda<1. \tag{11} \]

If, however, densities exist (denoted by the corresponding lowercase letters), then

\[ p(x,y)=f(x)f_1(y)\,[1+\lambda(2F(x)-1)(2F_1(y)-1)]; \tag{12} \]

in particular, in the normal case,

\[ p(x,y)=\frac{\exp\left[-\frac{x^2+y^2}{2}\right]}{2\pi}\,[1+4\lambda\Phi(x)\Phi(y)], \tag{13} \]

where

\[ \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_0^x e^{-t^2/2}\,dt . \]

3. Let us consider the following more general method of constructing functions from the Fréchet class \(k\{f(x), f_1(y)\}\), specified by the densities \(f(x)\) and \(f_1(y)\) on \(L=[a\le x\le b]\) and \(L_1=[a_1\le y\le b_1]\). Let \(\varphi(x)\) be an arbitrary function (not equal to a constant) bounded on \(L\), and let \(\varphi_1(y)\) be an arbitrary function bounded on \(L_1\). Then the quantities

\[ m=\int_a^b \varphi(x)f(x)\,dx,\qquad m_1=\int_{a_1}^{b_1}\varphi_1(y)f_1(y)\,dy, \]

\[ \sigma^2=\int_a^b[\varphi(x)-m]^2 f(x)\,dx,\qquad \sigma_1^2=\int_{a_1}^{b_1}[\varphi_1(y)-m_1]^2 f_1(y)\,dy \tag{14} \]

are meaningful.

Denote

\[ h=\sup_{x\in L}\frac{|\varphi(x)-m|}{\sigma},\qquad h_1=\sup_{y\in L_1}\frac{|\varphi_1(y)-m_1|}{\sigma_1}. \tag{15} \]

The one-parameter family of densities

\[ p_1(x,y,\lambda)=f(x)f_1(y)\left[1+\frac{\lambda}{hh_1}\, \frac{[\varphi(x)-m][\varphi_1(y)-m_1]}{\sigma\sigma_1}\right], \tag{16} \]

where \(-1\le \lambda\le 1\), belongs to the Fréchet class \(k\{f(x), f_1(y)\}\), since \(\varphi(x)-m\) and \(\varphi_1(y)-m_1\) are orthogonal to unity with weights \(f(x)\) and \(f_1(y)\), and the sum (16) is nonnegative.

We note that, by virtue of (14), \(h\ge 1\) and \(h_1\ge 1\), and

\[ \rho=\lambda/hh_1 \tag{17} \]

is the maximal correlation coefficient (5) between \(x\) and \(y\).

Integrating (16), we pass to the distribution functions

\[ P_1(x,y,\lambda)=F(x)F_1(y)+\frac{\lambda}{hh_1} \int_a^x \frac{\varphi(u)-m}{\sigma}\,dF(u) \int_{a_1}^y \frac{\varphi_1(v)-m_1}{\sigma_1}\,dF_1(v). \tag{18} \]

Since \(F(x)\) and \(F_1(y)\) are bounded functions, one may put \(\varphi(x)\equiv F(x)\), and \(\varphi_1(y)\equiv F_1(y)\); then, for continuous \(F(x)\) and \(F_1(y)\), we obtain that

\[ \frac{1}{2}=\int_a^b F(x)\,dF(x)=m=m_1=\sup_{x\in L}|F(x)-m|=\sup_{y\in L_1}|F_1(y)-m_1|, \]

\[ P_1(x,y,\lambda)=F(x)F_1(y)+\lambda(F^2(x)-F(x))(F_1^2(y)-F_1(y))= \]

\[ =F(x)F_1(y)[1+\lambda(F(x)-1)(F_1(y)-1)], \tag{19} \]

as in example (11). The assumption of continuity is not a restriction, for, if it is proved that (19) is a distribution function for

for arbitrary discontinuous \(F(x)\) and \(F_1(y)\), then (19) gives a distribution function also for arbitrary distribution functions \(F(x)\) and \(F_1(y)\).

1. If the densities \(p_k(x,y)\) belong to some Fréchet class, then a linear combination of them of the form \(\sum_k \varepsilon_k p_k(x,y)\), where \(\varepsilon_k > 0\) and \(\sum_k \varepsilon_k = 1\), also belongs to the same class. With the help of this remark and (16), one can construct new examples of densities and distribution functions belonging to a given class (the possibility of similar bilinear constructions was indicated in \({}^{6}\)).

Let us give several examples

\[ p(x,y)=f(x)f_1(y)\left[1+\sum_{k=1}^{\infty}\varepsilon_k \cos 2k\pi\bigl(F(x)-{}^{1}\!/_{2}\bigr)\cos 2k\pi\bigl(F_1(y)-{}^{1}\!/_{2}\bigr)\right], \tag{20} \]

where \(\sum_{k=1}^{\infty}|\varepsilon_k|\leq 1\), and, integrating (20), we obtain the corresponding family of distribution functions

\[ P(x,y)=F(x)F_1(y)+\frac{1}{4\pi^2}\sum_{k=1}^{\infty}\frac{\varepsilon_k}{k^2}\sin 2\pi k\bigl(F(x)-{}^{1}\!/_{2}\bigr)\sin 2\pi k\bigl(F_1(y)-{}^{1}\!/_{2}\bigr). \tag{21} \]

If in (18) we put \(\varphi(x)=F^k(x)\), \(\varphi_1(y)=F_1^k(y)\), and then consider the linear combination of the functions obtained, putting \(\varepsilon_k=\varepsilon^k\), \(k=1,2,\ldots,\ 0\leq \varepsilon \leq {}^{1}\!/_{2}\), then we arrive at the following expression, also giving functions from the Fréchet class:

\[ P_2(x,y,\varepsilon)=F(x)F_1(y)\left[1+\frac{l(\varepsilon)}{\varepsilon}\right] -\frac{F(x)}{\varepsilon}\,l[\varepsilon F_1(y)] \]
\[ -\frac{F_1(y)}{\varepsilon}\,l[\varepsilon F(x)] +\frac{1}{\varepsilon}\,l[\varepsilon F(x)F_1(y)], \tag{22} \]

where

\[ l(z)=-\int_0^z \frac{\ln(1-t)}{t}\,dt=\sum_{k=1}^{\infty}\frac{z^k}{k^2}, \quad \text{and} \quad P_2(x,y,0)=F(x)F_1(y). \]

In conclusion, let us give one example of discontinuous densities from the normal Fréchet class

\[ f(x,y,\lambda)= \begin{cases} \dfrac{\exp\left[-\dfrac{x^2+y^2}{2}\right]}{2\pi}(1+\lambda), & \text{for } xy>0,\\[1.2em] \dfrac{\exp\left[-\dfrac{x^2+y^2}{2}\right]}{2\pi}(1-\lambda), & \text{for } xy\leq 0,\\[0.8em] -1\leq \lambda \leq 1. \end{cases} \tag{23} \]

For the distribution (23), \(\lambda\) is the maximal correlation coefficient.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
17 VIII 1965

REFERENCES

\({}^{1}\) O. V. Sarmanov, DAN, 132, No. 2 (1960).
\({}^{2}\) M. Fréchet, Ann. Univ. Lyon, ser. III, A14, 53 (1951).
\({}^{3}\) M. Fréchet, C. R., 242, No. 20, 2426 (1956).
\({}^{4}\) E. Gumbel, C. R., 246, No. 19, 2717 (1958).
\({}^{5}\) O. V. Sarmanov, DAN, 121, No. 1 (1958).
\({}^{6}\) S. Kantorovitz, Bull. Res. Council. Israel, F.7, No. 1, 43 (1957).