Abstract Generated abstract
This note generalizes Craig’s theorem on the independence of quadratic forms in independent standard normal variables. It considers a quadratic polynomial Q and an arbitrary polynomial statistic P of a multivariate normal vector, and proves that P and Q are independent if and only if an orthogonal change of variables separates their dependence into disjoint coordinate sets. The result is also extended to nondegenerate multivariate normal distributions with zero mean by reducing the covariance matrix to the standard normal case through a linear transformation.
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UDC 519.281
MATHEMATICS
L. B. KLEBANOV
ON A GENERALIZATION OF CRAIG’S THEOREM
(Presented by Academician Yu. V. Linnik on June 8, 1970)
Let \(X=(x_1,\ldots,x_n)\) be an \(n\)-dimensional vector with independent components having the normalized normal distribution, and let \(Q_1(x_1,\ldots,x_n)\) and \(Q_2(x_1,\ldots,x_n)\) be two quadratic forms
\[ Q_1=X'AX,\qquad Q_2=X'BX. \]
Craig’s theorem is well known (see \((^1)\)): in order that the quadratic forms \(Q_1\) and \(Q_2\) be independent, it is necessary and sufficient that
\[ A\cdot B=0. \tag{1} \]
Let us formulate this result differently.
By an orthogonal transformation taking \(X\) into a vector \(y\) with independent and normalized normal components, \(Q_1\) can be transformed to the form
\[ Q_1=\sum_{i=1}^{r} a_i y_i^2,\qquad r\le n. \tag{2} \]
Craig’s result means that \(Q_1\) and \(Q_2\) are independent if and only if \(Q_2\) depends only on \(y_{r+1},\ldots,y_n\).
We have proved the following generalization of Craig’s result.
Theorem. Let \(X=(x_1,\ldots,x_n)\) be an \(n\)-dimensional vector with independent components distributed normally with mean \(0\) and variance \(1\). Let \(Q(x_1,\ldots,x_n)\) be a quadratic polynomial in \(x_1,\ldots,x_n\), and let \(P(x_1,\ldots,x_n)\) be an arbitrary polynomial statistic. For the independence of \(P\) and \(Q\) it is necessary and sufficient that there exist an orthogonal transformation \(A\)
\[ Ax=y, \]
such that \(Q\) depends only on \(y_1,\ldots,y_r\), while \(P\) depends only on \(y_{r+1},\ldots,y_n\).
In other words: if the polynomial \(Q\) is brought by an orthogonal transformation to the form
\[ Q=\sum_{j=1}^{s}\lambda_j y_j^2+\sum_{i=q}^{p}\mu_i y_i, \]
\[ q\le s+1,\qquad \lambda_i\ne 0,\qquad \mu_i\ne 0,\qquad \max(s,p)\ne r\le n, \]
then the polynomials \(P\) and \(Q\) are independent if and only if \(P\) depends only on \(y_{r+1},\ldots,y_n\).
Our result will essentially not change if it is assumed that \(X\) has a multivariate nondegenerate normal distribution with mean \(0\) and covariance matrix \(V\).
Indeed, the matrix \(V\) can be represented in the form
\[ V=T\cdot T', \]
where \(T\) is a real matrix, and the transformation \(X = Ty\) carries the exponent of the exponential corresponding to the distribution density of \(X\) from \(X'V^{-1}X\) into \(y'T'V^{-1}Ty = y'y\). Thus the general case reduces to the one already studied.
A. A. Zinger informed the author that he had proved an analogous result for the case of a quadratic form \(Q\).
The author is very grateful to Yu. V. Linnik for posing the problem and for his attention to the work.
Received
4 VI 1970
REFERENCES
- A. T. Craig, Ann. Math. Statist., 14, 195 (1943).