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

1. A. T. Craig, Ann. Math. Statist., 14, 195 (1943).