On some characterisations of the normal law

Abstract

Let X₁,...,X_n be independent variables and Y₁,...,Y_n be linear functions of X₁,...,X_n. In this paper, the conditions under which the equations E(Y_i|Y_p+1,...,Y_p+j)=0,i=1,..,p imply normality of X₁,...,X_n are examined. An important case considered is when E(Y₁|Y₂)=0 involving only two linear functions (with p=1, j=0), which provides a generalisation of the earlier results on the characterisation of the normal law by Darmois, Kagan, Linnik, Rao, Skitovich and others. The conditions imposed to ensure normality are of two types, one on the nature of the coefficients in the linear functions Y_i and another on the nature of the distributions of X₁,...,X_n, such as identical distribution, existence of moments etc.