On Normality VIA Conditional Normality

Abstract

Consider X=( Xo , X1 ... , Xp) and suppose [Formula: see text] Assume further that Xi's ( i=0,1, ... , p) are marginally identically distributed. Does this Imply normality of X ? Ahsanullah ( Metrika (1985), 32, 215-218) raised this question and resolved it in the affirmative for p = 1. This is, of course, not true for p > 1. We give a counter-example to that effect. Next we prove that exchangeability of the components Xo , .... Xp of X along with conditional normality of Xo (as stated above) indeed ensure normality of X.