On Normality VIA Conditional Normality

Abstract

Consider X=(X_o , X₁ ... , X_p) and suppose $X_{\circ }\hspace{0.17em}{|X}_i=x_i(i=1\hspace{0.17em},2,\mathrm{...},p)\hspace{0.17em}\hspace{0.17em}~N(a_{\circ }+{\Sigma }_1^p\hspace{0.17em}{a}_i{x}_i,{\sigma }^2).$ Assume further that X_i'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 X_o , .... X_p of X along with conditional normality of X_o (as stated above) indeed ensure normality of X.