Joint stable attraction of two sums of products (Q1102026)

If F is a bivariate distribution function, we say that F is in \(D(a_ 1,a_ 2)\) if F is in the domain of attraction of a bivariate stable distribution whose marginals are stable \((a_ 1)\) and stable \((a_ 2)\), respectively. The following two theorems are proved. Theorem 1. Suppose \((X_ 1,X_ 2)\) have joint distribution F in \(D(a_ 1,a_ 2)\), and \((X_ 1,X_ 2)\) is independent of \((Y_ 1,Y_ 2)\). Assume \(Y_ i\) is not degenerate at zero, and \(E| Y_ i|^{b_ i}\) is finite for some \(b_ i>a_ i\) (or \(b_ i=2\) if \(a_ i=2)\). Let H be the distribution of \((X_ 1Y_ 1,X_ 2Y_ 2)\). Then H is in \(D(a_ 1,a_ 2).\) Theorem 2. Suppose \((X_ 1,X_ 2)\) have joint distribution F in \(D(a_ 1,a_ 2)\), and \((Y_ 1,Y_ 2)\) have joint distribution G in \(D(b_ 1,b_ 2)\), with \((X_ 1,X_ 2)\) independent of \((Y_ 1,Y_ 2)\). Let H denote the distribution of \((X_ 1Y_ 1,X_ 2Y_ 2)\). Assume \(b_ 1>a_ 1\) but \(b_ 2<a_ 2\) (or \(b_ i=2\) if \(a_ i=2)\). Then H is in \(D(a_ 1,b_ 2)\). Also, if \(a_ 1<2\) or \(b_ 2<2\), the limiting law is that of independent stable random variables.