Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in R with non–zero mean

Let G = (G1,G2) be a Gaussian vector in R 2 with E(G1G2) 6= 0. Let c1, c2 ∈ R. A necessary and sufficient condition for the vector ((G1+c1α) , (G2+c2α) ) to be infinitely divisible for all α ∈ R is that Γi,i ≥ ci c j Γi, j > 0 ∀1≤ i 6= j ≤ 2. (0.1) In this paper we show that when (0.1) does not hold there exists an 0 < α0 < ∞ such that ((G1 + c1α) , (G2 + c2α) ) is infinitely divisible for all |α| ≤ α0 but not for any |α|> α0.