(a) Suppose n = 1 and let X and u be independent standard normal variates. Then X is nonzero with probability one and

because the distribution of

is Cauchy.

(b) The matrix X has full column rank with probability one if and only if

where λ_min(·) denotes the minimal eigenvalue of the argument.

exists if (and only if)

for any k-vector c with c′c = 1. In the sequel, let c be an arbitrary k-vector with unit length. Now,

where the first inequality uses the Cauchy–Schwarz inequality and the second inequality uses Magnus and Neudecker (1988), Theorem 11.4.

If X and u are independent,

, and

then

where ∥·∥₁ denotes the l¹ norm (i.e.,

for any

), the first inequality uses Jensen's inequality, and the last inequality uses the fact that existence of E(u|X) requires E(∥u∥₁) < ∞.

If

, and

then

where the second inequality uses the Cauchy–Schwarz inequality.

If

and

then

where the second inequality uses u′u ≤ ∥u∥₁² and the last inequality uses the fact that existence of E(u|X) requires E(∥u∥₁) < ∞. (This result contains the standard fixed-regressor result as a special case because Pr[λ_min(X′X) > δ] = 1 for some δ > 0 if X is nonrandom and of full column rank.)

Remark. The following chain of implications holds: (4) ⇒ (3) ⇒ (2) ⇒ (1).

Indeed, (4) ⇒ (3) because

by the Cauchy–Schwarz inequality. Finally, (2) ⇒ (1) because E [λ_min^−1/2(X′X)] = ∞ whenever Pr[λ_min(X′X) = 0] > 0.