3.23. Let Z(k) = (Zij(k», where i = 1, ... , p. j = 1. ... , q and k = 1. 2..... be a sequence of random matrices. Let one norm of a matrix A be N1(A) =

max i . j mod(a), and another he N2(A) = L',j a~ = tr AA'. Some alternative ways of defining stochastic convergence of Z(k) to B (p x q) are

(a) N1(Z(k) - B) converges stochastically to O.

(b) N2(Z(k) - B) converges stochastically to 0, and

(c) Zij(k) -=- bij converges stochastically to 0, i = 1, ... , p, j = 1,.,., q.

Prove that these three definitions are equivalent. Note that the definition of X(k) converging stochastically to 11 is that for every arhitrary positive I'i and e, we can find K large enough so that for k> K Pr{IX(k)-al <8}>I-s.