Using the notation of Exercises 2.28 and 2.30, show that if the eigenvalues decrease exponentially fast, say λj = λ

j

, with |λ| < 1, then nR has a non-degenerate limiting distribution for large n, with cumulants

κr (nR) ≃ 2 r−1 (r − 1)!λ

r/ (1 − λ

r).

Show that this result is false if λ is allowed to depend on n, say λj

= 1 − 1/j.