Suppose Xk is a noncentral Chi-squared variable with k degrees of freedom and noncentrality parameter 2 k

Suppose Xk is a noncentral Chi-squared variable with k degrees of freedom and noncentrality parameter δ2 k .

(i) Show that (Xk − k)/(2k)1/2 d

→ N(μ, 1) if δ2 k /(2k)1/2 → μ as k → ∞.

(ii) If ck,1−α is the 1 − α quantile of the Chi-squared distribution with k degrees of freedom, deduce that (cn,1−α − k)/√2k → z1−α.