Exercise 12.13 (Non-Stationary Poisson Process) Consider a counting process {N(t)} satisfying the properties (1) and (3) in Definition 12.2; but, in this exercise, let (t)

Exercise 12.13 (Non-Stationary Poisson Process) Consider a counting process {N(t)} satisfying the properties (1) and (3) in Definition 12.2; but, in this exercise, let λ(t) be a non-negative, deterministic function of time t.
Instead of (12.38), suppose that

for all k = 0, 1, . . . . Show that

P{N(th)k+1|N(t) =k} = X(t)h+o(h), t0,