Consider a delayed renewal process generated by the sequence of independent random variables X1, X2,... such that X1 has distribution function G(x) and Xi has distribution function F(x) for i ≥ 2.

If G(x) = F(x), then show that the renewal function U(x) = E(N[0,x])

satisfies the renewal equation U(x) = F(x) +  x 0

U(x − y)dF(y)

If G(x) is chosen to make the delayed renewal process stationary, then show that x µ = G(x) +  x 0 U(x − y)dG(y), (12.21)
where µ is the mean of F(x). If dH% (λ) = " ∞
0 e−λxdH(x) denotes the Laplace transform of the distribution function H(x) defined on (0, ∞), also verify the identity dG%(λ) = 1 − dF%(λ)
µλ .
Finally, prove that the Laplace transform of the density 1 µ [1 − F(x)]
matches dG%(λ).