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20) and uk = 0 for all k not multiple of d. The proof of the renewal theorem will be provided at the end of next section. 3. The renewal theorem in a continuous-time setting states that for a continuous-time recurrent aperiodic renewal process, for any h > 0, we have lim [Ψ (n + h) − Ψ (n)] = n→∞ h . 21) The same result holds true for a continuous-time recurrent periodic renewal process of period d, provided that h is a multiple of d. 6. Indeed, recall that for n ∈ N we have n Ψ (n) = k=0 uk (cf.

8. The component starts to work at time n = 0. Consider that the lifetimes (Xn )n∈N∗ of the component have a common geometric distribution on N∗ , of parameter p, 0 < p < 1, denoted by (hn )n∈N , h0 := 0, hn := p(1−p)n−1 , n ≥ 1. Consider also that the repair times (Yn )n∈N∗ have a common geometric distribution on N∗ , of parameter q, 0 < q < 1, denoted by (gn )n∈N , g0 := 0, gn := q(1 − q)n−1 , n ≥ 1. n 1. Show that the sequence Sn := i=1 (Xi + Yi ), n ∈ N∗, S0 := 0 forms a positive recurrent renewal chain and compute the characteristics of the chain: the waiting time distribution f = (fn )n∈N , the corresponding generating function, and the sequence (un )n∈N of probabilities that a renewal occurs at time n.

1 Markov Renewal Chains and Semi-Markov Chains 49 k (n) (n−1) Aij (k) := Air (l) Arj (k − l), n ≥ 2, k ∈ N, r∈E l=0 that is, A(0) := δI, A(1) := A and A(n) := A ∗ A(n−1) , n ≥ 2. Some important quantities for investigating the evolution of a Markov renewal chain are the probabilities P(Jn = j, Sn = k | J0 = i), i, j ∈ E, n ∈ N. They are the analog of the n-step transition functions of a Markov chain (see Appendix D). Recall that, for a finite Markov chain (Xn )n∈N of transition matrix p = (pij )i,j∈E , the n-step transition function (Appendix D) can be written as P(Xn = j | X0 = i) = pnij , for any n ∈ N, where pnij is the (i, j) element of the n-fold matrix product of p.

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A Berry-Esseen bound for U-Statistics by Gadasina L.V.

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