A Burns-Krantz type theorem for domains with corners by Baracco L., Zaitsev D., Zampieri G.

By Baracco L., Zaitsev D., Zampieri G.

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Similarly for τ0 we have P0 (τ0 > t) = e−λ0 t . 36) 0 and ψ0 (t, α) = E0 e−c(α) = Rt t e−λ0 t + 0 X(s)ds , τ0 > t + E0 e−c(α) Rt 0 X(s)ds λ0 e−λ0 x ψ1 (t − x, c(α))dx. 0 Define for β > 0 and i = 0, 1 ∞ ψˆi (β, α) = e−βt ψi (t, α)dt. 37) we get the system of equations ψˆ1 (β, α) = ψˆ0 (β, α) = λ1 1 ˆ λ1 +c(α)+β + λ1 +c(α)+β ψ0 (β, α) λ0 ˆ 1 λ0 +β + λ0 +β ψ1 (β, α) It follows that ψˆ1 (β, α) = λ1 + λ 0 + β . 38) where λ = λ0 + λ1 and b = 4λ0 c(α) − [λ + c(α)]2 . 38) with respect to α. 4. 4: The graph of the distribution function of L(10).

0 Hence ∞ E[Z(t)]e−βt dt 0 ∞ = −ϕ (0)e−βt dt 0 = = = 1 [1 − F ∗ (β)] β ∞ 0 ∞ ∞ 1 [1 − F ∗ (β)] 2β xe −βx ∞ F ∗ (β)n−1 n(n + 1) dF (x) 0 n=0 xe−βx dF (x) 0 xe−βx dF (x) . 1, and if E X1 e−βX1 < ∞ for some β > 0, then ∞ E[Z(t)]e −βt ∞ 0 xe−βx dF (x) . 10) Now we will derive the Laplace transform of the second moment of Z(t). 9) we obtain n Vn (α) n = Wn,j (α) j=1 Wn,i (α) i=1,i=j n n + n Wn,j (α) j=1 Wn,k (α) k=1,k=j Wn,i (α), i=1,i=j=k where Wn,j (α) = ∞ (n + 1 − j)2 x2 e−(α[n+1−j]+β)x dF (x) 0 if E X12 e−βX1 < ∞ for some β > 0,.

8. 1 Notations and Definitions Consider a locally finite point process on the positive half line [0, ∞). Denote the ordered sequence of points by 0 < S1 < S2 < . .. We will think of the points Sn as arrival times. We define S0 := 0, but this does not mean that we assume that there is a point in 0. , N (t) = sup{n ≥ 0 : Sn ≤ t}. Define for t ≥ 0 t Y (t) = N (s)ds. 0 If (N (t), t ≥ 0) is a renewal process, we call the stochastic process (Y (t), t ≥ 0) an integrated renewal process. 1) i=1 where N (t) Z(t) = Si .

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