Discounted branching random walks

Abstract

Let F(·) be a c.d.f. on [0,∞), f(s) = ∑^∞₀p_jsⁱ a p.g.f. with p₀ = 0, < 1 < m = Σ_jp_j < ∞ and 1 < ρ <∞. For the functional equation for a c.d.f. H(·) on [0,∞] we establish that if 1 – F(x) = O(x^–θ) for some θ > α =(log m)/(log p) there exists a unique solution H(·) to (∗) in the class C of c.d.f.’s satisfying 1 – H(x) = o(x^–α).We give a probabilistic construction of this solution via branching random walks with discounting. We also show non-uniqueness if the condition 1 – H(x) = o(x^–α) is relaxed.