Loud Shot Noise

We consider problems involving large or loud values of the shot noise process X(t) := ∑i: τᵢ ≤ t h(t - τᵢ), t ≥ 0, where h: 0, ∞) → 0, ∞) is nonincreasing and (τᵢ, i ≥ 0) is the sequence of renewal times of a renewal process. Results are obtained by extending the renewal sequence to all i ∈ Z and considering the stationary sequence (ξₙ) given by ξₙ = ∑i ≤ n h(τₙ - τᵢ). We show that ξₙ has a thin tail in the sense that under broad circumstances Pr\ξₙ > x + δ ξₙ > x\ → 0 as x → ∞, where δ > 0. We also show that Prmax(ξ₁, ⋯, ξₙ) ≤ uₙ\ - (Pr\ξ₀ ≤ uₙ\)ⁿ → 0 for real sequences (uₙ) for which lim n Pr\ξ₀ > uₙ\ < ∞.