Lower tail probability estimates for subordinators and nondecreasing random walks (Q1822127)

Let \(\{X_ i\}\) be i.i.d. non-negative random variables with mean \(\mu\), Laplace transform \(\psi\), and partial sums \(\{S_ n\}\). Let a be the infimum of the support of X. The asymptotics of \(\log (P(S_ n\leq nx_ n))\) are obtained for \(a<x_ n<\mu\) and, under additional conditions, it is shown that, if for some \(\delta >0\), \(a+\delta \leq x_ n<\mu\), \(P(S_ n\leq nx_ n)\sim \rho_ n\exp (-nR(\lambda_ n))\), where \(R(u)=-\log (\psi (u))+u\psi '(u)/\psi (u)\) and \(\lambda_ n\) solves \(- \psi '(\lambda_ n)/\psi (\lambda_ n)=x_ n\) (an explicit formula for \(\rho_ n\) is also given). Related work is contained in the papers by \textit{R. R. Bahadur} and \textit{R. Ranga Rao} [Ann. Math. Stat. 31, 1015-1027 (1960; Zbl 0101.126)], \textit{V. V. Petrov} [Teor. Veroyatn. Primen. 10, 310-322 (1965; Zbl 0235.60028)] and \textit{T. Höglund} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 49, 105-117 (1979; Zbl 0416.60030)]. Analogous results are then obtained for subordinators.