Asymptotic expansions for the distributions of stopped random walks and first passage times
This paper is concerned with the study of the \(d\)-dimensional random walk \(S_n = X_1 + X_2 + \dots + X_n\), where the \(X_i\) are i.i.d. \(d\)-dimensional random vectors subject to minor constraints. Let \(T_a = \inf \{n \geq n_a : ng(S_n/n) \geq a\}\), where \(n_a = o(a)\) and \(\theta = g(EX_i)\), \({\widehat \theta}_n = g(S^a_n/n)\) with \(g\) twice continuously differentiable in some neighborhood of \(\mu = E(X_i)\), and \(g'(\mu) > 0\). Also let \(\Delta_a = T_a {\widehat{\theta}}_{T_a} - a\). The authors first derive some lemmas on Edgeworth expansions and fluctuation theory before proving three theorems using asymptotic expansions of the Edgeworth type. These provide results for \(P\{T_a < n\}\), \(P\{T_a = n\), \(y_1 \leq \Delta_a \leq y_2\}\), and \(P\{Z_T < z\}\), where \(Z_T = (T_a \text{ var } X_i)^{-1/2} (S_{T_a} - T_a \mu)\). Full details of the complicated algebra required in the proofs are provided.
scientific article; zbMATH DE number 828373