Probabilities of large deviations for random walks with regular distribution of jumps. (Q1432214)

Let \(X_1,X_2,\dots\) be i.i.d. random variables with \(EX_1= a\), \(EX^2_1= 1\), \(E| X_1|^b< \infty\) with \(b\geq 2\) and let \(V(t)= P(X_1>t)= t^{-\beta}L(t)\) where \(L\) varies slowly at \(\infty\) and \(\beta> 2\), \(\beta\geq b\). Put \(S_n= X_1+\cdots+ X_n\), \(M_n= \max\{S_k:k\leq n\}\), \(V_n(x)= P(S_n> x)\), \(W_n(x)= P(M_n> x)\). Under further regularity conditions on the function \(L\) at \(\infty\), when \(a= 0\), the paper states finite asymptotic expansions for \(V_n\) as \(x\to\infty\), e.g. \[ V_n(x)= nV(x)\Biggl[1+{1\over 2}\beta(\beta+ 1)nx^{-2}(1+ o(1))\Biggr], \] uniformly in \(n\log x\leq cx^2\), and more general ones. Similar expansions are given for \(W_n(x)\), also when \(a< 0\). One- and two-term expansions for \(V_n(x)\) and \(P(\max(S_k- gk)> x)\) are given when \(V(t)= \exp(-\mu t)Y(t)\), where \(Y(t)\) behaves as the above \(V(t)\). These expansions are in terms of \(\varphi(\lambda)= E\exp(\lambda X_1)\), \(0\leq \lambda\leq\mu\) and \(\lambda(\alpha)= \sup_\lambda(\alpha\lambda- \log\varphi(\lambda))\).