Large deviations view points for heavy-tailed random walks
Let \(X,X_1,X_2,\dots\) be nonnegative i.i.d. random variables and \(S_n= X_1+\cdots+ X_n\) \((n\geq 1)\). Put \[ \overline\alpha:= -\limsup_{t\to\infty} {1\over t}\log P(\log X> t)\in [0,\infty], \quad \underline\alpha:= -\liminf_{t\to\infty} {1\over t}\log P(\log X> t)\in [0,\infty]. \] The main result of the present paper is the following Theorem. Assume that \(0<\overline\alpha< \infty\). Then for every \(x>\overline x:= \max(1,1/\overline\alpha)\) \[ \limsup_{n\to\infty} {1\over\log n}\log P(S_n> n^x)= -\overline\alpha x+ 1, \] and, if additionally \(\underline\alpha< \infty\), \[ \liminf_{n\to\infty} {1\over\log n}\log P(S_n> n^x)= -\underline \alpha x+ 1. \] This result supports a conjecture by \textit{N. Gantert} [Stat. Probab. Lett. 49, 113--118 (2000; Zbl 0963.60024)] in the case \(\overline\alpha= \underline\alpha\), whereas in the case \(1< \overline\alpha< \underline\alpha<\infty\) Ganter's conjecture fails to be true.
scientific article; zbMATH DE number 2128697