Limit theorems for continuous-time random walks in the double-array limit scheme
Let \(\{X_{n,j}\},\) \(n,j=1,2,\dots,\) be a double array of rowwise i.i.d. real-valued random variables, and \(\{T_{n,j}\}\), \(n,j=1,2,\dots \), a double array of i.i.d. nonnegative random variables. For \(t\geq 0\) consider the sequence of continuous-time random walks defined by \(S_{n}( t) =\sum_{j=1}^{N_{n}( t) }X_{n,j}\), where \(N_{n}( t) \) is a renewal process defined by the relation \(\sum_{j=1}^{N_{n}( t) }T_{n,j}\leq t<\sum_{j=1}^{N_{n}( t) +1}T_{n,j}\), \(t\geq 0,\) for any \(n\geq 1\). Continuing work on continuous-time random walk, mainly done by them, the authors extend asymptotic results obtained in the classical case of degenerate \(X\)- and \(T\)-arrays, that is, where both \(X_{n,j}\) and \(T_{n,j}\) do not depend on \(n\). See, e.g., \textit{V. Kolokoltsov, V. Korolev} and \textit{V. Uchaikin} [J. Math. Sci., New York 105, No. 6, 2569--2576 (2001; Zbl 0994.60010)] for previous work. An important auxiliary result of the paper consists of two general theorems on the stability of the representation of a distribution function as a special mixture of stable laws.
scientific article; zbMATH DE number 5144810
scientific article