Strong approximation of extended renewal processes
Let \(\{X_ n\}\) be a sequence of i.i.d. random vectors in \(R^ d\), \(d\geq 1\), with expectation \(\mu\) and finite \(\nu\) th moment, \(\nu >2\). Let h be a real function on \(R^ d\) which is homogeneous, \(h(\mu)>0\), and has continuous partial derivatives of the second order in a neighbourhood of \(\mu\). For \(0<q\leq 1\) define the extended renewal process \(N(t)=\min \{k:h(S_ k)\geq tk^{1-q}\}\), \(t\geq 0\), where \(S_ k=X_ 1+...+X_ k.\) The main result of the paper is a strong approximation of \((t/h(\mu))^{1/q}-N(t)\) by a time-transformed Wiener process \(cW((t/h(\mu))^{1/q})\). It is shown that the rate of approximation cannot be improved when \(\nu\geq 4\). Consequences of this theorem are functional and Chung-type laws of iterated logarithm, Bahadur-Kiefer type representation of N(t) in terms of partial sums, and the characterization of N(t) by four classes of deterministic curves in the sense of Révesz. Generalization of the results to the case of nonindependent and nonidentically distributed random vectors is also discussed. It is worthwhile to notice that this paper is the first one considering the case \(q<1\) when \(d>1\).
scientific article; zbMATH DE number 3879820