The point process approach for fractionally differentiated random walks under heavy traffic (Q713216)

Let \((X_i)_{i \geq 1}\) be a sequence of independent random variables, having common distribution function \(F\) which is centered and belongs to the domain of attraction of a stable law with index \(\alpha\), in \((1,2)\). Let \(S_n = \sum_{0\leq i < n} g_i X_{n-i}\), where \(g(x) = \sum_{i \geq 0}g_i x^i\) is a power series with radius of convergence \(1\). A heavy traffic approximation describes the limiting behaviour of some functional of the process \((S_n)\). In this paper, the authors prove a point process form of a heavy traffic approximation for the supremum functional. In the sequel, a maximal-type inequality is proved for the process.