Scaling limits of coupled continuous time random walks and residual order statistics through marked point processes (Q1940234)

Let \((J_1, X_1), (J_2,X_2),\dots\) be a sequence of i.i.d. \(\mathbb{R}^+\times \mathbb{R}^d\)-dimensional random vectors. No condition is imposed on the dependence structure between \(J_k\) and \(X_k\). For fixed \(t\geq 0\), denote by \(\nu_t\) the first passage time into \((t,\infty)\) of the zero-delayed standard random walk with increments \(J_n\). It was observed in [\textit{P. Straka} and \textit{B. I. Henry}, Stochastic Processes Appl. 121, No. 2, 324--336 (2011; Zbl 1219.60048)] (see also [\textit{A. Jurlewicz} et al., ``Fractional governing equations for coupled random walks'', Comput. Math. Appl. 64, No. 10, 3021--3036 (2012; \url{doi:10.1016/j.camwa.2011.10.010})]) that the processes \(\sum_{k=1}^{\nu_t-1}X_k\) and \(\sum_{k=1}^{\nu_t}X_k\), properly scaled, weakly converge, as \(t\to\infty\), to different limit processes. The authors of the present paper use a point process technique to explain this phenomenon. An essential ingredient of the authors' approach is a result proved in the paper about convergence of the residual order statistics. Among other things, this result solves a long-standing open problem.