There are two different phenomena that make the analysis of weakly dependent heavy-tailed models more subtle than in the independent case: clustering of big values and self-cancellation of groups of jumps. While clusters can often be managed with Skorokhod's topologies $M_1$ and $M_2$, self-cancelling jumps imply discontinuity of path-wise suprema, what excludes the use of the above-mentioned topologies. In the talk we shall discuss the potential applicability of a non-Skoro\-khod\-ian topology, introduced by the author ([2]) and called the $S$ topology. We shall start with motivating examples and the definition of the $S$ topology. Then we shall recall some naturally arising models (linear processes with heavy-tailed innovations - [1], ARCH or GARCH models in some range of parameters - [3]) for which the $S$ topology works. We shall conclude with some recent developments showing that the $S$ topology is, in fact, a linear and locally convex topology and thus can be used in the stochastic analysis on the Skorokhod space.\\[0.5cm] \noindent{\bf References} \begin{description} \item{[1]} R. Balan, A. Jakubowski and S. Louhichi (2014+) Functional Convergence of Linear Processes with Heavy-Tailed Innovations, {\em Journal of Theoretical Probability}, DOI 10.1007/s10959-014-0581-9. \item{[2]} A. Jakubowski (1997), A non-Skorohod topology on the Skorohod space, {\em Electronic Journal of Probability}, {\bf 2}, No 4, 1-21. \item{[3]} R-M. Zhang, CY. Sin and S. Ling (2015), On functional limits of short- and long-memory linear processes with GARCH(1,1) noises, {\em Stochastic Processes and their Applications} {\bf 125}, 482–-512. \end{description}