It is known due to Gin\'e and Marcus (1983) and Rosi\'nski and Woyczy\'nski (1986) that for any $\alpha \in (0,2)$, for any symmetric $\alpha$-stable L\'evy process $L$, and for any predictable real-valued process $\phi$ one has that
$$
\sup_{\lambda>0}\lambda^{\alpha}\mathbb P \Bigl(\sup_{t\geq 0}\Bigl|\int_0^{t}\phi_s d L_s\Bigr|>\lambda\Bigr) \eqsim_{L}\mathbb E \int_0^{\infty}|\phi_s|^{\alpha} d s,
$$
i.e.\ the weak $L^{\alpha}$-norm of the integral can be controlled by the $L^{\alpha}$-norm of the integrand.
The goal of the talk is to extend this result to both random measure setting and Banach space-valued integrals.
The talk is based on an ongoing work with Gergely Bod\'o (Universiteit van Amsterdam).