The probabilistic convergence problem of density functions related to $\partial_{x}^{3}+\partial_{x}^{-1}$

In this article, by using full randomization introduced by Hadama and Yamamoto (Probabilistic Strichartz estimates in Schatten classes and their applications to Hartree equation, J. Math. Phys. 67(2026), 35pp) and high-low frequency technique as well as the property of $\mathfrak{S}^{2}$,
we establish the probabilistic convergence of the density function
related to $\partial_{x}^{3}+\partial_{x}^{-1}$
on $\R$, which extends the Theorem 1.3 of Yan et al.
(Convergence problem of Ostrovsky equation with rough data and random data, Indiana Univ. Math. J. 71(2022), 1897-1921.).