On first-passage times for symmetric random walks without Lindeberg condition

We consider exit times for random walks with independent but not necessarily identically distributed increments. We are going to describe an asymptotic behavior of the probability that the random walk stays above the moving boundary for a long time. In the paper by D. Denisov, A. Sakhanenko, and V. Wachtel (Ann. Probab., 2018) an universal asymptotic formula for such probability was found in the case when the random walk satisfies the classical Lindeberg condition. Now we investigate a question if it is possible to find similar asymptotics for more general random walks when increments may have infinite variances, but the central limit theorem is still valid. We obtain such result for a class of walks with symmetrically distributed increments.