On the asymptotics of the distribution of the exit time beyond a non-increasing boundary for a compound renewal process

We consider a compound renewal process, which is also known as a cumulative renewal process, or a continuous time random walk. We suppose that the jump size has zero mean and finite variance, whereas the renewal-time has a moment of order greater than $3/2$. We investigate the asymptotic behaviour of the probability that this process is staying above a moving non-increasing boundary up to time $T$ which tends to infinity. Our main result is a generalization of a similar one for ordinary random walks obtained earlier by Denisov D., Sakhanenko A. and Wachtel V. in Ann. Probab., 2018.