On the asymptotics of the probability to stay above a non-increasing boundary for a non-homogeneous compound renewal process

We consider a non-homogeneous compound renewal process, which is also known as a cumulative renewal process, or a continuous time random walk. We suppose that the jump sizes have zero means and finite variances, whereas the renewal-times has moments 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 homogeneous compound renewal process, due to A. Sakhanenko, V. Wachtel, E. Prokopenko, A. Shelepova (2021).