Asymptotic $r$-log-convexity and P-recursive sequences

A sequence $\{ a_n \}_{n \ge 0}$ is said to be asymptotically $r$-log-convex if it is $r$-log-convex for $n$ sufficiently large. We present a criterion on the asymptotical $r$-log-convexity based on the asymptotic behavior of $a_n a_{n+2}/a_{n+1}^2$. As an application, we show that most P-recursive sequences are asymptotic $r$-log-convexity for any integer $r$ once they are log-convex. Moreover, for a concrete integer $r$, we present a systematic method to find the explicit integer $N$ such that a P-recursive sequence $\{a_n\}_{n \ge N}$ is $r$-log-convex. This enable us to prove the $r$-log-convexity of some combinatorial sequences.