  We study the rate of convergence of some recursive procedures based on some
"exact" or "approximate" Euler schemes which converge to the invariant measure
of an ergodic SDE driven by a L\'{e}vy process. The main interest of this work
is to compare the rates induced by exact and approximate Euler schemes. In our
main result, we show that replacing the small jumps by a Brownian component in
the approximate case preserves the rate induced by the exact Euler scheme for a
large class of L\'{e}vy processes.
