  This paper is concerned with long-time strong approximations of SDEs with
non-globally Lipschitz coefficients.Under certain non-globally Lipschitz
conditions, a long-time version of fundamental strong convergence theorem is
established for general one-step time discretization schemes. With the aid of
the fundamental strong convergence theorem, we prove the expected strong
convergence rate over infinite time for two types of schemes such as the
backward Euler method and the projected Euler method in non-globally Lipschitz
settings. Numerical examples are finally reported to confirm our findings.
