  We study the critical behavior of a random field O($N$) spin model with a
second-rank random anisotropy term in spatial dimensions $4<d<6$, by means of
the replica method and the 1/N expansion. We obtain a replica-symmetric
solution of the saddle-point equation, and we find the phase transition obeying
dimensional reduction. We study the stability of the replica-symmetric saddle
point against the fluctuation induced by the second-rank random anisotropy. We
show that the eigenvalue of the Hessian at the replica-symmetric saddle point
is strictly positive. Therefore, this saddle point is stable and the
dimensional reduction holds in the 1/N expansion. To check the consistency with
the functional renormalization group method, we obtain all fixed points of the
renormalization group in the large $N$ limit and discuss their stability. We
find that the analytic fixed point yielding the dimensional reduction is
practically singly unstable in a coupling constant space of the given model
with large $N$. Thus, we conclude that the dimensional reduction holds for
sufficiently large $N$.
