  An invariant random subgroup of the countable group {\Gamma} is a random
subgroup of {\Gamma} whose distribution is invariant under conjugation by all
elements of {\Gamma}. We prove that for a nonamenable invariant random subgroup
H, the spectral radius of every finitely supported random walk on {\Gamma} is
strictly less than the spectral radius of the corresponding random walk on
{\Gamma}/H. This generalizes a result of Kesten who proved this for normal
subgroups. As a byproduct, we show that for a Cayley graph G of a linear group
with no amenable normal subgroups, any sequence of finite quotients of G that
spectrally approximates G converges to G in Benjamini-Schramm convergence. In
particular, this implies that infinite sequences of finite d-regular Ramanujan
Schreier graphs have essentially large girth.
