  In this paper, we study the probabilistic caching for an $N$-tier wireless
heterogeneous network (HetNet) using stochastic geometry. A general and
tractable expression of the successful delivery probability (SDP) is first
derived. We then optimize the caching probabilities for maximizing the SDP in
the high signal-to-noise ratio (SNR) regime. The problem is proved to be convex
and solved efficiently. We next establish an interesting connection between
$N$-tier HetNets and single-tier networks. Unlike the single-tier network where
the optimal performance only depends on the cache size, the optimal performance
of $N$-tier HetNets depends also on the BS densities. The performance upper
bound is, however, determined by an equivalent single-tier network. We further
show that with uniform caching probabilities regardless of content
popularities, to achieve a target SDP, the BS density of a tier can be reduced
by increasing the cache size of the tier when the cache size is larger than a
threshold; otherwise the BS density and BS cache size can be increased
simultaneously. It is also found analytically that the BS density of a tier is
inverse to the BS cache size of the same tier and is linear to BS cache sizes
of other tiers.
