In the traditional analysis of Bloom filters, it's assumed that the hash functions are truly random functions, meaning that each hash function distributes each key uniformly and independently of each other key. This contrasts with the sorts of analyses that I've seen done elsewhere. For example, both the count sketch and count-min sketch only require 2-independent hash functions, as does a hash table that resolves collisions via chaining. (However, linear probing hash tables require 5-independence).
Has there been any work done into analyzing Bloom filters under the assumption that the hash functions used are, say, only $k$-independent? Or are any lower bounds known on the degree of independence required for Bloom filters to provide reasonable guarantees about false positives?