Streams of different seeded RNG partially correlated when the seed and the difference in seed is small

[sylvaticus]

Just noticing this behaviour. The streams of different seeded RNG is partially correlated when the seed (and the difference in seed) is small. Perhaps it is well known, just I didn't read about it in the package readme:

using Random, StableRNGs
rDiff(rngFunction,seedBase,seedDiff,repetitions) = norm(rand(rngFunction(seedBase),repetitions) .- rand(rngFunction(seedBase+seedDiff),repetitions))/repetitions

# Seed base 1000: ok
rDiff(StableRNG,1000,1,100000)          # 0.00129
rDiff(StableRNG,1000,10,100000)         # 0.00129
rDiff(StableRNG,1000,1000,100000)       # 0.00129

rDiff(MersenneTwister,1000,1,100000)    # 0.00129
rDiff(MersenneTwister,1000,10,100000)   # 0.00129
rDiff(MersenneTwister,1000,1000,100000) # 0.00129

# Seed base 10: Still ok
rDiff(StableRNG,10,1,100000)            # 0.00129
rDiff(StableRNG,10,10,100000)           # 0.00129
rDiff(StableRNG,10,1000,100000)         # 0.00129

rDiff(MersenneTwister,10,1,100000)      # 0.00129
rDiff(MersenneTwister,10,10,100000)     # 0.00129
rDiff(MersenneTwister,10,1000,100000)   # 0.00129

# Seed base 1: We start seeing problems for StableRNG..
rDiff(StableRNG,1,1,100000)             # 0.00125  <--
rDiff(StableRNG,1,10,100000)            # 0.00129
rDiff(StableRNG,1,1000,100000)          # 0.00129

rDiff(MersenneTwister,1,1,100000)       # 0.00129
rDiff(MersenneTwister,1,10,100000)      # 0.00129
rDiff(MersenneTwister,1,1000,100000)    # 0.00129

# Seed base 0: Unexpected results for for StableRNG..
rDiff(StableRNG,0,1,100000)             # 0.00105 <----------
rDiff(StableRNG,0,2,100000)             # 0.00116 <-----
rDiff(StableRNG,0,5,100000)             # 0.00123 <---
rDiff(StableRNG,0,10,100000)            # 0.00126 <--
rDiff(StableRNG,0,1000,100000)          # 0.00129

rDiff(MersenneTwister,0,1,100000)       # 0.00130 <-
rDiff(MersenneTwister,0,2,100000)       # 0.00129
rDiff(MersenneTwister,0,5,100000)       # 0.00129
rDiff(MersenneTwister,0,10,100000)      # 0.00129
rDiff(MersenneTwister,0,1000,100000)    # 0.00129

[rfourquet]

I'm not well versed in RNG theory, but I regret not having added at least a basic "scrambling" of the user-provided seed, like split-mix hash or whatever. It's probably not a problem for the intended use case of this RNG, but It could be worth it adding some warning in the README. And this should definitely be kept in mind if there is a version 2.0 of this package one day, or if another RNG is added to this package.