Self-Similar Distributions

A more common random fractal construction involves statistical self-similarity: instead of specifying exact scalings, at each iteration the scaling of each piece is selected randomly from a set range.
Suppose the scaling factors are selected randomly from the range [1/4, 1/2]. A square with diagonal corners (a, b) and (a+L, b+L) is replaced by four squares arranged as shown here. Each of the scaling factors r1, r2, r3, and r4 is selected randomly from [1/4, 1/2].
For example, here are three realizations of the first iterate of this random process; below each is the sixth iterate.
Click here for an animation of the first six steps of the left example.
Statistical self-similarity refers to the fact that sub-pieces of each piece have the same distribution of sizes.
With probability 1, the dimension of this set is given by the randomized Moran equation:
E(r1d + r2d + r3d + r4d) = 1.
For example, construct a random Cantor set by at each stage either
replacing each interval with two intervals scaled by 1/3, with probability 1/4, or
replacing each interval with two intervals scaled by 1/4, with probability 3/4
Then the randomized Moran equation becomes
1 = E(r1d + r2d)
= (1/4)⋅((1/3)d + (1/3)d) + (3/4)⋅((1/4)d + (1/4)d)
= (1/2)⋅(1/3)d + (3/2)⋅(1/4)d
Solving numerically, we find d ≈ 0.529053

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