Cantor Singular Distribution

Worked Example: Finite Step Construction

Let us consider the finite construction at n=2, tracking the possible endpoints X_n = Σ(k=1 to n) 2B_k / 3^k where B_k is in {0, 1}.

Depth n = 2 has 2^2 = 4 atoms.
B_1 = 0, B_2 = 0 => X_2 = 2(0)/3 + 2(0)/9 = 0
B_1 = 0, B_2 = 1 => X_2 = 2(0)/3 + 2(1)/9 = 2/9
B_1 = 1, B_2 = 0 => X_2 = 2(1)/3 + 2(0)/9 = 2/3
B_1 = 1, B_2 = 1 => X_2 = 2(1)/3 + 2(1)/9 = 8/9

Thus, P(X_2 = x) = 1/4 for x in {0, 2/9, 2/3, 8/9}.
The finite step CDF at x = 2/3 is P(X_2 <= 2/3) = 1/4 + 1/4 + 1/4 = 3/4 = 0.75.

Meanwhile, the continuous Cantor-limit CDF at x = 2/3 evaluates to 0.5, which bounds the maximum difference error to 2^(-2) = 0.25.

Interactive Experiment

Explore the convergence of the discrete finite approximations to the singular continuous limit using the module. Follow these steps:

1. Open the Cantor Singular Distribution module.
2. Observe the default state with Depth set to 4 and Probe x at 1/3. Notice how the Limit CDF and Finite CDF align.
3. Change Depth to 1 and Probe x to 1/9. Observe that the Finite CDF is 0.5 because 1/9 >= 0, but it is smaller than 2/3, while the Limit CDF equals 0.25.
4. Adjust Depth to higher values (e.g., 6 or 8) and observe how the step-like Finite CDF tightly bounds and approaches the smooth fractal limit curve.

Limitations

The finite-step model is purely discrete, utilizing exactly 2^n distinct uniformly weighted atoms (or a single atom at n=0). It is not itself a singular-continuous distribution. We must consider the infinite limit n → ∞ to obtain the true singular distribution, which possesses no discrete atoms yet entirely avoids regions of positive density (its derivative is zero almost everywhere). Because floating-point limits restrict recursion depth, actual measurements compute the recursive definition up to arbitrary machine precision iterations instead of infinite steps.

References

The following reference formalizes the exact limiting behavior of the Cantor singular distribution and its CDF.

• Cantor Singular Distribution Limits - Detailed analysis on construction from ternary sums and resulting singular measures.