Green's Open Problem 28

References:

• Green, Ben. "100 open problems." (2024).
• Mathoverflow/339137 asked by user Sil
• MathStackexchange/3325163 asked by user Emmanuel Amiot

def Green28.IsUniformOnSupport (X : PMF ℤ) : Prop

True if a PMF on $\mathbb{Z}$ is uniformly distributed on its support.

Equations

• Green28.IsUniformOnSupport X = ∃ (s : Finset ℤ) (hs : s.Nonempty), X = PMF.uniformOfFinset s hs

noncomputable def Green28.indepSum (X Y : PMF ℤ) : PMF ℤ

The discrete convolution of two PMFs on $\mathbb{Z}$, representing the distribution of the sum of two independent random variables.

Equations

• Green28.indepSum X Y = do let x ← X let y ← Y PMF.pure (x + y)

theorem Green28.green_28 : True ↔ ∀ (X Y : PMF ℤ), X.support.Finite ∧ Y.support.Finite ∧ IsUniformOnSupport (indepSum X Y) → IsUniformOnSupport X ∧ IsUniformOnSupport Y

Suppose that $X, Y$ are two finitely-supported independent random variables taking integer values, and such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly distributed on their ranges?