Conjectures

On this page, I occasionally list some conjectures that appeared in my papers. If you are able to prived a solution, please let me know.

Maximum Number of Oscillations of a Gaussian Convolution

Consider a scalar random variable \(Y=X+Z\) where \(X\) and \(Z\) are independent. Assume that \(Z\) is a standard normal, and \(X\) is a bounded symmetric random variable such that \(|X| \le A\) for some \(A>0\). Denote by \(f_Y\) the probability density function of \(Y\). For obvious reasons, we refer to \(f_Y\) as a Gaussian convolution. Now denote by \(\mathrm{N}(S,f_Y-\kappa)\) the number of zeros of a function \(f_Y-\kappa\) on the set \(S \subseteq \mathbb{R}\). The conjecture can now be stated as follows:

\[ \max_{ X: |X| \le A } \max_{ \kappa: \kappa>0} \mathrm{N}( \mathbb{R},f_Y-\kappa)= O(A) \quad (*). \]

The first maximization in \((*)\) is over all symmetric distributions on \(X\) that have support on the interval \([-A,A]\).