Berry-Esseen type theorem for Monotonic independence

Question

The central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases to infinity. Under stronger assumptions, the Berry–Esseen theorem, or Berry–Esseen inequality, gives a more quantitative result, because it also specifies the rate at which this convergence takes place by giving a bound on the maximal error of approximation between the normal distribution and the true distribution of the scaled sample mean.

In free probability, analogue of Central Limit Theorem is known where Wigner's semi-circle law plays the role of Normal distribution. Berry–Esseen type theorem for free random variables is due to Vladislav Kargin 2007.

In monotone probability, analogue of Central Limit theorem is also known where arc-sine law plays the role of normal distribution.

But the Berry-Esseen type theorem seems to be missing.. Is it just an open problem or are there known reasons for which such a result won't be possible....thanks....

Answer 1

A Berry Esseen type theorem is indeed plausible. However, the rate of convergence cannot be of order $n^{1/2}$ as in the classical free version of it. In this case one can find an explicit example (centered monotone Poisson) whose rate of convergence is $n^{1/4}$ with respect to the Kolmogorov distance.

Under some mild conditions in ongoing work (with M. Salazar and J-C. Wang) , we can give estimates to obtain a Berry Esseen type theorem. However, for the moment, the exact rate of convergence is an open problem.