arXiv subjects
Publications and source records attributed to Ilya Tyurin.
We prove that for any pair of i.i.d. random variables $X,Y$ with finite moment of order $a \in (0,2]$ it is true that $E |X-Y|^a \leq E |X+Y|^a$. Surprisingly, this inequality turns out to be related with bifractional Brownian motion. We extend this result to Bernstein functions and provide some counter-examples.
We investigate the convergence rate in the Lyapunov theorem when the third absolute moments exist. By means of convex analysis we obtain the sharp estimate for the distance in the mean metric between a probability distribution and its zero bias transformation. This bound allows to derive new estimates of the convergence rate in terms of Kolmogorov's metric as well as the metrics $ζ_r$ (r=1,2,3) introduced by Zolotarev. The estimate for $ζ_3$ is optimal. Moreover, we show that the constant in the classical Berry-Esseen theorem can be taken as 0.4785. In addition, the non-i.i.d. analogue of this theorem with the constant 0.5606 is provided.