Non-classical error bounds in the central limit theorem

The classical central limit theorem states the uniform convergence of the distribution functions of the standardized sums of independent and identically distributed square integrable real-valued random variables to the standard normal distribution function. While first versions of the central limit theorem are already due to Moivre (1730) and Laplace (1812), a systematic study of this topic started at the beginning of the last century with the fundamental work of Lyapunov (1900, 1901). Meanwhile, extensions of the central limit theorem are available for a multitude of settings. This includes, e.g., Banach space valued random variables as well as substantial relaxations of the assumptions of independence and identical distributions. Furthermore, explicit error bounds are established and asymptotic expansions are employed to obtain better approximations. Classical error estimates like the famous bound of Berry and Esseen are stated in terms of absolute moments of the random summands and therefore do not reflect a potential closeness of the distributions of the single random summands to a normal distribution. Non-classical approaches take this issue into account by providing error estimates based on, e.g., pseudomoments. The latter field of investigation was initiated by work of Zolotarev in the 1960's and is still in its infancy compared to the development of the classical theory. For example, non-classical error bounds for asymptotic expansions seem not to be available up to now ...

Long-term Forecasting of Frame Alignment Losses for Circuit Emulation Implementation

The article provides a method for long-term forecast of frame alignment losses based on the bit-error rate monitoring for structure-agnostic circuit emulation service over Ethernet in a mobile backhaul network. The developed method with corresponding algorithm allows to detect instants of probable frame alignment losses in a long term perspective in order to give engineering personnel extra time to take some measures aimed at losses prevention. Moreover, long-term forecast of frame alignment losses allows to make a decision about the volume of TDM data encapsulated into a circuit emulation frame in order to increase utilization of the emulated circuit. The developed long-term forecast method formalized with the corresponding algorithm is recognized as cognitive and can act as a part of network predictive monitoring system.

Stability and error analysis for a numerical scheme to approximate elastic flow

Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, Univ., Dissertation, 2015