Exact convergence rates in central limit theorems for a branching random walk with a random environment in time

Chen [Ann. Appl. Probab. {\bf 11} (2001), 1242--1262] derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process.We extend Chen's results to a branching random walk under weaker moment conditions. For the branching Wiener process, our results sharpen Chen's by relaxing the second moment condition used by Chen to a moment condition of the form \E X (\ln^+X )^{1+\lambda}< \infty. In the rate functions that we find for a branching random walk, we figure out some new terms which didn't appear in Chen's work.The results are established in the more general framework, i.e. for a branching random walk with a random environment in time.The lack of the second moment condition for the offspring distribution and the fact that the exponential moment does not exist necessarily for the displacements make the proof delicate; the difficulty is overcome by a careful analysis of martingale convergence using a truncating argument. The analysis is significantly more awkward due to the appearance of the random environment.Comment: Corrected version of https://hal.archives-ouvertes.fr/hal-01095105. arXiv admin note: text overlap with arXiv:1504.01181 by other author

One-Bit Compressed Sensing by Greedy Algorithms

Sign truncated matching pursuit (STrMP) algorithm is presented in this paper. STrMP is a new greedy algorithm for the recovery of sparse signals from the sign measurement, which combines the principle of consistent reconstruction with orthogonal matching pursuit (OMP). The main part of STrMP is as concise as OMP and hence STrMP is simple to implement. In contrast to previous greedy algorithms for one-bit compressed sensing, STrMP only need to solve a convex and unconstraint subproblem at each iteration. Numerical experiments show that STrMP is fast and accurate for one-bit compressed sensing compared with other algorithms.Comment: 16 pages, 7 figure

Axial dynamic load identification of hydraulic turbine based on Chebyshev orthogonal polynomial approximation

An analysis method is proposed to identify axial dynamic loads acting on the Francis turbine based on Chebyshev orthogonal polynomial expansion theory. Dynamic loads are expressed as functions of time and polynomial coefficients. The dynamic load identification model is constructed through discretized integral convolution of the loads, such as the Duhamel integral. However, the discretized numerical integral has a time-cumulative error problem that decreases the recognition accuracy of the dynamic load. Compared with the traditional method, the algorithm proposed in this paper constructs the relationship between the modal displacement and force using polynomial orthogonality and derivative relation between displacement and velocity or acceleration. The new method could avoid the Duhamel integral and time-cumulative error problem. This algorithm not only requires less measuring point information, but is also highly efficient. Compared with genetic algorithm identification, orthogonal polynomial algorithm is not easy falling into local convergence, and does not require multiple repetitions positive analysis trial to evaluate individual fitness value. Numerical simulations demonstrate that the identification and assessment of dynamic loads are effective and consistent when the proposed method is used