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IDENTICALLY DISTRIBUTED UNCORRELATED RANDOM VARIABLES NOT FULFILLING THE WLLN
Landers, Dieter,Rogge, Lothar Korean Mathematical Society 2001 대한수학회보 Vol.38 No.3
It is shown that for each 1 < p < 2 there exist identically distributed uncorrelated random variables $X_n\; with\;E({$\mid$X_1$\mid$}^p)\;<\;{\infty}$, not fulfilling the weak law of large numbers (WLLN). If, however, the random variables are moreover non-negative, the weaker integrability condition $E(X_1\;log\;X_1)\;<\;{\infty}$ already guarantees the strong law of large numbers.
Probabilistic vibration and lifetime analysis of regenerated turbomachinery blades
Berger, Ricarda,Rogge, Timo,Jansen, Eelco,Rolfes, Raimund Techno-Press 2016 Advances in aircraft and spacecraft science Vol.3 No.4
Variances in turbomachinery blades caused by manufacturing, operation or regeneration can result in modified structural behavior. In this work, the scatter of geometrical and material properties of a turbine blade and its influence on structure performance is discussed. In particular, the vibration characteristics and the lifetime of a turbine blade are evaluated. Geometrical variances of the surface of the blades are described using the principal component analysis. The scatter in material properties is considered by 16 varying material parameters. Maximum vibration amplitudes and the number of load cycles the turbine blade can withstand are analyzed by finite element simulations incorporating probabilistic principles. The probabilistic simulations demonstrate that both geometrical and material variances have a significant influence on the scatter of vibration amplitude and lifetime. Dependencies are quantified and correlations between varied input parameters and the structural performance of the blade are detected.