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Heavy-tailed input: LSD
Published in Arup Bose, Koushik Saha, Random Circulant Matrices, 2018
Proof. Recall that Zj=Γj−1/α=(∑t=1jEt)−1/α, where {Ei} is a sequence of i.i.d. exponential random variables with mean 1. Hence, by the Law of the Iterated Logarithm (see Section 11.2 of Appendix), for almost all ω and for arbitrary ε > 0 there exist j0(ω) so that for j ≥ j0(ω), (1j+(2+ε)jloglogj)1α<Zj(ω)<(1j−(2+ε)jloglogj)1α.
Dynamics of stochastic FitzHugh–Nagumo system on unbounded domains with memory
Published in Dynamical Systems, 2023
We now introduce the following Hilbert spaces which are respectively endowed with the norms induced on by For the random term ω, we define a shift operator ϑ on Ω (where Ω is defined in previous section) by Then is a measure preserving transformation group on , that is, is a parametric dynamical system. By the law of the iterated logarithm, there exists a -invarant set of full P measure such that for , Put and let be a group on given by Then both and are parametric dynamical systems. We will define a continuous random cocycle for system (9) over and .
Adaptive fully sequential selection procedures with linear and nonlinear control variates
Published in IISE Transactions, 2023
Shing Chih Tsai, Jun Luo, Guangxin Jiang, Wei Cheng Yeh
In the frequentist approach, many simulation techniques have been taken into consideration, e.g., Common Random Numbers (CRNs) in the fully sequential selection procedure of Kim and Nelson (2001) and other Variance Reduction Techniques (VRTs) in both multi-stage and fully sequential selection procedures of Tsai and Kuo (2012) and Tsai et al. (2017). Meanwhile, new theoretical building blocks have been established to improve the algorithm efficiency (e.g., constructing a variance-dependent Brownian motion process in Hong (2006)), and to solve new types of R&S problems (e.g., conducting a asymptotic analysis for steady-state simulation in Kim and Nelson (2006b)), as well as to remove the widely used IZ assumption (e.g., proposing Law of the Iterated Logarithm-based continuation boundaries in Fan et al. (2016)). With the fast development of computer technology, parallel computing environments have also been adopted to solve large-scale R&S problems, e.g., the asymptotic parallel selection procedure in Luo et al. (2015) and the good selection procedure in Ni et al. (2017) and Zhong et al. (2022). On the other hand, under the Bayesian approach, instead of delivering a guarantee of correct selection of the best system, Bayesian-type procedures often consider the efficiency as the primary goal to design various (nearly) optimal sample allocation rules under different objective functions. Typically, this is achieved by maximizing the posterior PCS or maximizing the value of information, for example, the optimal computing budget allocation family procedures in Chen et al. (2000), Lee et al. (2012), and Gao et al. (2017), the value of information procedures in Chick and Inoue (2001), Chick et al. (2010), and Qu et al. (2015), and the approximate dynamic control policies in Frazier et al. (2008), Ryzhov (2016), and Peng et al. (2018). Notice that the optimality of the sampling efficiency is usually derived in the asymptotic sense as the total number of samples goes to infinity.
Boundary stabilisation to non-stationary solutions for deterministic and stochastic parabolic-type equations
Published in International Journal of Control, 2019
By the law of the iterated logarithm, it follows that hence, for we have as r → ∞ (for details, see Lemma 3.4 in Barbu & Da Prato, 2012).