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The invariance principle known as the functional central limit theorem is the fundamental knowledge for understanding the convergence of a Markov chain to a diffusion process and has been applied to many areas of financial economics. The derivation of that principle from the properties of stochastic process is nasty (Billingsley, 1999). Following the approach of Gikhman and Skorokhod (2000) this study uses the properties of probability measures to simplify the proof of the principle. If the sequences of finite distributions on weakly converge then they are tight. Using this properties, the invariance principle was directly proved. As applications of this principle, I derived the statistic of unit root process and the convergence of the Cox et al. (1979) binomial option pricing model to the continuous time Black Scholes (1972) option pricing model.

When a parameter of interest is nondifferentiable in the probability, the existing theory of semiparametric efficient estimation is not applicable, as it does not have an influence function. Song (2014) recently developed a local asymptotic minimax estimation theory for a parameter that is a nondifferentiable transform of a regular parameter, where the transform is a composite map of a continuous piecewise linear map with a single kink point and a translation-scale equivariant map. The contribution of this paper is twofold. First, this paper extends the local asymptotic minimax theory to nondifferentiable transforms that are a composite map of a Lipschitz continuous map having a finite set of nondifferentiability points and a translation-scale equivariant map. Second, this paper investigates the discontinuity of the local asymptotic minimax risk in the true probability and shows that the proposed estimator remains to be optimal even when the risk is locally robustified not only over the scores at the true probability, but also over the true probability itself. However, the local robustification does not resolve the issue of discontinuity in the local asymptotic minimax risk.

Considers neurons, neural networks and neural fields from the viewpoint of abstract automata. Introduces Abstract neural automata (ANA) to explain and to provide a mathematical description of neural functions and theory. Surveys some current literature including that concerned with Boltzmann machines and the author’s own view of an associative Boltzmann neural model. Provides definitions and theorems to support the author’s theses on cognition, the human brain and the role of ANA in the understanding of neural networks.