It is well known that, under fairly general conditions, the waiting time distribution of Markovian queueing systems decays exponentially fast. This property is used to estimate or approximate very small cell loss probability of high-speed information/communication networks. It has been reported, however, that the input traffic of the recent LAN and WAN are observed to have stronger correlation which can result in longer tail of a waiting time distribution. In this article, we describe how the correlation structure of the input traffic affect the tail behavior of the waiting time distributions. To unify the explanations, we investigate the correlated random walks that are frequently encountered in the analysis of queueing phenomena.

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Rissanen has introduced stochastic complexity to define the amount of information in a given data sequence relative to a given hypothesis class of probability densities, where the information is measured in terms of a logarithmic loss associated with universal data compression. This paper proposes the notion of extended stochastic complerxity (ESC) by generalizing Rissanen's stochastic complexity into the decision-theoretic setting where a general real-valued function is used as a hypothesis and a general loss function is used as a distortion measure. We thereby demonstrate the effectiveness of ESC in design and analysis of learning algorithms in sequential prediction and function-estimation scenarios. As an application of ESC to sequential prediction, this paper shows that a sequential realization of ESC produces a sequential prediction algorithm called the aggregating strategy, for which the worst-case cumulative prediction loss is asymptotically minimal. As an application of ESC to function-estimation, this paper shows that a batch-approximation of ESC induces a batchlearning algorithm called the minimum L-complexity algorithm (MLC), for which an upper bound on the statistical risk is least to date. Through ESC we give a unifying view of designing the most effective learning algorithms in fundamental learning issues.

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Among various applications of symplectic mappings, numerical application is selected. A representation of a symplectic mapping based on a Weinstein generating function is introduced, and it will be shown that the representation is closely related to time-reversibility and leads to such practical schemes as the implicit midpoint method and so on. Moreover, the symmetric Runge-Kutta (abbreviated to RK) method, which has been known as possessing reversibility, is studied from a view point of linear symplecticity. A few dynamical properties intrinsic to linear symplectic RK methods are considered to find that similar properties are inherited by two kinds of RK methods separately when the objects to be integrated are enlarged to nonlinear systems. Finally, two results with respect to numerical reproduction of a phase portrait are dealt with by use of symplectic RK methods.

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