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We obtain the strong approximation of the sojourn time progress for a two-stage tandem queue in heavy traffic, that is, the traffic intensity $\rho_1=\rho_2=1$. The sojourn time is the period from a customer's arrival to her departure, and the strong approximation is a function of Brownian motion.
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This paper establishes limsup type law of the iterated logarithm of the occupation measure, using the asymptotic equivalence relation between the occupation measure and the number of excursion process of a symmetric Cauchy process. Furthermore, by using the density theorem and the economic coverage method, it derives the exact Hausdorff measure for the range of a symmetric Cauchy process in \mathbb{R}.
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In this paper, semiparametric estimation of a regression function in the third order partially linear autoregressive model with first order autoregressive errors is mainly studied. We suppose that the regression function has a parametric framework, and use the conditional least squares method to obtain the parameter estimators. Then semiparametric estimators of the regression function can be given by combining with the nonparametric kernel function adjustment. Furthermore, under certain conditions, the consistency of the estimators is proved. Finally, simulation research is presented to evaluate the
effectiveness of the proposed method.
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Kolmogorov-Smirnov (KS), Cramer-von Mises (CM) and Anderson-Darling (AD) test, which are based on empirical distribution function (EDF), are well-known statistics in testing univariate normality. In this paper, we focus on the high dimensional case and propose a family of generalized EDF based statistics to test the high-dimensional normal distribution by reducing the dimension of the variable. Not only can we approximate the corresponding critical values of three statistics by Monte Carlo method, we also can investigate the approximate distributions of proposed statistics based on approximate formulas in univariate case under null hypothesis. The Monte Carlo simulation is carried out to demonstrate that the performance of proposed statistics is more competitive than existing methods under some alternative hypotheses. Finally, the proposed tests are applied to real data to illustrate their utility.
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Model of option pricing driven by Brownian motion is the most classical model. However, it can not describe long-term property and invariance in a short period of time of asset price. In this article, option pricing model driven by sub-fractional Brownian motion is studied under time-transform with dividend-paying. Firstly, the model of diffusion B-S model of sub-fractional Brownian motion is build, and get option pricing formula with dividends. Secondly, statistical simulation is used by real data in finance and show that new model can reflect real financial assets.
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In this paper, we consider the optimal dividend problem in the spectrally positive L\'{e}vy model with regime switching. By an auxiliary optimal problem, the principle of dynamic programming and the fluctuation theory of L\'{e}vy processes, we show that optimal strategy is a modulated barrier strategy. The value function and the optimal dividend barrier are obtained by iteration.
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This paper is based on ``Pao-Lu Hsu's lecture'' (2019/3/22) at Peking University and the subsequent expansion of his reports. It begins with some recollections benefited of the author from Professor Hsu, and ends with thanking to a group of professors at Peking University for their support and help over the past decades. The middle part is the theme of the talk. It gives first an overview of personal cross research. Then, from a challenge of computing, the author reports on the study looking for a larger class of complex matrices which have real spectrum. This was done mainly in the last year. It involves the fields of computation, probability, statistical mechanics and quantum mechanics Next, the paper introduces the latest development of algorithms, which is another illustration of the intersection between probability theory and computational mathematics. As the end, it also outlines the understanding of the cross study.
