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This paper deals with the maximum likelihood estimator for the parameter of first-order autoregressive models driven by the stationary Gaussian noises (Colored noise) together with an input. First, we will find the optimal input that maximizes the Fisher information, and then, with the method of the Laplace transform, both the asymptotic properties and the asymptotic design problem of the maximum likelihood estimator will be investigated. The results of the numerical simulation confirm the theoretical analysis and show that the proposed maximum likelihood estimator performs well in finite samples. [ABSTRACT FROM AUTHOR]

We study the projection of an element of fractional Gaussian noise onto its neighbouring elements. We prove some analytic results for the coefficients of this projection. In particular, we obtain recurrence relations for them. We also make several conjectures concerning the behaviour of these coefficients, provide numerical evidence supporting these conjectures, and study them theoretically in particular cases. As an auxiliary result of independent interest, we investigate the covariance function of fractional Gaussian noise, prove that it is completely monotone for H > 1 / 2 , and, in particular, monotone, convex, log-convex along with further useful properties. [ABSTRACT FROM AUTHOR]

Phytoplankton movement patterns and swimming behavior are important and basic topics in aquatic biology. Heavy tail distribution exists in diverse taxa and shows theoretical advantages in environments. The fat tails in the movement patterns and swimming behavior of phytoplankton in response to the food supply were studied. The log-normal distribution was used for fitting the probability density values of the movement data of Oxyrrhis marina. Results showed that obvious fat tails exist in the movement patterns of O. marina without and with positive stimulations of food supply. The algal cells tended to show a more chaotic and disorderly movement, with shorter and neat steps after adding the food source. At the same time, the randomness of turning rate, path curvature and swimming speed increased in O. marina cells with food supply. Generally, the responses of phytoplankton movement were stronger when supplied with direct prey cells rather than the cellfree filtrate. The scale-free random movements are considered to benefit the adaption of the entire phytoplankton population to varied environmental conditions. Inferentially, the movement pattern of O. marina should also have the characteristics of long-range dependence, local self-similarity and a system of fractional order. [ABSTRACT FROM AUTHOR]