We introduce the function Z(x;ξ,ν):=∫x−∞φ(t−ξ)⋅Φ(νt)dt, where φ and Φ are the pdf and cdf of N(0,1), respectively. We derive two recurrence formulas for the effective computation of its values. We show that with an algorithm for this function, we can efficiently compute the second-order terms of Bonferroni-type inequalities yielding the upper and lower bounds for the distribution of a max-type binary segmentation statistic in the case of small samples (where asymptotic results do not work), and in general for max-type random variables of a certain type. We show three applications of the method -- (a) calculation of critical values of the segmentation statistic, (b) evaluation of its efficiency and (c) evaluation of an estimator of a point of change in the mean of time series.
A delay stochastic method is introduced to control a certain class of chaotic systems. With the Lyapunov method, a suitable kind of controllers with multiplicative noise is designed to stabilize the chaotic state to the equilibrium point. The method is simple and can be put into practice. Numerical simulations are provided to illustrate the effectiveness of the proposed controllable conditions.