Research On Random Spline Function

Random spline is the extension of random polynomial, adding another useful mathmatic instrument for studing the things by random changing patten but containing special law in the nature.In the general,the things by random changing patten in the natrure is discribed by random function,but the random funtion is complex ,and very hardly grasp,we hope to approximate random functin by the random spline as continuous funtion is approximated by polynomial,thus, the essence in the nature which could be ovserved is revealed. So,study the essence,the inherent character of random spline and the approximation theory should be a very important subjuct.The concept of random spline is given in paper[1],and proof the approximation theory of such spline function to random function,this paper study the probability character of random spline.At first , this paper discuss several probability properties of the random spline——the expectation,variance and the correlation of random spline ,show the expectation of the random spline which with the kinds of distribution,and the geometric meaning of those properties of the random spline.Futhermore, ,discuss the random splne which with two-order moment by the opinion of functional, deduce the inner product space of space H ,analyze the property of mean-square convergence,and the character of mean-square limit.Then,analyze random spline with the property of two-order moment prosses or not? Study the property of mean-square coutinuous, mean-square derivative, mean-square integral,and the relationship of these properties.Afterwords, based on the meaning of space H ,two approximation theoties of random spline to the random funcion below the meaning of mean-square convergence and by probability convergence are given.At last, regarding the random function which with the Wiener process , we study the random interpolation spline funtion still with the property of the Wiener process or not? By analyzing random spline with 1,we know ,as the random funtion with the Wiener process ,though the random interplation splne funtion with 1 is the independent increments process, not the Wiener process.