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Again on the derivatives of random functions

Authors
Publication Date
Keywords
  • 1998-99
Disciplines
  • Mathematics

Abstract

In this paper, two properties are shown to be valid for the derivatives of random functions: 1° Let $X(t,w)$ and $Y(t,w)$ be two random functions, defined in an interval I, which are mean derivables of order $p$ in the point $t_0 \in \ I$, then: The sum $X(t,w) + Y(t,w)$ is mean derivable of order $p$ in the point $t_0$ and we have: $(X+Y)'(t_0,w)=X'(t_0,w)+Y'(t_0,w)$. 2° Let $X(t,w)$ and $Y(t,w)$ be two indipendent random functions, defined in an interval I, which are mean derivables of order $p$ in the point $t_0 \in \ I$, then: The product $X(t,w)\cdot \ Y(t,w)$ is mean derivable of order $p$ in the point $t_0$ and we have: $(X \cdot \ Y)'(t_0,w)=X'(t_0,w) \cdot \ Y'(t_0,w)$. In a preceding paper we have shown two theorems on mean square derivatives of the random functions. In this paper we prove two theorems concerning the mean derivatives of order $p$ of the random functions.

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