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# Analytic Properties of Random Processes

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Properties  ::  Continuity  ::  Differentiation  ::  Integration

## Continuity

Let . Recall that a function is continuous at if . is continuous if it is continuous for every .

Continuity w.p. 1 is usually quite strong, in fact stronger than is typically needed for analysis.

We say that is continuous in probability at if (i.p.) as . That is,

We say that is mean-square continuous at if (m.s.) as . That is,

An important property: A second-order random process is mean-square continuous at if and only if is continuous at , that is

For a W.S.S. process, we have the following: A W.S.S. r.p. is mean-square continuous if and only if is continuous at . This follows since , and

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 07). Analytic Properties of Random Processes. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lecture7_2.htm. This work is licensed under a Creative Commons License