Statistical Properties of the Integral of a Binary Random Process

Statistical properties of the output <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z(t)</tex> of a finite time integrator are discussed. The input process considered is a binary random process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y(t)</tex> having successive axis-crossing intervals which are statistically independent. Transform expressions are derived for the first- and second-order transition probability densities of the integrated process, and it is shown how these results may be extended to three or more dimensions. Four processes are considered as examples. The integrated process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z(t)</tex> is shown to be a projection of a Markov process in three dimensions. The other two components are the original binary process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y(t)</tex> , and an "associated ramp process" <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> . Various statistical properties of this ramp process are considered and it is shown that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> is Markovian in one dimension. The first-order probability density and the transition probability density are discussed. Also, the transition probability density for the joint process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[x(t), y(t), z(t)]</tex> is given. Finally, in Appendix II, results are given for the first passage and recurrence time probability densities of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> , together with a relation between these two density functions.