upcrossings and downcrossings
Inequalities involving the number of times at which a stochastic process passes upwards or downwards through a bounded interval play an important role in the theory of stochastic processes. Whether or not a sequence of real numbers converges to a limit can be expressed in terms of the finiteness of the number of upcrossings or downcrossings, leading to results such as the martingale convergence theorem and regularity of martingale sample paths. As the main applications are to stochastic processes, we suppose that is a real-valued stochastic process with totally ordered time index set , usually a subset of the real numbers. However, for the definitions here, it is enough to consider to be a sequence of real numbers, and the dependence on any underlying probability space is suppressed.
For real numbers , the number of upcrossings of across the interval is the maximum nonnegative integer such that there exists times satisfying
and for which . The number of upcrossings is denoted by . Note that this is either a nonnegative integer or is infinite. Similarly, the number of downcrossings, denoted by , is the maximum nonnegative integer such that there are times satisfying (1) and such that .
Note that between any two upcrossings there is a downcrossing. Given satisfying (1) and , we can set and for . Then, from which we conclude that . Similarly, we have . Hence, the number of upcrossings and the number of downcrossings of an interval differ by at most .
For a finite index set , the number of upcrossings can be computed as follows. Set and define and by
The number of upcrossings of is then equal to the maximum such that (see Figure 1).
- 1 David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, 1991.
|Title||upcrossings and downcrossings|
|Date of creation||2013-03-22 18:49:33|
|Last modified on||2013-03-22 18:49:33|
|Last modified by||gel (22282)|