upcrossings and downcrossings

Inequalities involving the number of times at which a stochastic processMathworldPlanetmath 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 convergesPlanetmathPlanetmath 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 martingaleMathworldPlanetmath sample paths. As the main applications are to stochastic processes, we suppose that (Xt)tβˆˆπ•‹ is a real-valued stochastic process with totally ordered time index setMathworldPlanetmathPlanetmath 𝕋, usually a subset of the real numbers. However, for the definitions here, it is enough to consider Xt to be a sequence of real numbers, and the dependence on any underlying probability spaceMathworldPlanetmath is suppressed.

For real numbers a<b, the number of upcrossings of X across the interval [a,b] is the maximum nonnegative integer n such that there exists times sk,tkβˆˆπ•‹ satisfying

s1<t1<s2<t2<β‹―<sn<tn (1)

and for which Xsk<a<b<Xtk. The number of upcrossings is denoted by U⁒[a,b]. Note that this is either a nonnegative integer or is infiniteMathworldPlanetmathPlanetmath. Similarly, the number of downcrossings, denoted by D⁒[a,b], is the maximum nonnegative integer n such that there are times sk,tkβˆˆπ•‹ satisfying (1) and such that Xsk>b>a>Xtk.

FigureΒ 1: Process with 3 upcrossings of the interval [a,b].

Note that between any two upcrossings there is a downcrossing. Given sk,tk satisfying (1) and Xsk<a<b<Xtk, we can set skβ€²=tk and tkβ€²=sk+1 for k=1,…,n-1. Then, Xskβ€²>b>a>Xtkβ€² from which we conclude that D⁒[a,b]β‰₯U⁒[a,b]-1. Similarly, we have U⁒[a,b]β‰₯D⁒[a,b]-1. Hence, the number of upcrossings and the number of downcrossings of an interval differ by at most 1.

For a finite index set 𝕋={1,2,…,N}, the number of upcrossings can be computed as follows. Set t0=0 and define s1,s2,… and t1,t2,… by


The number of upcrossings of [a,b] is then equal to the maximum n such that tn<∞ (see Figure 1).


  • 1 David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, 1991.
Title upcrossings and downcrossings
Canonical name UpcrossingsAndDowncrossings
Date of creation 2013-03-22 18:49:33
Last modified on 2013-03-22 18:49:33
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Definition
Classification msc 40A05
Classification msc 60G17
Related topic MartingaleConvergenceTheorem
Related topic ConvergenceOfASequenceWithFiniteUpcrossings
Defines upcrossing
Defines downcrossing