Patterns for the waiting time in the context of discrete-time stochastic processes

The aim of this study is to extend the scope and applicability of the level-crossing method to discrete-time stochastic processes and generalize it to enable us to study multiple discrete-time stochastic processes. In previous versions of the level-crossing method, problems with it correspond to the fact that this method had been developed for analyzing a continuous-time process or at most a multiple continuous-time process in an individual manner. However, since all empirical processes are discrete in time, the already-established level-crossing method may not prove adequate for studying empirical processes. Beyond this, due to the fact that most empirical processes are coupled; their individual study could lead to vague results. To achieve the objectives of this study, we first find an analytical expression for the average frequency of crossing a level in a discrete-time process, giving the measure of the time experienced for two consecutive crossings named as the “waiting time.” We then introduce the generalized level-crossing method by which the consideration of coupling between the components of a multiple process becomes possible. Finally, we provide an analytic solution when the components of a multiple stochastic process are independent Gaussian white noises. The comparison of the results obtained for coupled and uncoupled processes measures the strength and efficiency of the coupling, justifying our model and analysis. The advantage of the proposed method is its sensitivity to the slightest coupling and shortest correlation length.

Figure 1

The average frequency of the upcrossings of the level $x$ for various values of correlation $\rho$ in a discrete-time stationary Gaussian process.

Figure 2

A schematic illustration of the radial (a) and angular (b) level crossings. Panel (a): A cut of a whole sphere limited to the positive axis of the Cartesian coordinates. The dashed solid line sketches a path taken by a vector stochastic process named $\left\{\mathbf{X}\right\}$. Panel (b): A cone where its axis overlaps with one of the Cartesian axes. Note that for both panels the black points are created when the path crosses the surface of the shapes, and the dashed and solid parts of the line indicate whether its inside or outside the shapes.

Figure 3

Sensitivity of the two dimensional level-crossing analysis on coupling with short correlation lengths. Panels (a) and (b) are the result of comparison between two uncoupled white noises and two coupled white noises, respectively, in the radial and angular level-crossing regimes.