Discrete wave-analysis of continuous stochastic processes

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he behaviour of a continuous-time stochastic process in the neighbourhood of zero-crossings and local maxima is compared with the behaviour of a discrete sampled version of the same process.For regular processes, with finite crossing-rate or finite rate of local extremes, the behaviour of the sampled version approaches that of the continuous one as the sampling interval tends to zero. Especially the zero-crossing distance and the wave-length (i.e., the time from a local maximum to the next minimum) have asymptotically the same distributions in the discrete and the continuous case. Three numerical illustrations show that there is a good agreement even for rather big sampling intervals.For non-regular processes, with infinite crossing-rate, the sampling procedure can yield useful results. An example is given in which a small irregular disturbance is superposed over a regular process. The structure of the regular process is easily observable with a moderate sampling interval, but is completely hidden with a small interval.
Original languageEnglish
Pages (from-to)83-105
JournalStochastic Processes and their Applications
Issue number1
Publication statusPublished - 1973

Subject classification (UKÄ)

  • Probability Theory and Statistics

Free keywords

  • stationary processes
  • crossing problems
  • wave-length
  • sampling of continuous processes
  • maxima of Gaussian processes


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