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On exit times of a multivariate random walk and its embedding in a quasi Poisson process. (English) Zbl 1105.60065

This paper investigates a class of renewal processes marked by a multivariate sequence of independent random variables. In other words, the entire process forms a multivariate marked random walk. A single-variate random walk has been a prominent topic in probability for decades. Various problems are related to random walk’s behavior about critical levels, such as first passage times, when the random walk crosses constant lines, or lines with positive slopes. Let \(\mathcal F=\{\tau_0, \tau_1,..\}\) be a delayed renewal process marked by a multivariate random walk \((\mathcal G, \mathcal F)\). The authors derive the joint distribution of first passage time \(\tau_{\rho}\), pre-exit time \(\tau_{\rho-1}\), and the respective values of multivariate random walk \((\mathcal G, \mathcal F)\) at \(\tau_{\rho}\) and \(\tau_{\rho-1}\) in a closed form. Section 4 deals with a multivariate random walk embedded in a quasi-Poisson process \(\Pi\). The formalism of Section 4 is further enhanced in Section 5, where the information on \(\Pi\) is interpolated in vicinities of first passage and pre-exit times. These results generalize and refine earlier results [see R. Agarwal, J. H. Dshalalow and D. O’Regan, J. Math. Anal. Appl. 293, No. 1, 1–13 (2004; Zbl 1052.60036), ibid. 293, No. 1, 14–27 (2004; Zbl 1047.60005) and J. H. Dshalalow, J. Appl. Probab. 38, No. 3, 707–721 (2001; Zbl 0996.60097)].

MSC:

60K05 Renewal theory
60K10 Applications of renewal theory (reliability, demand theory, etc.)
60K25 Queueing theory (aspects of probability theory)
60G25 Prediction theory (aspects of stochastic processes)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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