36
Views
2
CrossRef citations to date
0
Altmetric
Articles

On the limit properties of the last exit time and the first crossing point for the stationary dependent chi-sequences

& ORCID Icon
Pages 1237-1250 | Received 20 Nov 2020, Accepted 30 Apr 2021, Published online: 18 May 2021

References

  • Albin, J. M. P. 1987. On extremal theory for non differentiable stationary processes. PhD thesis, University of Lund.
  • Albin, J. P. M. 1990. On extremal theory for stationary processes. The Annals of Probability 18 (1):92–128. doi: 10.1214/aop/1176990940.
  • Albin, J. M. P., and D. Jarušková. 2003. On a test statistic for linear trend. Extremes 6 (3):247–58. doi: 10.1023/B:EXTR.0000031181.16968.17.
  • Albin, P., E. Hashorva, L. Ji, and C. Ling. 2016. Extremes and limit theorems for difference of chi-type processes. ESAIM: Probability and Statistics 20:349–66. doi: 10.1051/ps/2016018.
  • Aronowich, M., and R. J. Adler. 1986. Extrema and level crossings of χ2 processes. Advances in Applied Probability 18:901–20.
  • Burnashev, M. V., and G. K. Golubev. 2015. On limit distributions of the time of first passage over a high level. Problems of Information Transmission 51 (2):148–64. doi: 10.1134/S0032946015020064.
  • Hashorva, E., and L. Ji. 2015. Piterbarg theorems for chi-processes with trend. Extremes 18 (1):37–64. doi: 10.1007/s10687-014-0201-1.
  • Hüsler, J. 1977. On limit distributions of first crossing points of Gaussian sequences. Stochastic Processes and Their Applications 6 (1):65–75. doi: 10.1016/0304-4149(77)90018-7.
  • Hüsler, J. 1979a. Almost sure limiting behaviour of first crossing points of Gaussian sequences. Stochastic Processes and Their Applications 8 (3):315–21. doi: 10.1016/0304-4149(79)90006-1.
  • Hüsler, J. 1979b. The limiting behaviour of the last exit time for sequences of independent, identically distributed random variables. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 50 (2):159–64.
  • Hüsler, J. 1980. Limit distribution of the last exit time for stationary random sequences. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 52 (3):301–308.
  • Hüsler, J. 1981. The law of the iterated logarithm for the last exit time of independent random sequences. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 57 (3):397–405.
  • Jarušková, D., and V. I. Piterbarg. 2011. Log-likelihood ratio test for detecting transient change. Statistics & Probability Letters 81 (5):552–59. doi: 10.1016/j.spl.2011.01.006.
  • Leadbetter, M. R., G. Lindgren, and H. Rootzen. 1983. Extremes and related properties of random sequences and processes. New York: Springer-Verlag.
  • Lindgren, G. 1989. Slepian models for χ2-process with dependent components with application to envelope upcrossings. Journal of Applied Probability 26:36–49.
  • Ling, C., and Z. Tan. 2016. On maxima of chi-processes over threshold dependent grids. Statistics 50 (3):579–95. doi: 10.1080/02331888.2015.1083021.
  • Liu, P., and L. Ji. 2017. Extremes of locally stationary chi-square processes with trend. Stochastic Processes and Their Applications 127 (2):497–525. doi: 10.1016/j.spa.2016.06.016.
  • Malinovskii, V. K. 2018. Approximations in the problem of level crossing by a compound renewal process. Doklady Mathematics 98 (3):622–25. doi: 10.1134/S1064562418070232.
  • Piterbarg, V. I. 1994. High excursions for nonstationary generalized chi-square processes. Stochastic Processes and Their Applications 53 (2):307–37. doi: 10.1016/0304-4149(94)90068-X.
  • Piterbarg, V. I. 1996. Asymptotic methods in the theory of gaussian processes and fields. Providence, RI: AMS.
  • Sharpe, K. 1978. Some properties of the crossing process generated by a stationary χ2-process. Advances in Applied Probability 10:373–91.
  • Song, W., J. Shao, and Z. Tan. 2021. Almost sure central limit theorems for the maxima of Gaussian functions. Communications in Statistics-Theory and Methods. Advance online publication. doi: 10.1080/03610926.2020.1871019.
  • Tan, Z., and E. Hashorva. 2013a. Exact asymptotics and limit theorems for supremum of stationary χ-processes over a random interval. Stochastic Processes and Their Applications 123 (8):2983–98. doi: 10.1016/j.spa.2013.03.009.
  • Tan, Z., and E. Hashorva. 2013b. Limit theorems for extremes of strongly dependent cyclo-stationary χ-processes. Extremes 16 (2):241–54. doi: 10.1007/s10687-013-0170-9.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.