Convergence of Longuet-Higgins Series for Stationary Gaussian Markov Processes of First Order
Roman Nickolaevich Miroshin · Theory of Probability and Its Applications · 1981
Previous article Next article Convergence of Longuet-Higgins Series for Stationary Gaussian Markov Processes of First OrderR. N. MiroshinR. N. Miroshinhttps://doi.org/10.1137/1126008PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] E. Wong, Some results concerning the zero-crossings of Gaussian noise, SIAM J. Appl. Math., 14 (1966), 1246–1254 10.1137/0114099 34:6875 0147.15905 LinkGoogle Scholar[2] Mark Kac and , David Slepian, Large excursions of Gaussian processes, Ann. Math. Statist., 30 (1959), 1215–1228 22:6024 0089.34101 CrossrefGoogle Scholar[3] E. Wong, The distribution of intervals between zeros for a stationary Gaussian process, SIAM J. Appl. Math., 18 (1970), 67–73 10.1137/0118008 41:1112 0221.60024 LinkGoogle Scholar[4] S. O. Rice, Mathematical analysis of random noise, Bell System Tech. J., 24 (1945), 46–156 6,233i 0063.06487 S. O. Rice, Mathematical analysis of random noise, Bell System Tech. J., 23 (1944), 282–332 6,89b 0063.06485 CrossrefGoogle Scholar[5] M. S. Longuet-Higgins, The distribution of intervals between zeros of a stationary random function, Philos. Trans. Roy. Soc. London Ser. A, 254 (1961/1962), 557–599 28:1654 0107.35201 CrossrefGoogle Scholar[6] R. N. Miroshin, Convergence of Rice and Longuet-Higgins series for a Wong process, Theory Prob. Appl., 21 (1976), 863–866 0374.60049 LinkGoogle Scholar[7] J. Doob, Stochastic processes, John Wiley & Sons Inc., New York, 1953viii+654 15,445b 0053.26802 Google Scholar[8] R. N. Miroshin, Second-order Markov and reciprocal stationary processes, Theory Prob. Appl., 24 (1979), 845–851 0441.60038 LinkGoogle Scholar[9] R. N. Miroshin, Asymptotics of the second factorial moment of the number of upward crossings of the line $kt+a$ by a Gaussian stationary process, Vestnik Leningrad. Univ., 23 (1968), 109–120, (In Russian.) 38:3916 Google Scholar[10] H. P. McKean, A winding problem for a resonator driven by a white noise, J. Math. Kyoto Univ., 2 (1963), 227–235 27:6312 0119.34701 CrossrefGoogle Scholar[11] G. Bateman and , A. Erdelyi, Tables of Integral Transforms, McGraw-Hill, New York, 1954 Google Scholar[12] I. S. Gradshtein and , I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1980xv+1160 81g:33001 Google Scholar[13] M. A. Evgrafov, Asymptotic estimates and entire functions, Translated by Allen L. Shields. Russian Tracts on Advanced Mathematics and Physics, Vol. IV, Gordon and Breach, Inc., New York, 1961x+181 31:1376 0121.30202 Google Scholar[14] Jack Cuzick, Local nondeterminism and the zeros of Gaussian processes, Ann. Probability, 6 (1978), 72–84 58:7813 0374.60051 CrossrefGoogle Scholar[15] Jack Cuzick, A lower bound for the prediction error of stationary Gaussian processes, Indiana Univ. Math. J., 26 (1977), 577–584 55:11364 0367.60040 CrossrefGoogle Scholar[16] Harald Cramér, , M. R. Leadbetter and , R. J. Serfling, On distribution function—moment relationships in a stationary point process, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 18 (1971), 1–8 44:7633 0195.47603 CrossrefGoogle Scholar[17] V. A. Gaganov, Estimates of the factorial moments of the number of crossings of the zero level by a Gaussian stationary process, Vestnik Leningrad. Univ. No. 1 Mat. Meh. Astronom., (1973), 136–138, 157, (In Russian.) 48:9829 0309.60028 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails On some solutions to the Chapman-Kolmogorov integral equationVestnik St. Petersburg University: Mathematics, Vol. 40, No. 4 | 1 Dec 2007 Cross Ref A Survey of Recent Progress on Level-Crossing Problems for Random ProcessesCommunications and Networks | 1 Jan 1986 Cross Ref The Use of Rice SeriesR. N. MiroshinTheory of Probability & Its Applications, Vol. 28, No. 4 | 28 July 2006AbstractPDF (1011 KB) Volume 26, Issue 1| 1981Theory of Probability & Its Applications1-217 History Submitted:03 October 1978Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1126008Article page range:pp. 97-117ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics