Distribution of time intervals determining the first instant when a stationary process reaches the set level
This paper describes a method of determining the rigorous analytical expression for the distribution of time instants when a stationary process differentiable in the root-mean-square sense attains threshold for the first time. It has been shown that this expression is a solution of the partial integral-differential equation corroborating earlier results.
S. O. Rice, “Mathematical Analysis of Random Noise,” BSTJ 23, No. 3 (1944); 24, No. 1 (1945).
B. R. Levin, Theoretical Foundations of the Statistical Radio Engineering, Book 1 (Sov. Radio, Moscow, 1969) [in Russian].
Ya. A. Fomin, The Theory of Surges of Random Processes (Svyaz’, Moscow, 1980) [in Russian].
B. R. Levin and Ya. A. Fomin, “Approximate determination of the distribution function of the length of the envelop surge for the sum of deterministic signal and normal stationary noise below the threshold level,” Radiotekhnika, No. 5 (1963).
B. R. Levin and Ya. A. Fomin, “Distribution of the duration of the threshold level exceedance in stationary random sequences,” in Proc. of XXI All-Union Scientific Session devoted to the 70th anniversary of radio invention by A. S. Popov: Division of the Theory of Information (1965).
D. V. Evgrafov, “The Integro-Differential Equation for Distribution of the Absolute Maximum of a Gaussian Stationary Process,” Izv. Vyssh. Uchebn. Zaved., Radioelektron. 46(5), 25 (2003); Radioelectron. Commun. Syst. 46(5), 18 (2003).
D. V. Evgrafov, “Distributions of the Absolute Maximum of a Stationary Process Differentiable in the Mean-Square Sense,” Izv. Vyssh. Uchebn. Zaved., Radioelektron. 46(6), 76 (2003); Radioelectron. Commun. Syst. 46(6), 53 (2003).
D. V. Evgrafov, “Distribution of the absolute maximum in the theory of signal detection,” in MOU, NAOU. Trudy Akademii (MOU, NAOU, Kiev, 2005), No. 65, pp. 86–89.
V. F. Shukailo, “On distribution of the absolute maximum of the stationary random process,” Radiotekh. Elektron. 13, No. 6, 996 (1968).