Level-crossing problems for random processes

In a variety of practical problems involving random processes, it is necessary to have statistical information on their level-crossing properties. This paper presents a survey of known results on certain aspects of this problem and provides a basis for further study in the area. The goal has been to give a broad view of the problems considered in the literature and a brief indication of the techniques used in their solution. Much material of a more or less historical nature has been included since, to the authors' knowledge, no other survey of this nature exists.

[1]  S. Rice The Distribution of the Maxima of a Random Curve , 1939 .

[2]  P. Levy Sur certains processus stochastiques homogènes , 1940 .

[3]  M. Kac A correction to “On the average number of real roots of a random algebraic equation” , 1943 .

[4]  S. Rice Mathematical analysis of random noise , 1944 .

[5]  G. Uhlenbeck,et al.  On the Theory of the Brownian Motion II , 1945 .

[6]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[7]  M. Kac Random Walk and the Theory of Brownian Motion , 1947 .

[8]  P. Erdös,et al.  On the number of positive sums of independent random variables , 1947 .

[9]  M. Kac,et al.  On the Theory of Noise in Radio Receivers with Square Law Detectors , 1947 .

[10]  S. O. Rice,et al.  Statistical properties of a sine wave plus random noise , 1948, Bell Syst. Tech. J..

[11]  M. Kac On distributions of certain Wiener functionals , 1949 .

[12]  M. Kac On Some Connections between Probability Theory and Differential and Integral Equations , 1951 .

[13]  A. Siegert On the First Passage Time Probability Problem , 1951 .

[14]  F. W. Lehan Expected Number of Crossings of Axis by Linearly Increasing Function Plus Noise , 1951 .

[15]  F. David A note on the evaluation of the multivariate normal integral , 1953 .

[16]  D. Darling,et al.  THE FIRST PASSAGE PROBLEM FOR A CONTINUOUS MARKOFF PROCESS , 1953 .

[17]  R. Plackett A REDUCTION FORMULA FOR NORMAL MULTIVARIATE INTEGRALS , 1954 .

[18]  J. McFadden URN Models of Correlation and a Comparison with the Multivariate Normal Integral , 1955 .

[19]  P. Schultheiss,et al.  Short‐Time Frequency Measurement of Narrow‐Band Random Signals by Means of a Zero Counting Process , 1955 .

[20]  C. Dunnett,et al.  Approximations to the probability integral and certain percentage points of a multivariate analogue of Student's t-distribution* , 1955 .

[21]  L. Takács Investigation of waiting time problems by reduction to Markov processes , 1955 .

[22]  P. Erdös,et al.  A limit theorem for the maximum of normalized sums of independent random variables , 1956 .

[23]  D. S. Palmer Properties of random functions , 1956 .

[24]  G. Baxter Wiener process distributions of the “arc-sine law” type , 1956 .

[25]  F. Spitzer A Combinatorial Lemma and its Application to Probability Theory , 1956 .

[26]  M. Longuet-Higgins,et al.  The statistical distribution of the maxima of a random function , 1956, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[27]  Carl W. Helstrom The distribution of the number of crossings of a Gaussian stochastic process , 1957, IRE Trans. Inf. Theory.

[28]  M. Donsker,et al.  On the distribution of the supremum functional for processes with stationary independent increments , 1957 .

[29]  S. O. Rice,et al.  Distribution of the duration of fades in radio transmission: Gaussian noise model , 1958 .

[30]  J. McFadden The fourth product moment of infinitely clipped noise , 1958 .

[31]  M. Longuet-Higgins On the intervals between successive zeros of a random function , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[32]  Arnold J. F. Siegert,et al.  A systematic approach to a class of problems in the theory of noise and other random phenomena-III: Examples , 1957, IRE Trans. Inf. Theory.

[33]  U. Grenander,et al.  Statistical analysis of stationary time series , 1958 .

[34]  E. Parzen,et al.  Principles and applications of random noise theory , 1959 .

[35]  G. Turin,et al.  Principles and applications of random noise theory , 1959 .

[36]  D. Slepian,et al.  Large Excursions of Gaussian Processes , 1959 .

[37]  P. Mazur On the theory of brownian motion , 1959 .

[38]  V. Volkonskii,et al.  Some Limit Theorems for Random Functions. II , 1959 .

[39]  J. Gani,et al.  THE CONTENT OF A DAM AS THE SUPEREMUM OF AN INFINITELY DIVISIBLE PROCESS , 1960 .

[40]  V. Ivanov On the Average Number of Crossings of a Level by the Sample Functions of a Stochastic Process , 1960 .

[41]  D. Slepian First Passage Time for a Particular Gaussian Process , 1961 .

[42]  E. Bulinskaya On the Mean Number of Crossings of a Level by a Stationary Gaussian Process , 1961 .

[43]  J. A. McFadden The axis crossings of a stationary Gaussian Markov process , 1961, IRE Trans. Inf. Theory.

[44]  George Weiss,et al.  The Passage Problem for a Stationary Markov Chain , 1961 .

[45]  Attilio J. Rainal Zero-crossing intervals of Gaussian processes , 1962, IRE Trans. Inf. Theory.

[46]  M. Kac Probability Theory: Its Role and Its Impact , 1962 .

[47]  H. McKean A winding problem for a resonator driven by a white noise , 1962 .

[48]  D. Slepian The one-sided barrier problem for Gaussian noise , 1962 .

[49]  H. Cramér On the maximum of a normal stationary stochastic process , 1962 .

[50]  G. F. Newell,et al.  Zero Crossing Probabilities for Gaussian Stationary Processes , 1962 .

[51]  D. Burkholder Successive Conditional Expectations of an Integrable Function , 1962 .

[52]  S. Berman A Law of Large Numbers for the Maximum in a Stationary Gaussian Sequence , 1962 .

[53]  M. Longuet-Higgins The distribution of intervals between zeros of a stationary random function , 1962, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[54]  S. Gupta Bibliography on the Multivariate Normal Integrals and Related Topics , 1963 .

[55]  S. Gupta Probability Integrals of Multivariate Normal and Multivariate $t^1$ , 1963 .

[56]  J. Keilson The First Passage Time Density for Homogeneous Skip-free Walks on the Continuum , 1963 .

[57]  B. A. Chartres A Geometrical Proof of a Theorem Due to D. Slepian , 1963 .

[58]  Kiyosi Itô The expected number of zeros of continuous stationary Gaussian processes , 1963 .

[59]  V. Zolotarev The First Passage Time of a Level and the Behavior at Infinity for a Class of Processes with Independent Increments , 1964 .

[60]  D. Blackwell,et al.  A Remark on the Coin Tossing Game , 1964 .

[61]  S. Berman Limit Theorems for the Maximum Term in Stationary Sequences , 1964 .

[62]  Nelson M. Blachman FM reception and the zeros of narrow-band Gaussian noise , 1964, IEEE Trans. Inf. Theory.

[63]  Andrew J. Viterbi,et al.  The Effect of Sequential Decision Feedback on Communication over the Gaussian Channel , 1965, Inf. Control..

[64]  M. R. Leadbetter On crossings of arbitrary curves by certain Gaussian processes , 1965 .

[65]  M. R. Leadbetter,et al.  The Moments of the Number of Crossings of a Level by a Stationary Normal Process , 1965 .

[66]  H. Cramér A Limit Theorem for the Maximum Values of Certain Stochastic Processes , 1965 .

[67]  J. McFadden,et al.  Certain Properties of Gaussian Processes and Their First‐Passage Times , 1965 .

[68]  M. R. Leadbetter,et al.  The variance of the number of zeros of a stationary normal process , 1965 .

[69]  S. M. Cobb,et al.  The distribution of intervals between zero crossings of sine wave plus random noise and allied topics , 1965, IEEE Trans. Inf. Theory.

[70]  M. Shur On the Maximum of a Gaussian Stationary Process , 1965 .

[71]  N. Ylvisaker The Expected Number of Zeros of a Stationary Gaussian Process , 1965 .

[72]  H. Robbins,et al.  Moments of Randomly Stopped Sums , 1965 .

[73]  B. A. Rogozin On Some Classes of Processes with Independent Increments , 1965 .

[74]  A. Borovkov On the First Passage Time for One Class of Processes with Independent Increments , 1965 .

[75]  L. Takács On the distribution of the supremum for stochastic processes with interchangeable increments , 1965 .

[76]  E. Shtatland On the Distribution of the Maximum of a Process with Independent Increments , 1965 .

[77]  Andrew J. Viterbi,et al.  Principles of coherent communication , 1966 .

[78]  H. Teicher,et al.  ON SECOND MOMENTS OF STOPPING RULES , 1966 .

[79]  B. A. Rogozin On Distributions of Functionals Related to Boundary Problems for Processes with Independent Increments , 1966 .

[80]  M. R. Leadbetter On Crossings of Levels and Curves by a Wide Class of Stochastic Processes , 1966 .

[81]  H. Cramér On the intersections between the trajectories of a normal stationary stochastic process and a high level , 1966 .

[82]  E. Shtatland The Distribution of the Time the Maximum is Achieved for Processes with Independent Increments , 1966 .

[83]  N. Ylvisaker On a Theorem of Cramer and Leadbetter , 1966 .

[84]  Y. Belyaev On the Number of Intersections of a Level by a Gaussian Stochastic Process. I , 1966 .

[85]  L. Shepp Radon-Nikodym Derivatives of Gaussian Measures , 1966 .

[86]  E. Wong Some Results Concerning the Zero-Crossings of Gaussian Noise , 1966 .

[87]  Y. Belyaev On the Number of Level Crossings by a Gaussian Stochastic Process, II , 1967 .

[88]  J. McFadden,et al.  Multivariate normal integrals for highly correlated samples from a wiener process , 1967, Journal of Applied Probability.

[89]  L. Breiman First exit times from a square root boundary , 1967 .

[90]  L. Shepp A First Passage Problem for the Wiener Process , 1967 .

[91]  J. Beekman Gaussian-Markov processes and a boundary value problem , 1967 .

[92]  M. R. Leadbetter A note on the moments of the number of axis-crossings by a stochastic process , 1967 .

[93]  harald Cramer,et al.  Stationary And Related Stochastic Processes , 1967 .

[94]  D. Ylvisaker A Note on the Absence of Tangencies in Gaussian Sample Paths , 1968 .

[95]  C. Qualls On a Limit Distribution of High Level Crossings of a Stationary Gaussian Process , 1968 .

[96]  N. A. Strakhov,et al.  An upper bound on the zero-crossing distribution , 1968 .

[97]  D. Gusak,et al.  On the First Passage Time Across a Given Level for Processes with Independent Increments , 1968 .

[98]  L. Stone On the Distribution of the Maximum of a Semi-Markov Process , 1968 .

[99]  Y. Belyaev On the Number of Exits Across the Boundary of a Region by a Vector Stochastic Process , 1968 .

[100]  J. Pickands Upcrossing probabilities for stationary Gaussian processes , 1969 .

[101]  L. Stone On the Distribution of the Supremum Functional for Semi-Markov Processes with Continuous State Spaces , 1969 .

[102]  Carl W. Helstrom,et al.  A bound on the probability that a Gaussian process exceeds a given function (Corresp.) , 1969, IEEE Trans. Inf. Theory.

[103]  T. Malevich Asymptotic Normality of the Number of Crossings of Level Zero by a Gaussian Process , 1969 .

[104]  B. A. Rogozin,et al.  On Joint Distributions of Random Variables Associated with Fluctuations of a Process with Independent Increments , 1969 .

[105]  Yu. M. Ryzhov Limit Distributions of Some Functionals of a Stationary Gaussian Process , 1969 .

[106]  D. Gusak On the Joint Distribution of the First Exit Time and Exit Value for Homogeneous Processes With Independent Increments , 1969 .

[107]  Y. Belyaev,et al.  Characteristics of Excursions above a High Level for a Gaussian Process and Its Envelope , 1969 .

[108]  H. Robbins,et al.  Boundary Crossing Probabilities for the Wiener Process and Sample Sums , 1970 .

[109]  L. Stone,et al.  Distribution of time above a threshold for Markov processes , 1970 .

[110]  P. B. M. Roes On the expected number of crossings of a level in certain stochastic processes , 1970 .

[111]  E. Wong The Distribution of Intervals Between Zeros for a Stationary Gaussian Process , 1970 .

[112]  C. Qualls On the joint distribution of crossing of high multiple levels by a stationary gaussian process , 1970 .

[113]  S. Berman Asymptotic Independence of the Numbers of High and Low Level Crossings of Stationary Gaussian Processes , 1971 .

[114]  L. Shepp First Passage Time for a Particular Gaussian Process , 1971 .

[115]  S. Berman Excursions above high levels for stationary Gaussian processes. , 1971 .

[116]  L. J. Savage,et al.  Inequalitites on the probability content of convex regions for elliptically contoured distributions , 1972 .

[117]  William C. Lindsey,et al.  SYNCHRONIZATION SYSTEMS in Communication and Control , 1972 .

[118]  S. Berman A class of limiting distributions of high level excursions of Gaussian processes , 1972 .