Records in the presence of a linear trend

Records from the sequence Yn = Xn + cn, n ≧ 1 are analyzed, where {Xn } is a strictly stationary random sequence. We prove various limit theorems for the record rate, record times, and record values. The situation when {Xn } is a stationary Gaussian process is considered with special attention given to Gaussian ARMA sequences. Data for the 400 and 800 metre races are used to illustrate these results.

[1]  Mark C. K. Yang.  On the distribution of the inter-record times in an increasing population , 1975 .

[2]  L. Rogers,et al.  The k-record processes are i.i.d. , 1984 .

[3]  P. Deheuvels.  The complete characterization of the upper and lower class of the record and inter-record times of an I.I.D. sequence , 1983 .

[4]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[5]  J. Kingman.  Subadditive Ergodic Theory , 1973 .

[6]  L. de Haan,et al.  On extreme-value theory in the presence of a trend , 1987, Journal of Applied Probability.

[7]  R. Shorrock.  A limit theorem for inter-record times , 1972 .

[8]  Ned Glick,et al.  Breaking Records and Breaking Boards , 1978 .

[9]  J. E. Miller,et al.  A non-Gaussian state space model and application to prediction of records , 1986 .

[10]  S. Resnick.  Record Values and Maxima , 1973 .

[11]  W. Stout.  Almost sure convergence , 1974 .

[12]  D. B. Preston.  Spectral Analysis and Time Series , 1983 .

[13]  R. Shorrock,et al.  Record values and inter-record times , 1973, Journal of Applied Probability.

[14]  I. Ibragimov,et al.  Independent and stationary sequences of random variables , 1971 .

[15]  S. Shapiro,et al.  An Analysis of Variance Test for Normality (Complete Samples) , 1965 .

[16]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[17]  Ken-ichi Yoshihara,et al.  Note on the law of the iterated logarithm for stationary processes satisfying mixing conditions , 1971 .

[18]  R. Shorrock.  On record values and record times , 1972, Journal of Applied Probability.

[19]  Records from improving populations , 1985 .

[20]  Extremal processes and record value times , 1973 .

[21]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[22]  S. Resnick.  Weak Convergence to Extremal Processes , 1975 .

[23]  T. Liggett.  An Improved Subadditive Ergodic Theorem , 1985 .