The supremum of a process with stationary independent and symmetric increments

Let Xt, t [greater-or-equal, slanted] 0, be a process with stationary independent and symmetric increments. If the tail of the Levy spectral measure in the representation of the characteristic function is of regular variation of index -[alpha], for some 0 u) ~ P(Xt, > u), for u --> [infinity],for each t > 0.

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

[2]  On the Joint Distribution of a Process with Stationary Increments and Its Maximum , 1969 .

[3]  J. Doob.  Stochastic processes , 1953 .

[4]  S. Berman.  Sojourns and extremes of a diffusion process on a fixed interval , 1982, Advances in Applied Probability.

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

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

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

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

[9]  Sojourns and Extremes of Gaussian Processes , 1974 .

[10]  S. Berman.  Limit theorems for sojourns of stochastic processes , 1985 .

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

[12]  Sojourns and extremes of Fourier sums and series with random coefficients , 1983 .

[13]  S. Berman,et al.  The Maximum of a Gaussian Process with Nonconstant Variance: A Sharp Bound for the Distribution Tail , 1989 .

[14]  V. Zolotarev.  Addendum: Asymptotic Behavior of the Distributions of Processes with Independent Increments , 1965 .

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

[16]  Simeon M. Berman,et al.  Sojourns and Extremes of Stationary Processes , 1982 .

[17]  Simeon M. Berman.  AN ASYMPTOTIC FORMULA FOR THE DISTRIBUTION OF THE MAXIMUM OF A GAUSSIAN PROCESS WITH STATIONARY INCREMENTS , 1985 .