Unilateral estimate for the supremum distribution of certain processes (Q1110904)

The author presents an estimate of the probability \(P\{\) \(\sup_{0\leq s\leq t}X(s)>u\}\), \(u>0\), for a stochastically continuous separable process \(\{\) X(t), \(t\geq 0\}\) with stationary increments satisfying \(X(0)=0\), E X(t)\(=0\) and the following three conditions: \[ E| X(t)- X(s)| \leq Kq(t-s),\quad 0\leq s\leq t, \] \[ P\{X(t)-X(s)>q(t- s)x\}<AP\{X(1)>x\},\quad 0\leq s\leq t,\quad x\geq 0, \] \[ P\{X(1)>x\}\sim Cx^{\beta}\exp (-Bx^{\gamma})\quad as\quad x\to \infty, \] where q is a continuous, increasing function on [0,\(\infty)\) such that \(q(0)=0\), \(q(1)=1\), and \(A,B,C,K,\gamma >0\) and \(\beta\) are constants. This estimate was obtained by \textit{S. M. Berman} [Ann. Probab. 2, 999- 1026 (1974; Zbl 0298.60026)] for Gaussian processes. The result is applied to obtain an upper function for X(t).