ABSTRACT
Let X be a random vector in R n , and f(t), 0 ≤ t ≤ 1, a measurable function into R n ; and define the real stochastic process X t = (X, f(t)) where (·,·) is the inner product function. It is assumed that X has a distribution that is invariant under all orthogonal transformations of R n , and that ||f(t)|| = 1, where || · || represents the Euclidean norm. In this case, it can be shown that the marginal distributions of X t , 0 ≤ t ≤ 1, are identical, although the process is not necessarily stationary. This chapter contains a detailed analysis of the tail of the distribution of max0≤t≤1 X t and the asymptotic form of the sojourn time distribution above u, for u → ∞.
