Extreme value theory for stochastic integrals of Legendre polynomials

For t ≥ 0, let X(t) = (X0(t), . . . , Xp(t)) , where Xi(t) denotes the integral of the ith order Legendre polyonimal with respect to the same Brownian motion described by the corresponding standard deviation (0 ≤ i ≤ p). We obtain the exact tail behavior of P ( sup0≤t≤h |X(t)| > u ) as u → ∞, and the limit distribution of sup0≤t≤T |X(t)| as T → ∞. These processes naturally arise in the context of polynomial regression models if one is interested in the limiting distribution of a test statistic designed to detect changes in the underlying regression parameters. AMS 2000 Subject Classification: Primary 62J02, Secondary 62J12, 60G70.