Asymptotic behavior of the local score of independent and identically distributed random sequences

Let (Xn)n[greater-or-equal, slanted]1 be a sequence of real random variables. The local score is Hn=max1[less-than-or-equals, slant]i<j[less-than-or-equals, slant]n (Xi+...+Xj). If (Xn)n[greater-or-equal, slanted]1 is a "good" Markov chain under its invariant measure, the Xi are centered, we prove that converges in distribution to B1* when n-->+[infinity], where B1*=max0[less-than-or-equals, slant]u[less-than-or-equals, slant]1 Bu and (Bu,u[greater-or-equal, slanted]0) is a standard Brownian motion, B0=0. If (Xn)n[greater-or-equal, slanted]1 a sequence of i.i.d. random variables, and Var(X1)=[sigma]2>0, we prove the convergence of to [sigma][xi][delta]/[sigma] where [xi][gamma]=max0[less-than-or-equals, slant]u[less-than-or-equals, slant]1 {(B(u)+[gamma]u)-min0[less-than-or-equals, slant]s[less-than-or-equals, slant]u(B(s)+[gamma]s)}. We approximate the probability distribution function of [xi][gamma] and we determine the asymptotic behavior of P([xi][gamma][greater-or-equal, slanted]a), a-->+[infinity].