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

Let (Xn)n⩾1 be a sequence of real random variables. The local score is Hn=max1⩽i<j⩽n (Xi+⋯+Xj). If (Xn)n⩾1 is a “good” Markov chain under its invariant measure, the Xi are centered, we prove that Hn/n converges in distribution to B1∗ when n→+∞, where B1∗=max0⩽u⩽1 |Bu| and (Bu,u⩾0) is a standard Brownian motion, B0=0. If (Xn)n⩾1 a sequence of i.i.d. random variables, E(X1)=δ/n and Var(X1)=σ2>0, we prove the convergence of Hn/n to σξδ/σ where ξγ=max0⩽u⩽1 {(B(u)+γu)−min0⩽s⩽u(B(s)+γs)}. We approximate the probability distribution function of ξγ and we determine the asymptotic behavior of P(ξγ⩾a), a→+∞.