A central limit theorem for the optimal selection process for monotone subsequences of maximum expected length
Let \((N_s)_{s \geq 0}\) be a Poisson process with intensity 1 and with occurrence times \(0<T_1<T_2< \cdots \) a.s., \(T_0=0\). Let \((X_k)_{k=1,2, \ldots}\) be a sequence of i.i.d. uniform random variables on \([0,1]\), independent of the \(T_j\)'s. The random vectors \((T_k,X_k)\) are observed sequentially. For a given horizon \(t\), consider the objective to select in sequential order, without recall on preceding observation, a subsequence \((T_{k_1},X_{k_1}),(T_{k_2},X_{k_2}),\ldots\) satisfying \(X_{k_1}\leq X_{k_2} \leq \cdots\) for \(0<k_1<k_2<\cdots\) and \(T_{k_i}\leq t \), consisting of as many elements as possible. Denote by \(L^t_u\) the number of the selected values up to time \(u\) under the \(t\)-optimal strategy. The authors study the stochastic process \((L^t_u)_{0 \leq u \leq t}\). They prove a functional central limit theorem for this process as \(t\) tends to infinity.
scientific article; zbMATH DE number 2177810