A functional limit theorem for observations that change with time (Q1770898)

Let \(\{T_k:k\geq 1\}\) be a unit's arrival sequence generated by a Poisson process \(\{N(t): t\geq 0\}\) having continuous intensity function \(\lambda(t)\). Let \((X_1,X_2,\dots)\) be an i.i.d. sequence of nonnegative random variables independent of \(\{N(t)\}\). Furthermore, let \(h(t)\) \((t\geq 0)\) be a nonnegative nonincreasing function with \(h(0)= 1\). Put \[ m(t)= \int^t_0\lambda(s)\,ds,\quad L_r(t)= \int^t_0 h^r(t- s)\lambda(s)\,ds,\quad \sigma^2_2(t) = L_2(t) EX^2- {(L_1(t)EX)^2\over m(t)}. \] The authors prove, among others, that under some conditions the process \[ \Biggl\{{1\over\sigma_2(n)} \Biggl\{\sum^{N(t)}_{k=1} h(t- T_k)X_k- {N(t)L_1(t)EX\over m(t)}\Biggr\};\,0\leq t\leq 1\Biggr\} \] converges weakly to some Gaussian process.