2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74619Motivated by regularities observed in time series of returns on speculative assets, we develop an asymptotic theory of GARCH(1,1) processes {y_k} defined by the equations y_k=σ_kε_k, σ_k^2=ω+αy_{k-1}^2+βσ_{k-1}^2 for which the sum α+βapproaches unity as the number of available observations tends to infinity. We call such sequences near-integrated. We show that the asymptotic behavior of near-integrated GARCH(1,1) processes critically depends on the sign of γ:=α+β-1. We find assumptions under which the solutions exhibit increasing oscillations and show that these oscillations grow approximately like a power function if γ\leq 0 and exponentially if γ>0. We establish an additive representation for the near-integrated GARCH(1,1) processes which is more convenient to use than the traditional multiplicative Volterra series expansion.Published at http://dx.doi.org/10.1214/105051604000000783 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)Probability62M10, 91B84. (Primary)text