Quantile lines of stationary process

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Quantile lines of any stationary process are parallel and constant. But for different procesess I've obtained different behavior of quantile lines. First case was process after Lamperti transormation of alpha-stable motion. In that case quantile lines are parallel from the beginning. The second case was Orstein-Uhlenbeck process(simulated from stochastic differential equation), which is stationaty and I've obtained quantile lines which becomes parallel after some time. Why am I observing such behavior?