Limit Behaviour of Equivalent Processes

Question

Let $(X_t)_{t\ge 0}$ and $(Y_t)_{t\ge 0}$ be two real-valued stochastic processes on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that their finite-dimensional distributions coincide.
Assume furthermore that $$ \limsup\limits_{t\to \infty} \; X_t \; = \; +\infty \quad \text{ and } \quad \liminf\limits_{t\to \infty} \; X_t \; = \; -\infty$$ almost surely.

I expect that in this case it also holds $$ \limsup\limits_{t\to \infty} \; Y_t \; = \; +\infty \quad \text{ and } \quad \liminf\limits_{t\to \infty} \; Y_t \; = \; -\infty$$ almost surely. Is this generally true?