Denote by $conv(X_n,X_{n+1},...)$ the set of convex combinations of the random variables $(X_n)_{n \in \mathbb{N}}$, i.e. it contains the elements $\sum_{i=n}^\infty \lambda_i X_i$ for $\lambda_k \geq 0$, $\sum_{i=n}^\infty \lambda_i$ and all but finitely many $\lambda_i$ are not zero.
Komlós Lemma states the follwing
Lemma: Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables. Then there exist convex combinations $X'_n \in conv(X_n,X_{n+1},...)$ and a random variable $X$, such that $$X'_n \rightarrow X \text{, $P$-a.s.}$$
My question is now: If I have two sequences $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ is it possible to choose the same convex combinations such that $X'_n \rightarrow X'$ and $Y'_n \rightarrow Y'$ for some random variables $X'$ and $Y'$? By same convex combinations I mean that the $\lambda's$ in the convex combinations are the same for every $X'_n$ and $Y'_n$.
Thanks a lot in advance!