# Convergence equivalent random sequences

1. Suppose we have a sequence of independent random vars $X_n$ and consider a sequence of truncated random variables $Y_n=X_n1_{X_n\le n}$ s.t. $E[Y_n]=0$.

2. We know that $X_n$'s and $Y_n$'s are convergence equivalent:

$$\sum_{n=1}^\infty P(X_n\ne Y_n)<\infty$$

1. $\sum X_n$ converges to a nondegenerate random variable $X$.

2. Finally, for any $\epsilon>0$, $P(|\sum_{m=k}^\infty(X_m-Y_m)|>\epsilon)\rightarrow 0$ as $k\rightarrow \infty$.

In this settings we can say that for $\epsilon>0$ there exists $k$ s.t.

$$P(\sum_{m=k}^\infty X_m\le x)-\epsilon\le P(\sum_{m=k}^\infty Y_m\le x)\le P(\sum_{m=k}^\infty X_m\le x)+\epsilon$$

However, can we give a more precise bound on $P(\sum_{m=k}^\infty Y_m\le x)$. Especially, I'm interested in comparing the moments of limiting distributions.

Thank's!