Statement about convergence in probability

Question

If $X_1, X_2, ..., X_n, ...$ are same distributed and independent random variables and $n \mathbb P (|X_1| > n) \to 0$, $a_n = \mathbb E\left[X_1I_{|X_1| < n}\right]$, does this imply that $X_1+X_2+...+X_n \over n$ $- a_n \to^{\mathbb P} 0$? How to prove it?

Answer 1

For a fixed $n$, let $X'_{k,n}:= X_kI_{\{\lvert X_k\rvert\leqslant n\}}$ and $X''_{k,n}:= X_kI_{\{\lvert X_k\rvert\leqslant n\}}$.

1. Since $X_k=X'_{k,n}+X''_{k,n}$ it suffices to prove that $n^{-1}\sum_{k=1}^n X'_{k,n}-\mathbb E\left[X'_{k,n}\right]\to 0$ in probability and $n^{-1}\sum_{k=1}^n X''_{k,n}\to 0$ in probability.
2. For the first part, look at the moment of order two. You will need an estimate for $\mathbb E\left[{X'}_{1,n}^2\right]$.
3. The event $\{n^{-1}\sum_{k=1}^n X''_{k,n}\neq 0\}$ is contain in $\bigcup_{k=1}^n\{ \lvert X_k\rvert\gt n\}$.