Suppose $Z , \epsilon \sim N(0, 1)$ are independent Gaussian random variables. Let $a \ll 1$ be a small positive number. Let $W = a \cdot Z + \epsilon$. It can be show that \begin{align} \mathbb{E} [ W^2 (Z^2 - 1)] = 2 a^2. \end{align} Now suppose I truncate random variable $W$. I was wondering what truncation level $R$ should be such that \begin{align} \mathbb{E} [ W^2 \cdot \mathbf{1}\{ | W| > R\} \cdot (Z^2 - 1)] \leq a^2. \end{align} Can we set $R$ to be a constant? This seems a easy problem, but computing the expectation is seems not trivial.

Suppose $Z , \epsilon \sim N(0, 1)$ are independent Gaussian random variables. Let $a \ll 1$ be a small positive number. Let $W = aZ + \epsilon$. It can be show that \begin{align} \mathbb{E} [ W^2 (Z^2 - 1)] = 2 a^2. \end{align} Now suppose I truncate random variable $W$. I was wondering what truncation level $R$ should be such that \begin{align} \mathbb{E} [ W^2 \cdot \mathbf{1}\{ | W| > R\} \cdot (Z^2 - 1)] \leq a^2. \end{align} Can we set $R$ to be a constant? This seems an easy problem, but computing the expectation seems not trivial.