Covariance between $X_i$ and $Y=\sum_{k=-\infty}^{\infty} a_i X_i$

Question

Let $X_i$, $i\in \mathbb{Z}$, be independent and identically Gaussian variables with mean $0$ and variance $\sigma_i^2$, $X_i\sim\mathcal{N}(0,\sigma_i^2)$, then $Y=\sum_{i=-\infty}^{+\infty}a_iX_i\sim\mathcal{N}(0,\sum_{i=-\infty}^{+\infty}a_i^2\sigma_i^2)$. How to compute the covariance between $X_i$ and $Y$?

Thanks you in advance for your answer.

Tuan

Answer 1

By using: $$\text{Cov}(X_i,\sum_ja_jX_j)=\sum_ja_j\text{Cov}(X_i,X_j)$$