# Sampling Distribution of Sample Correlation Coefficient

For a linear process $X_t=\mu+\sum_j\varphi_jW_{t-j}$

where $W_t$ is white noise and $\mathbb E(W_t^4)<\infty$ ,

$$\begin{pmatrix} \hat\rho(1) \\ \hat\rho(2) \\ \vdots \\ \hat\rho(k) \\ \end{pmatrix} \sim AN \begin{pmatrix} \begin{pmatrix} \rho(1) \\ \rho(2) \\ \vdots \\ \rho(k) \\ \end{pmatrix},\frac{1}{n}V \end{pmatrix}$$

Where $$V_{i,j}=\sum_{h=1}^{\infty}(\rho(h+i)+\rho(h-i)-2\rho(i)\rho(h))\times(\rho(h+j)+\rho(h-j)-2\rho(j)\rho(h))$$

And $AN$ denotes Asymptotic Normal.

• My first question is , why is the variance of $\rho=\frac{1}{n}V$ , that is , why have we multiplied $\frac{1}{n}$ with $V$ ?

• How does the formula of $V_{i,j}$ come ?

• It's Theorem 7.2.1. from Time Series: Theory and Methods by P. Brockwell and R. Davis, a very thorough proof is given in section 7.3 there. – Zhanxiong Jun 29 '15 at 13:11