# Good reference on sample autocorrelation?

I'm not a statistician but I'm writing my thesis on mathematical finance and I think it would be neat to have a short section about independence of stock returns. I need to get better understanding about some assumptions (see below) and have a good book to cite.

I have a model for stock prices $S$ in which the daily ($t_i - t_{i-1}=1$) log-returns

$$X_n = \ln\left(\frac{S(t_n)}{S(t_{n-1})}\right), \ \ n=1,...,N$$

are normally distributed with mean $\mu-\sigma^2/2$ and variance $\sigma^2$. The autocorrelation function with lag 1 is

$$r = \frac{\mathrm{Cov}(X_1,X_2)}{\mathrm{Var}(X_1)}$$

which I estimate by

$$\hat{r} = \frac{(n+1)\sum_{i=1}^{n-1} \bigl(X_i - \bar{X} \bigr)\bigl(X_{i+1} - \bar{X} \bigr)}{n \sum_{i=1}^{n}\bigl(X_i - \bar{X} \bigr)^2}$$

where

$$\bar{X} = \frac{1}{n}\sum_{i=1}^N X_i$$

Now I understand that under some some assumptions it holds that

$$\lim_{n \rightarrow \infty} \sqrt{n}\hat{r} \in N(0,1)$$

I would be very glad if someone could point me towards a good book which I can cite in my thesis and read about these assumptions (I guess it has something to do with the central limit theorem).

Crossposting at:

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Please don't simultaneously cross post. –  Andy W May 1 '12 at 15:16

Van der Vaart's time series lecture notes are excellent, and chapter 5 has exactly the sort of theorem you're looking for.

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Really excellent notes! Very witty too: "Many financial time series exhibit an exponential trend, not always in the right direction for the owners of the corresponding assets." –  Dimitriy V. Masterov May 1 '12 at 19:46