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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

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

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).

Thank you in advance!

Crossposting at:

Statistics: http://stats.stackexchange.com/questions/27465/good-reference-on-sample-autocorrelation

Quantative Finance: http://quant.stackexchange.com/questions/3390/good-reference-on-sample-autocorrelation

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It's poor form to cross post (as you have already been told in both other forums). I don't really care but some people do. As far as your question, in my time series class, we have been told that stock returns (daily change) is almost always white noise, i.e., a trivial model, i.e., they are iid normal with mean 0 and constant variance. And, in the project I did earlier in the semester, this is exactly what I found to be true in the data I looked at. It's so simple, my professor is considering not allowing stock data. Unless I misunderstand, I think what you're doing is way too complex. –  Graphth May 1 '12 at 15:45

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