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

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


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

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:

Mathematics: https://math.stackexchange.com/questions/139408/good-reference-on-sample-autocorrelation

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

  • 1
    $\begingroup$ Don't cross-post right away like this. Think about what you want to learn from this question. Do you need something only a statistician could answer? Do you need an application to finance? Do you need to write a proof? They can't all be goals, so not all sites should receive this question. $\endgroup$ May 1, 2012 at 15:23
  • $\begingroup$ Your claim that lim_{n->inf} sqrt(n) r_hat is supposed to be an element of the normal distribution does not make sense mathematically. A distribution is not a set. If you mean its support, that is the real line, an element of which your limit surely is. But there is little value in that claim. Besides, regarding autocorrelation basically autoregressive time series and fractional Brownian motion (and numerical approximation thereof) come to my mind. But I'm not sure wether either topic fits your background and time frame. $\endgroup$
    – Konsta
    May 2, 2012 at 21:37

1 Answer 1


Three good references are the

  1. Asymptotic theory for econometricians, H. White
  2. Stochastic Limit Theory, Davidson
  3. Asymptotic Theory of Statistical Inference for Time Series, Taniguchi and Kakizawa

They are roughly in order of complexity.

The crux of the matter is to balance the requirements of finiteness of higher moments of $X$ with its dependence structure, or, put it differently, to balance the thickness of the tails with the memory of the process.

Don't peek into the abyss for too long.


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