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

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


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

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

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

  • 3
    $\begingroup$ Please don't simultaneously cross post. $\endgroup$
    – Andy W
    May 1, 2012 at 15:16

3 Answers 3


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

  • $\begingroup$ 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." $\endgroup$
    – dimitriy
    May 1, 2012 at 19:46

The assumptions are :we have i.i.d. observations (Xi, Yi), i = 1, . . . , n, for (X, Y ). Note that the the first 'i' means independent thus daily prices would be inapplicable due to their auto-correlative structure unless filtered by an ARIMA model with any necessary deterministic variables such as Pulses, Level Shifts , Seasonal Pulses and/or Local Time trends to render the process "identically distributed". You might want to look at some of my comments in Correlating volume timeseries . Pay close attention to the Yule paper (1926) that is cited as there continues to be a total oversell of the cross-correlation when dealing with non-Gaussian data, ignoring the critical assumptions that you wisely ask for. Your question is dead-on and deserves attention. You might also review a recent Masters Thesis at GSU Interval Estimation for the Correlation Coefficient by Aekyung Jung [email protected] at http://digitalarchive.gsu.edu/cgi/viewcontent.cgi?article=1109&context=math_theses. Hope this helps .


It might be worth checking out:

Analysis of Financial Time Series (2010) by Ruey S. Tsay


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