I would be interested in knowing if the fact that returns are Gaussian is disproved on all time frames, or if, for example, the 5 minute intra-day time frame could exhibits Gaussian returns assuming there are no micro-structure issues (low volume).

More generally is there any relationship between the time frame of equity returns and their generating distribution? Any known research in this direction?

  • $\begingroup$ Does this question help you? $\endgroup$
    – Bob Jansen
    Jun 15, 2015 at 16:55
  • $\begingroup$ What does the Central Limit Theorem say ? $\endgroup$
    – nbbo2
    Jun 15, 2015 at 18:08
  • 3
    $\begingroup$ @noob2 please be constructive in comments. Rhetorical questions are never as clear as a gentle explanation of your point, and they can be offensive. Thanks $\endgroup$
    – SRKX
    Jun 16, 2015 at 6:38
  • $\begingroup$ Sorry, no offense intended, I just meant it as a friendly hint of the point Alexander Didenko makes below. I'll be more circumspect in future. $\endgroup$
    – nbbo2
    Jun 16, 2015 at 14:02
  • $\begingroup$ Yes, the variance ratio test is based upon the non-Gaussian scaling of aggregated data. Long story short, it's not always Gaussian, but nor is it completely unpredictable. "An Introduction to High-Frequency Finance" goes into the scaling properties of high-frequency data in some detail. You might also be interested in the "Epps effect" which distorts the autocorrelation and IID properties of high-frequency trade data. $\endgroup$ Jun 18, 2015 at 17:32

3 Answers 3


My main reference will be "Dan Xu, Christian Beck - Transition from lognormal to chi-square superstatistics for financial time series"

Non-equilibrium statistical mechanics (more specifically, superstatistics) gives some ideas of explaining the relation between time frame and its distribution: "...to regard the time series as a superposition of local Gaussian process weighted with a process of a slowly changing variance parameter"

In their article, the authors found out empirically that: "Chi-square superstatistics appears to best suitable for daily price change (assuming independent variation of volatility parameter in each interval), whereas on much smaller time scales of minutes, lognormal superstatistics seems preferrable"

There are couples of related articles on this topic:

  1. M. Ausloos and K. Ivanova - Dynamical model and nonextensive statistical mechanics of a market index on large time windows

  2. Katz, Y.A.; Tian, L. - Superstatistical fluctuations in time series of leverage returns

  3. S.M.D. Queirós and C. Tsallis - On the connection between financial processes with stochastic volatility and nonextensive statistical mechanics

  4. C. Becka, E.G.D. Cohen - Superstatistics

I'm not expert in this field, but hope the idea may help.


Surely, there is; search for aggregational gaussianity in Google Scholar or ScienceDirect.

In fact, 5 minutes returns are leptokurtic and fat-tailed; then as you increase timeframe, returns become more and more normal. Yearly data is almost normal, if you have enough points.

  • $\begingroup$ Don't leptokurtic and fat-tailed mean the same thing? Usually, it's better to give paper names rather that advising to go on some search engine and search for a term (although here the term is useful and part of the answer). The second part of your answer would benefit from citations as these are quite important "assertions" (I'm not saying they're wrong). $\endgroup$
    – SRKX
    Jun 16, 2015 at 6:36

If high frequency returns are iid and the mean and variance are finite and vthe variance is greater than zero then the Central Limit theorem holds Then, regardless of the distribution of the high returns, when aggregated over time the aggregated returns will tend in distribution to a Normal distribution. The Lindeberg-Lévy-Feller version of the Central Limit Theorem gives a generalization of this result to independent random variables. If the higher frequency returns have fat tailed distributions with density such that \begin{equation} f(x)\sim \begin{cases} B_{-}|x|^{-(1+a)}\quad \text{as} \quad x \rightarrow - \infty \\ B_{+}|x|^{-(1+a)}\quad \text{as} \quad x \rightarrow \ \infty, \end{cases} \end{equation} where $0<a<2$ and $B_{-}$ and B_{+} are appropriate constants then the aggregated distributions will tend to $\alpha$-stable by the Generalized Central limit Theorem. One should note that with the sample sizes available no normality test has power against the alternative of an $\alpha$-stable distribution.


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