Scaling of the distribution of fluctuations
The following two papers analyze the scaling property of index returns and individual stock returns :

Scaling of the distribution of fluctuations of financial market indices

Scaling of the distribution of price fluctuations of individual companies
In this post, I will briefly summarize the main contents of the papers.
Scaling of the distribution of fluctuations of financial market indices :
It is well known that index returns and stock return distributions observed in reality are far from Gaussian. It was Mandelbrot who analyzed cotton prices and observed time scaling property – distribution of returns for various choices of time intervals have similar functional forms. He called it a Levy stable distribution. With the available high frequency data, various empirical studies have shown that distribution of returns decay faster than predicted by a Levy distribution. Hence a truncated Levy distribution has been proposed – the central part obeys Levy. The exponential truncation ensures that there is a finite second moment. This means that truncated Levy process with i.i.d. random variables should converge to Gaussian.
So, a natural question that arises is – At what time scale does the return distribution move away from a particular functional form towards Gaussian ?
The authors analyze 13 years of S&P index, 14 years of NIKKEI index data and 18 years of HangSeng index data at various time scales and observe the following :

At 1 minute scales, return distributions are outside the Levy stable range. Only the central region obeys stable Levy distribution

For time scales from 5 minutes to 1 day, the functional form does not vary significantly

Analyzing the various moments for normalized returns for time scales ranging from 5 minutes to 1 day, the authors find that the estimated moments are nowhere close to Gaussian moments

Beyond 4 day time scales, the moments of returns start slowly converging towards Gaussian.

Data is simulated from a statistically independent power law distribution and is used to generate time series at various time scales. The moments plot for this simulated time series quickly converges to Gaussian whereas the observed index data does not converge to Gaussian even at a 16 day scale.

The authors hypothesize the scaling behavior observed is because of the time dependency of returns. To verify the hypothesis, a shuffled time series is analyzed . The results show that the rate of convergence to Gaussian is faster

Scaling behavior does not hold if time dependencies are removed. The breakdown of the scaling behavior of the distribution of returns upon shuffling the time series suggests that longrange volatility correlations which persist up to several months many be one possible reason for the observed scaling behavior
The paper uses Hill estimator for analyzing the return distribution parameters. The idea is extremely simple. Here’s a sample code that computes the power law index for a certain known distribution :
https://gist.github.com/anonymous/a6c07b99d193a9830f20.js
Scaling of the distribution of price fluctuations of individual companies:
This paper use the same principles and tools with the difference being the dataset. The other question that this paper analyses is, “ Why should be the distribution of returns for individual companies and for the S&P 500 index have the same asymptotic form ? This finding is unexpected, since the S&P500 returns are weighted sum of the returns of 500 companies. Hence, we would expect, the S&P 500 returns to be distributed approximately as Gaussian, unless there are significant dependencies between the returns of different companies. The authors observe the following :

At 1 minute scales, return distributions are outside the Levy stable range. Only the central region obeys stable Levy distribution

For time scales from 5 minutes to 4 days, the functional form does not vary significantly

Analyzing the various moments for normalized returns for time scales ranging from 5 minutes to 1 4days, the authors find that the estimated moments are nowhere close to Gaussian moments

When the cross correlations amongst stocks are replaced by random noise, the constructed index converges to Gaussian rapidly