assurance and web 2.0 , e-insurance, Financial Services industry, and "social brain"

In 2008, global insurance premiums grew by 3.4%  to reach $4.3 trillion. The financial crisis (since sept. 2008; an highly improbable. event?) has shown that the insurance sector is sufficiently capitalised: the majority of insurance companies had enough capital to absorb this impact and only a very small number turned to government for support.

Modeling and analysis of financial markets and of risks are often based on the Gaussian distribution but 50 years ago, Benoît Mandelbrot discovered  that changes in prices do not follow this distribution: changes in prices are rather modeled better by Lévy alpha-stable distributions.

An increasing variety of outcomes are being identified to have heavy tail distributions, including income distributions, financial returns, insurance payouts, reference links on the web, etc... A particular subclass of heavy tail distributions are power-laws.

See: http://reference.wolfram.com/mathematica/ref/StableDistribution.html

For example of a power-law, the scale of change (volatility), depends on the length of the time interval to a power a bit more than 1/2.

Power-law tail behavior and the summation scheme of Levy-stable distributions ( alpha- stable distribution) is the basis for their frequent use as models when fat tails above a Gaussian distribution are observed. However, recent studies suggest that financial asset returns exhibit tail exponents well above the Levy-stable regime (0 < alpha < = 2 ). A paper ( http://ideas.repec.org/p/wpa/wuwpem/0305003.htm) illustrates that widely used tail index estimates (log-log linear regression and Hill) can give exponents well above the asymptotic limit for alpha close to 2, resulting in overestimation of the tail exponent in finite samples. The reported value of the tail exponent alpha around 3 may very well indicate a Levy-stable distribution with alpha around 1.8.

One of the most important subclass of heavy tail distributions are power-laws, which means that the probability density function is a power. One of the most important properties of power-laws is its scale invariance. The universality of power laws (with a particular scaling exponent) has an origin in the dynamical processes (self-organized systems) that autogenerate the power-law relation. Risk is not only a tail distribution but also the consequences of "social brain" (i.e. Richardson's Law for the severity of violent social conflicts). 
For info on  "social brain", see the work of Robin I.M. Dunbar: 
http://ex-ample.blogspot.com/2011/06/robin-ian-macdonald-dunbar-cambridge.html

The terms long-range dependent, self-similar and heavy-tailed... cover a range of tools from different disciplines that may be used in the important science of determining the probability of rare events, which is the basis of insurance industry.

I think that e-insurance is not only online insurance. By  "integrating" of studies (econometrics, statistics, simulation on large corpus, social aspects...) and sharing of encrypted media (cloud computing; see: http://ex-ample.blogspot.com/2011/06/example-exemple-boxnet-html5-sharing.html),  insurance companies can leverage on Web 2.0 technology  to online evaluate financial informations and deliver insurance products to the people, and to the networks of social media platform who need protection against risks.

Rem: if statistics are too complex (as Churchill already said: "There are lies, damn lies - and statistics."), this problem is in fact simple. I try to explain this behavior with an example: if we consider the sizes of files transferred from a web-server, then the distribution is heavy-tailed, that is, there are a very large number of small files transferred but, crucially, the "small" number of very large files transferred remains a major component of the volume downloaded.