How to prove that data is following log-normal or T-distribution rather normal? Instead of just showing the graphs, is there any test to prove it mathematically?
 A: The only complete test you can use to test among three choices is to use Bayesian model selection.  AIC and BIC are approximations of the posterior density under some restrictive assumptions.  If the AIC and BIC have a very clear winner and two clear losers, then you could use them.  If they are close you should consider two Bayesian alternatives.
The first is to test the three distributions as three separate simple hypotheses.  The second would be to have an infinite number of hypotheses, but test to see if the data is inside the Pearson families of distributions.  The log-normal, the normal and Student's t distribution are all members of the Pearson family of distributions or the Burr family.
For the Pearson family, you can find specific examples of the method at:
H.M. Markowitz and N. Usemen. The likelihood of various stock market return distributions, part 1: principles of inference. Journal of Risk and Uncertainty. 1996. 13 pp. 207-219
and
H.M. Markowitz and N. Usemen. The likelihood of various stock market return distributions, part 2: empirical results. Journal of Risk and Uncertainty. 1996. 13pp. 221-247
Generally speaking, to test H!: data follows normal versus H2: data follows log-normal versus H3: data follows Student's t you would construct a model selection process as in Markowitz or either of the two links below.
There is a decent discussion of it at http://alumni.media.mit.edu/~tpminka/statlearn/demo/
A more detailed approach is at:
http://www-stat.wharton.upenn.edu/~edgeorge/Research_papers/ims.pdf
A: It is easy to show which model fits a data set better. It is also easy to show which data model is better (e.g., BIC). However, that does not allow one to state that even the best fitting model is normal, lognormal or whatever; it only allows one to state which models are improbable. However, it is possible to identify a physical process that has known characteristics. For example, it is often stated that photons are best modeled using Gaussian noise.
A: There are a variety of ways you might measure fit, or test between two alternatives.
Generally you can't prove that the distribution is something (you may be able to say "it's not drawn from a normal distribution" but you can't just say "... it is drawn from a lognormal"). There are always alternatives that will be arbitrarily close to whatever you're trying to say it is ... and so there would be some alternative "between" the data and the thing you might want to say that your sample was drawn from. (Consider the simpler case of trying to say the population mean height is 175 cm. Lets say you take a sample of size 10^5 and you get 175.07 cm. "Great," you say "this is perfectly consistent with 175". And it may be, but it's even more consistent with 175.1 cm, and more consistent still with say 175.05 (which is between 175 and the sample value). So you can't tell it's not from one of those alternatives. A similar thing goes on with distributions, but there's a much wider range of ways of being a bit like some given distribution but different from it.
You can say when the data are inconsistent with one distribution. You can also say when the data are more consistent (in some sense) with some other distribution. So it may be that you can say the data are more consistent with the lognormal than with the normal; that's quite a different proposition to proving it's lognormal.
However, note that when you're comparing a distribution with some distribution class that encompasses yours (like normal vs t) then you have one nested in the other -- the t, with its three parameters will always fit at least as well as the normal which fixes the parameter which the t can allow to vary. You need some way of comparing things that shouldn't be expected to fit equally well. 
