Normal and power law distributions

[erszega]

Is it correct to say that independent random events (additively) lead to a normal distribution, and dependent random events (multiplicatively) lead to a power law distribution?

The following might be trivial, but it was quite interesting to find for me, someone with a very limited knowledge of mathematics or statistics:

Take a matrix of random numbers r(i,j), where 0 < r(i,j) < 1.

Let S(i) = int( ( r(i,1) + r(i,2) +...+ r(i,j) )*n ), with S(i) >= S(i-1) if i >= i-1, that is, S is a sorted list of the integer parts of the sums of random numbers multiplied by an integer.

Let P(i) = int( ( r(i,1) * r(i,2) * ... * r(i,j) )*n ), with P(i) >= P(i-1) if i >= i-1, that is P is a sorted list of the integer parts of the products of random numbers multiplied by an integer.

Hypothesis (based simply on observation of graphs of S(i) and P(i)):

the higher the values of i and j, the more S(i) approximates normal distribution, and P(i) approximates a power-law distribution.

Is this right?

 

[erszega]

Apologies, I meant the frequency of each S(i) or P(i).
It may be better to describe what I did in the following way:
I used a spreadsheet, and created a table, consisting of, say, 10,000 rows and 5 columns, of random numbers (using the spreadsheet's random number function). Then I added (or multiplied) the random numbers in each row. I multiplied the sums (or products) by, say, 100, and took the integer parts (to create "bins"). Then I sorted the "bins", calculated the frequency (number of occurencies in the list) of each "bin", and then put a graph on the frequency list. I hope this makes sense.