# Why if a random variable is power law distributed I should consider ln N (with ζ=1) in place of sqrt(N)?

In this paper

I don't get Preposition 2. In particular firm size can be described as: $$\DeclareMathOperator{\E}{\mathbb{E}} dS_{it}=0S_{it}+\sigma S_{it}\; dW_{it}$$ or according to the paper $$dS_{it}=\sigma S_{it} \varepsilon_{it}$$ where $$\sigma$$ is the same among different firms. So if GDP is given by: $$Y_t= \sum_{i=1}^N S_{it}$$ And $$\sigma_{GDP}= \left( \sum_{i=1}^N \sigma^2 \cdot \left(\frac{S_{it}}{Y_t} \right)^2 \right)^{\frac{1}{2}}$$ Where $$h=\left[ \sum_{i=1}^N \left(\frac{S_{it}}{Y_t} \right)^2 \right]^{\frac{1}{2}}$$ is the square root of the sales herfindahl of the economy. So if we consider an island economy with $$N$$ firms whose sizes are drawn from a distribution with finite variance and with same volatility (Gibrat's law for variance), as $$N \rightarrow \infty$$: $$\sigma_{GDP}= \frac{\E X_t[S_{it}^2]^{\frac{1}{2}}}{\E X_t[S_{it}]} \frac{\sigma}{\sqrt{N}}$$ Second proposition is about the failure of $$1/\sqrt{N}$$ argument when the firm size is power law distributed. According to the paper if we consider a series of island economies indexed by $$N\geq 1$$, each economy has $$N$$ firms whose growth rate volatility is $$\sigma$$ (Gibrat's law for variance) and whose size $$S_1,...,S_N$$ are drawn from a power law distribution: $$P(S>x)=ax^{-\zeta}$$ For $$x>a^{\frac{1}{\zeta}}$$ with exponent $$\zeta \geq 1$$, as $$N \rightarrow \infty$$: $$\sigma_{GDP}= \frac{\upsilon_{\zeta}}{\ln {N}}\sigma$$ for $$\zeta=1$$, or: $$\sigma_{GDP}= \frac{\upsilon_{\zeta}}{N^{1-1/ \zeta}}\sigma$$ for $$1< \zeta <2$$ or: $$\sigma_{GDP}= \frac{\upsilon_{\zeta}}{N^{1/2}}\sigma$$ for $$\zeta \geq 2$$. I don't understand:

1. How can the process $$dS_{it}=\sigma S_{it} \varepsilon_{it}$$ be described by a power law distribution;
2. What is $$\upsilon_{\zeta}$$, in the paper they say it is a random variable, but I don't get the link with this random variable and the firm size differential equation;
3. I am trying to implement with Excel a little example of the paper Propositions in order to understand: if i fix $$\sigma=12%$$ I can simulate $$dW_{it}$$ with =NORMINV(RAND()) and $$dS_{it}$$ with =0.12*NORMINV(RAND()). So starting from $$S_{i0}=100$$ for every $$i$$ I can simulate $$S_i$$ at time $$t$$ for a given number of firms and verify the first preposition. How could I proceed to verify the second proposition, if it is possible, with Excel?