Pareto distribution ($\mathrm{Pareto}(\alpha ,\beta )$)

In the shape–minimum parameterization, the density is $\alpha {\beta }^{\alpha }{x}^{-\alpha -1}{1}_{\{x>\beta \}}$, with $\alpha ,\beta >0$. The survival function is $(\beta /x{)}^{\alpha }$ for $x\ge \beta$. Large shape concentrates the distribution near its lower bound. This convention is useful for uniform-Pareto conjugacy.

For nonnegative independent identically distributed summands with polynomial upper tail $\mathbb{P}(X_1\ge x)=x^{-\beta }$ for $x\ge 1$, positivity gives $\mathbb{P}(S_n\ge na)\ge (na{)}^{-\beta }$ for fixed $a>0$ and large $n$. Since this probability is also at most one, its logarithm divided by $n$ tends to zero. No positive exponential moment exists, so the exponential-moment form of the Cramér theorem cannot be applied. The bound proves a logarithmic rate and does not assert an exact tail asymptotic.