I have the following problem. Assume I have $n$ independent Pareto random variables $X_1,...,X_n$, with the CDF of $X_i$ being $Pr(X_i \leq x_i) = F(x_i) = 1 - (\frac{b_i}{x_i})^{\alpha_i}$. For convenience of notation, I use the complementary CDF $Pr(X_i \geq x_i) = G_i(x_i) = 1 - F_i(x_i) = (\frac{b_i}{x_i})^{\alpha_i}$.

Let $X = min(X_1,...,X_n)$ be the minimum of $X_1,...,X_n$ and let $H(X_1,...,X_n) = \frac{n}{\frac{1}{X_1} + ... + \frac{1}{X_n}}$ be the harmonic mean.

**I want to show that the distribution $Pr(X \leq x | H(X_1,...,X_n) \geq \tau)$ follows a Pareto Distribution.**

I know how to do this if we do not condition on the harmonic mean. It is straightforward to show that $X = \min(X_1,...,X_n)$ has a Pareto distribution with complementary CDF given by $$G(x) = \prod_{i=1}^n (\frac{b_i}{x})^{\alpha_i} = (\prod_{i=1}^n b_i^{\alpha_i}) \cdot (\frac{1}{x})^{\alpha_1+...\alpha_n} $$

However, when we condition on the harmonic mean being above a certain threshold, I'm not sure where to start tackling this problem.