# convergence in probability of harmonic mean

I have $X_1,...,X_n$ iid as gamma($\alpha,\beta$).

Defined also is the harmonic mean $Y_n=\frac{n}{\sum{x_i^{-1}}}$.

I'm trying to figure out if $Y_n$ converges in probability to some constant $c$.

My intuition says c=0, but I do not really know any ways to solve for convergence in probability outside of Chebychev's Inequality.

• Consider applying any techniques you know to the random variables $1/X_i$ and notice that $1/Y_n$ is the mean of the $1/X_i$. – whuber Jan 15 '17 at 21:02
• Looks like $1/Y_n=\frac{\sum x_i^{-1}}{n}$ converges almost surely to $E(1/x_1)$. So, $Y_n$ converges to $1/E(1/x_1)$. But I am still not understanding proving convergence in probability directly. – Mark Jan 17 '17 at 16:43