Definition Let $W$ be a chi-squared random variable with $m$ degrees of freedom and let $V$ be a chi-squared random variable with $n$ degrees of freedom, where $W$ and $V$ are independently distributed. Then $\frac{W/m}{V/n}$ has $F_{m,n}$ distribution.
Limiting case: $\frac{W}{m}$ has $F_{m,\infty}$ distribution.
I am wondering how $\frac{V}{n}$ goes to $1$ as $n$ goes to $\infty$. In my view, $\frac{V}{n}$ should go to $0$ instead, because $n$ goes to $\infty$ in the denominator. Thanks in advance!
Best Answer
Remember the variable $V$ is not a fixed distribution; it varies with $n$. In fact the chi-square distribution with $n$ degrees of freedom is distributed like $\sum_{i=1}^n Z_i^2$, where $Z_1,\ldots,Z_n$ are iid standard normal variables. So $V/n$ has expectation $1$. You can also look up, or calculate, that the variance of $V$ is $2n$, so the variance of $V/n$ is $2/n$, which implies that $V/n$ converges in probability to one as $n\to\infty$.