# Proportion of categories in multinomial

Given a multinomial random variable with support $\{x_1,x_2,x_3\}$, in which each category occurs with strictly positive probabilities $p_1$,$p_2$,$p_3$ respectively.

Is is true that $\frac{N_1}{N_2}$ approaches $\frac{p_1}{p_2}$ as $n\to \infty$? (where $N_i$ is the number of occurences of category $x_i$ and $n$ is the number of trials)

Intuitively this makes senses for me, but somehow I'm not totally comfortable. How could I prove this?

• Law of large numbers, plus perhaps Slutsky's theorem. – kimchi lover Sep 24 '17 at 0:19