This problem is from a Ph.D Qualifying Exam on mathematical statistics(also related to probability theory).
Let $(X_1,\cdots,X_k)$ be a random vector with multinomial distribution of $n$ trials and cell probabilities $(p_1,\cdots,p_k)$.
Let $X_j=\sum_{l=1}^n X_j^l$ where $X_j^l=1$ if the $l$-th observation falls in cell $j$ and $0$ otherwise. Define $$W_n=\frac{\sum_{l=1}^{n}(X_1^l-\bar{X_1})(X_2^l-\bar{X_2})}{\sqrt{\sum_{l=1}^{n}(X_1^l-\bar{X_1})}\sqrt{\sum_{l=1}^{n}(X_2^l-\bar{X_2})}}.$$
Check if $W_n$ converges in probability as $n$ increases.
My attempt: If all the $X_1^l$'s and $X_2^l$'s were independent, the result would be obvious by WLLN. However, in multinomial distribution they are not independent. Rather, the random vectors $X^l=(X_1^l,\cdots,X_k^l)$ are iid and I'd like to apply something like WLLN for random vectors. Does anyone have ideas?