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?

  • $\begingroup$ One possible way to get started is to figure out the first and second moments of each summation. Another thing to note is that $X_a^lX_b^l=0$ whenever $a\neq b$ $\endgroup$ – probabilityislogic Sep 2 '18 at 11:35

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