Some remarks on the chi-squared test with both margins fixed

Under the hypothesis of independence, the chi-squared test statistic for independence in a two-way contingency table follows an asymptotic chi-squared distribution under both a multinomial and a product-multinomial models. Alalouf(1987) showed the same result holds for the third case where both margins are fixed. In this paper an intuitively easier way of proof using the conditional limit theorems is suggested and some points are discussed.