A Combinatorial Central Limit Theorem

Key Points

• This research aims to establish sufficient conditions for the asymptotic normality of sums involving permutations of a random vector.
• Considered a random vector taking $n!$ permutations of $(1, \, \cdots, \, n)$ with equal probabilities.
• Developed conditions for the normality of $S_n = \sum^n_{i=1} c_n(i, Y_{ni})$.
• Explored specific cases where $c_n(i,j) = a_n(i)b_n(j)$ to obtain stronger results.
• Theorem 3 presents sufficient conditions for asymptotic normality of sums involving random vectors.
• Theorem 4 extends a result by Wald, Wolfowitz, and Noether in a special case.
• Theorem 1 simplifies a condition originally proposed by Noether.