Malaysian Journal of Mathematical Sciences, December 2024, Vol. 18, No. 4
Discretized Normal Approximation for the Number of Descents
Siripraparat, T. and Neammanee, K.
Corresponding Email: [email protected]
Received date: 28 February 2024
Accepted date: 11 July 2024
Abstract:
This paper explores the probability approximation of the number of descents in a random permutation. It is known that the distribution of the number of descents can be approximated by a normal distribution under Kolmogorov distance. Subsequently, an explicit constant is provided in the bound of $13.42$. The objective is to prove a similar result but using a stronger distance, namely, the total variation distance. The Stein's method and the exchangeable pair transformation has been used to give a bound for discretized normal approximation for the distribution of number of descents under the total variation distance. The result obtained gave an improved constant of $10.71$.
Keywords: the number of descents; normal approximation; discretized normal approximation; Kolmogorov distance; total variation distance; Stein's method; exchangeable pair
