This research explored the number of returns to the origin within the framework of a symmetric simple random walk. Our primary objective was to approximate the distribution of return events to the origin by utilizing the half-normal distribution, which is chosen for its appropriateness as a limit distribution for nonnegative values. Employing the Stein's method in conjunction with concentration inequalities, we derived an exponential non-uniform bound for the approximation error. This bound signifies a significant advancement in contrast to existing bounds, encompassing both the uniform bounds proposed by Döbler ^[1] and polynomial non-uniform bounds presented by Sama-ae, Chaidee, and Neammanee ^[2], and Siripraparat and Neammanee ^[3].

The statistics defined on symmetric permutation groups, particularly descent and inversion, have been extensively investigated due to their wide-ranging applications in areas such as card shuffling and sorting algorithms. Descent and inversion were shown to satisfy the central limit theorem by Tanny (1973) and Bender (1973), respectively. Since then, many mathematicians studied the error bounds associated with these approximations. Uniform bounds were first established by Fulman in 2004, while Chuntee and Neammanee (2013) and Sumritnorrapong et al. (2018) derived the non-uniform bounds. The latest work of non-uniform bounds from Sumritnorrapong et al. (2018) was not practical since their main theorems are valid for large $n$ and $z$ ( $n\ge 7.07\times {10}^6$ and $|z|\ge 8\sqrt{3}$ for descent and $n\ge 1.9\times {10}^8$ and $|z|\ge 24$ for inversion). In this paper, we extended the theorem to hold for arbitrary $n\in \mathbb{N}$ and $z\in \mathbb{R}$. Moreover, our constants were sharper than previously seen. The approach in this work was done by combining Stein's method with the exchangeable pair technique.