Optimal bounds for Cauchy approximations for the winding distribution of planar Brownian motion
Let \(Z(t)=X(t)+iY(t)\), \(t\geq 0\), be a \(\mathbb C\)-valued Brownian motion starting from \(Z(t)=1\). Denote the winding number by \(\theta(t)=\arg Z(t)\) and let \( F_t(x)= P(\theta(t)/\log \sqrt t <\alpha/2)\) be its distribution function with respect to the natural logarithmic scaling. \textit{F. Spitzer} [Trans. Am. Math. Soc. 87, 187-197 (1958; Zbl 0089.13601)] proved that \( \lim_{t\to\infty} F_t(x) =G_0(x)\), the standard Cauchy distribution function. Then \textit{G. Pap} and \textit{M. Yor} [Period. Math. Hung. 41, 213-226 (2000; Zbl 1074.60507) identified functions \(G_j\) and coefficients \(a_j\) for which the following uniform bounds are valid \[ \sup_{x\in{\mathbb R}}\left|F_t(x)-G_0(x)-\sum_{j=1}^{k-1}a_jG_j(x)\delta_t^j \right|\leq c_k\delta_t^k,\quad t\geq 2,\;k\in\mathbb N, \] where \(\delta_t=1/\log\sqrt t\). In this further work, nonuniform bounds, which are shown to be optimal, are given. They are of the form \[ \sup_{x\in{\mathbb R}}(1+|x|^{2[k/2]+1}) \left|F_t(x)-G_0(x)-\sum_{j=1}^{k-1}\widetilde a_j(t)G_j(x)\delta_t^j \right|\leq c_k\delta_t^k,\quad t\geq 2,\;k\in\mathbb N, \] with explicit expressions for the coefficients \(\widetilde a_j(t)\). As a corollary, for \(t\geq 2\), \[ \sup_{x\geq 0} |(1-F_t(x))/(1-G_0(x)) -1|\leq c\delta_t. \] Optimal nonuniform bounds for the approximations of the density are also derived.
nonuniform asymptotic expansions of distribution functions
scientific article; zbMATH DE number 1960097