Speed of convergence in the Central Limit Theorem for the determinantal point process with the Bessel kernel

We consider a family of linear operators, diagonalized by the Hankel transform. The Fredholm determinants of these operators, restricted to $L_2[0,R]$, are expressed in a convenient form for asymptotic analysis as $R\to \mathrm{\infty }$. The result is an identity, in which the determinant is equal to the leading asymptotic multiplied by an asymptotically small factor, for which an explicit formula is derived. We apply the result to the determinantal point process with the Bessel kernel, calculating the speed of the convergence of additive functionals with respect to the Kolmogorov-Smirnov metric.