Before revision:  Our saddlepoint density and tail area approximation feature relative error of order O( m^{-1 n(T-1)  with n being the cross-sectional dimension and T the time-series dimension.
After revision:  Our saddlepoint density and tail area approximation feature relative error of order O( 1/( n(T-1) with n being the cross-sectional dimension and T the time-series dimension.