Analytic form of the limiting random-walk distribution for extrapolated CBS estimates

Derive the exact analytic form of the limiting probability distribution of the converged random-walk estimate \tilde{e}_\infty used to assign uncertainties to complete basis set (CBS) extrapolations. The procedure begins from two adjacent extrapolated values e_X and e_{X-1} and samples e_{X+1} uniformly within the interval [e_X − |e_X − e_{X-1}|, e_X + |e_X − e_{X-1}|]; it then iterates by sampling each subsequent e_{X+k} uniformly within [\tilde{e}_{X+k−1} − |\tilde{e}_{X+k−1} − e_{X+k−2}|, \tilde{e}_{X+k−1} + |\tilde{e}_{X+k−1} − e_{X+k−2}|] until the interval width tends to zero. Determine the exact distribution of \tilde{e}_\infty in the limit of infinitely many independent trajectories.

Background

The paper introduces a random-walk-based method to estimate uncertainties for results extrapolated to the complete basis set (CBS) limit in quantum chemistry. Starting from two consecutive CBS extrapolations, the method defines bounding intervals for hypothetical larger-basis extrapolations and samples uniformly within these intervals, iterating until the bounds shrink to a negligible width. Aggregating many independent trajectories yields an empirical distribution of converged values (\tilde{e}_\infty), from which confidence intervals are assigned.

Numerical tests show the distributions are nearly symmetric but clearly non-Gaussian and not well represented by simple forms such as the Laplace distribution. The authors explicitly state that they were unable to determine the exact analytic form of the limiting distribution, noting the difficulty arises from strong interdependence among successive randomization steps. An analytic characterization would provide a closed-form foundation for confidence interval assignment and deepen understanding of the method’s statistical properties.