A Truncation Approach for Fast Computation of Distribution Functions

In this paper, we propose a general approach for improving the efficiency of computing distribution functions. The idea is to truncate the domain of summation or integration.

💡 Research Summary

The paper introduces a general truncation framework designed to accelerate the numerical evaluation of distribution functions such as cumulative distribution functions (CDFs) and probability mass functions (PMF). Traditional exact methods require summing or integrating over an infinite or very large domain, which becomes computationally prohibitive especially when the tail of the distribution contributes negligibly to the final value. The authors propose to identify a truncation point that guarantees the omitted tail probability does not exceed a pre‑specified tolerance ε. For discrete distributions the truncation point k is the smallest integer satisfying Σ_{i≥k} P(X=i) ≤ ε (and analogously for the lower tail). For continuous distributions the truncation points x₁ and x₂ are defined by ∫{−∞}^{x₁} f(x)dx ≤ ε and ∫{x₂}^{∞} f(x)dx ≤ ε.

The theoretical contribution consists of rigorous error bounds. By leveraging Markov’s inequality, Chebyshev’s inequality, and distribution‑specific tail approximations (e.g., the error‑function bound for the normal distribution, incomplete gamma bounds for the gamma distribution), the authors derive explicit formulas for ε‑controlled truncation. They also introduce a “correction factor” that can be added back to the truncated result to further tighten the error, either by direct evaluation of the omitted tail or by consulting pre‑computed tables.

Algorithmically, the paper distinguishes between discrete and continuous cases. In the discrete case, all terms whose probability mass falls below a threshold are filtered out before the summation, leaving only k terms to be processed. In the continuous case, an adaptive quadrature scheme is combined with truncation: the interval