An Upper Bound on the Error Induced by Saddlepoint Approximations-Applications to Information Theory

Area of Science:

• Information Theory
• Probability Theory
• Statistical Inference

Background:

• Accurate bounds on cumulative distribution functions (CDFs) are crucial for analyzing communication channel performance.
• Saddlepoint approximations offer efficient ways to estimate CDFs, especially in large deviation regimes.
• Analytical calculation of dependence testing (DT) and meta-converse (MC) bounds on decoding error probability (DEP) is often intractable.

Purpose of the Study:

• To introduce a novel upper bound on the difference between a CDF and its saddlepoint approximation for sums of independent and identically distributed random variables.
• To apply this bound to derive new analytical bounds for the DT and MC bounds in information theory.
• To investigate the practical utility of these bounds through numerical experiments.

Main Methods:

• Derivation of an upper bound for the absolute difference between a CDF and its saddlepoint approximation.
• Application of the derived bound to establish new upper and lower bounds for the DT and MC bounds.
• Numerical evaluation of the proposed bounds using standard communication channel models.

Main Results:

• A precise upper bound is established for the approximation error of CDFs of sums of random variables.
• New upper and lower bounds for the dependence testing (DT) and meta-converse (MC) bounds on decoding error probability (DEP) are derived.
• Numerical results demonstrate the effectiveness of the bounds for binary symmetric, AWGN, and α-stable noise channels.

Conclusions:

• The developed upper bound provides a valuable tool for analyzing and bounding decoding error probabilities in information theory.
• The new bounds offer improved analytical tractability for studying channel coding limits.
• The numerical experiments validate the theoretical findings across various channel types.