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Arbitrarily tight bounds on cumulative distribution function of Beckmann distribution

Beckmann distribution is a versatile mathematical model, which can be applied in performance analyses of radio frequency communications, free-space optical communications and underwater optical communications. However, the cumulative distribution function (CDF) of Beckmann distribution does not have a closed-form expression, which makes it challenging to derive closed-form outage probability expression of communications over channels involving Beckmann random variables. In this paper, we derive closed-form upper and lower CDF bounds of Beckmann distribution, and the bounds can be arbitrarily tight by properly choosing the parameters of the bounds. Compared to the numerical estimation of the double-fold integral expression of the Beckmann CDF, using the closed-form bounds to estimate the CDF is not only faster, but also has less space complexity. More importantly, the analytical CDF bounds explicitly quantify the largest possible discrepancy between the approximate CDF and the exact CDF, while the discrepancy of the numerical estimation is unknown.