Taillowerbounds

Taillowerbounds are inequalities that establish a nontrivial lower bound on the probability that a random variable deviates from its typical value by at least a specified amount. They are used to guarantee that rare events have at least a minimum likelihood, in contrast to more familiar upper-tail bounds that restrict how unlikely such events can be. In formal terms, for a nonnegative random variable X and a threshold t, a taillowerbound asserts that P(X ≥ t) ≥ g(t) for some function g(t) that is positive on a region of interest. Analogous statements may be made for the lower tail, P(X ≤ t) ≥ h(t), for thresholds t below a central value.

Common tools for deriving taillowerbounds include moment-based inequalities such as the Paley–Zygmund inequality, which, for a nonnegative X with finite second moment, yields P(X ≥ θ E[X]) ≥ (1−θ)^2 E[X]^2 / E[X^2] for 0 ≤ θ < 1. These bounds often require knowledge of moments in addition to the distribution’s support. In some settings, two-sided large deviation principles provide asymptotic taillowerbounds that match known upper bounds up to subexponential factors.

Applications of taillowerbounds appear in risk assessment, reliability engineering, and the analysis of randomized algorithms, where it may be important to prove that extreme events occur with at least a certain probability. Limitations include the fact that nontrivial taillowerbounds can be difficult to obtain for heavy-tailed or highly skewed distributions, and in some models no meaningful lower bound exists over all thresholds.

See also: tail bounds, Chernoff bounds, Hoeffding inequalities, Paley–Zygmund inequality, large deviations.