On Sharpest Tail Bounds for Functions of Tail Bounded Random Variables

Abstract: Consider $n$ real/complex, independent/dependent random variables with respective tail bounds and $g$ a measurable function of the r.v.'s. Consider $f$ the "sharpest" tail bound of $g$ (sharpest in the sense that if $f$ were any less, then for some $X_1,...,X_n$ satisfying the conditions, $g(X_1,...,X_n)$ would not satisfy $f$). Significant research has been done to approximate $f$ often with high accuracy. These results are often of the form that for $g$ in this family and tail bounds of $X_k$ in this family, $f$ is bounded by some $f'$ with high accuracy. However, the question "what would it take to find $f$ exactly?" has received little attention, apparently even for simple cases. This is the question we try to answer. For $X_1,...,X_n$ required to be mutually independent, first the $X_k$ are simplified to be monotone on $(0,1)$ WLOG. This strengthens convergence in distribution to convergence a.e. (Skorokhod's representation theorem) and allows defining shift operators, which help reduce the space of r.v.'s one searches to find $f$ and/or the maximum measure of a subset. We do find $f$ in some special cases; however $f$ rarely has a closed form. For $X_1,...,X_n$ dependent/not necessarily independent, another reduction in the space of r.v.'s one searches to find $f$ is done.

• The paper introduces a framework to characterize and construct the sharpest tail bounds for functions of tail-bounded random variables.
• It employs shift operators and boundary reductions to optimize tail probability estimates in both independent and dependent settings.
• The approach provides exact representations and algorithmic strategies with significant implications for risk quantification and high-dimensional data analysis.

Sharpest Tail Bounds for Functions of Tail Bounded Random Variables

Problem Context and Motivation

Obtaining tight concentration inequalities for functions of random variables is a longstanding problem with broad implications for probability theory, statistics, and high-dimensional data analysis. Traditionally, research has focused on deriving high-quality upper bounds for the probability that a function $g(X_1, ..., X_n)$ of random variables exceeds a threshold, given tail or moment restrictions on the $X_k$. However, the absolute sharpest (i.e., smallest possible, or "least") tail bound—such that reducing it would invalidate the inequality for some permissible configuration of $X_k$—had not previously been characterized in generality, or even for many nontrivial concrete cases. This paper systematically investigates the structure of such optimal tail bounds, provides explicit representations in special but instructive scenarios, and supplies a rigorous framework for constructing or approximating sharpest bounds in higher dimensions and under general dependence structures.

Formalization of "Sharpest" Tail Bounds

A tail bound $f$ for $g(X_1, ..., X_n)$ is called sharpest if:

• For all configurations of $X_k$ meeting their respective tail constraints and (in the independent case) independence, $g(X_1,...,X_n)$ satisfies the tail probability bound $P(g(X_1,...,X_n) \geq t) \leq f(t)$.
• For any strictly smaller $f'$, there exist admissible $X_k$ making $X_k$0 for some $X_k$1.

The sharpest tail is thus a uniformly optimal upper bound over the permitted class of random vectors, for all thresholds simultaneously.

Reduction to Structured Random Variables: Neat and Radially Neat Constructions

The work demonstrates that, for both the independent and dependent variable settings, sharpest tail bounds may always be realized by suitably structured random variables—namely, "neat" or "radially neat" random variables. These are defined on $X_k$2 (with Lebesgue measure), are monotone in the appropriate sense, and their value at $X_k$3 is explicitly constructed to saturate the tail constraint as much as possible.

For independent variables, the existence and uniqueness of such neat representations (left- or right-continuous) follow from basic measure-theoretic and probabilistic results, including Skorokhod's representation theorem and properties of monotone rearrangements.

For a given tail upper function $X_k$4, the "maximal" neat variable $X_k$5 is constructed:

$X_k$6

with $X_k$7. For functions $X_k$8 that are componentwise monotone and subsets $X_k$9 that are suitably regular, maximizing the measure of $X_k$0 is reduced to maximizing mass on extremal "corners."

Main Structural Results: Shift Operators and Boundary Reduction

Independent Variables: Shift Operators

For product spaces and independent $X_k$1, the sharpest upper mass for $X_k$2 is achieved by shifting as much mass as possible onto those points where $X_k$3 is maximized, under the constraints that each $X_k$4's univariate marginal obeys its prescribed tail bound. This motivates the construction of shift operators ($X_k$5, and $X_k$6) which determine, for each $X_k$7, the rightmost/leftmost point in $X_k$8 that a neat random variable can occupy without violating the tail bound.

The shift operator construction, in effect, translates the tail-optimization into a variational problem on the allowed product measure space, where tail constraints are enforced pointwise.

Dependent Variables: Boundary Reductions

When arbitrary dependence is permitted, the optimization can be recast in terms of measures on $X_k$9 with prescribed tail constraints on each coordinate. The sharpest bound for the measure of a closed set $f$0 depends only on its "southwest boundary," the minimal subset required to "capture" all incoming mass from below. This is formalized through mass retractions and yields the reduction:

$f$1

where $f$2 is the (componentwise) absolute value map and $f$3 extracts minimal points relative to the positive orthant ordering.

Key Theorems and Exact Results

• For componentwise nondecreasing $f$4 and right tail-bounded, independent $f$5:

$f$6

where $f$7 is the vector of maximal neat RVs for each tail bound.

• For multilinear, absolute, or two-sided tail bounds: analogous exact representations are proved using appropriate shift operators.
• For small, discrete $f$8 (as in two-point or finite settings): the optimal measure (and thus the sharpest tail) can be obtained explicitly by checking finitely many candidate grids and choosing the maximal measure, facilitated by the grid and closure operations detailed in the text.
• For dependent variables, the use of mass retractions and reduction to the southwest boundary yields both conceptual simplification and (for $f$9) a closed-form characterization of the sharpest attainable upper bound, involving infima and suprema over slices of $g(X_1, ..., X_n)$0.

Implications and Technical Innovations

The methodology in this paper provides, for the first time, exact (or structurally exact) expressions for the best possible tail bounds in a wide range of settings. This includes settings previously accessible only via non-sharp inequalities (e.g., Hoeffding, Bernstein, generic moment bounds). The results have several notable technical consequences:

• Sharpest bounds might lack closed forms: In all but the simplest cases, the sharpest possible tail bound, as a function of the underlying $g(X_1, ..., X_n)$1 and $g(X_1, ..., X_n)$2, cannot be represented in elementary closed form. The paper documents cases where semi-closed forms (involving special functions) are attainable, but in general, the sharpest function $g(X_1, ..., X_n)$3 is piecewise defined by suprema over feasible measure packings.
• Reduction in search space: By leveraging shift operators and boundary reductions, the search for extremal distributions is dramatically narrowed, making both computation and conceptual understanding of optimal bounds feasible in cases previously assumed too complex for analysis.
• Asymptotic tightness: The dependent variable outer bounds can often closely approximate the independent case, especially for large $g(X_1, ..., X_n)$4 or when the dependencies are weak/non-pathological.

Contrasting Statements and Numerical Strength

Unlike the majority of concentration inequalities in the literature—which provide universal explicit (but not always tight) upper bounds, with possibly suboptimal constants—the bounds characterized here are not improvable under the stated hypotheses. In the special case of sums of independent, sub-Gaussian variables, the sharpest bound is achieved exactly by their cumulative distribution function (e.g., for Gaussians, by the standard normal tail formula).

Numerically, for $g(X_1, ..., X_n)$5 monotone and $g(X_1, ..., X_n)$6 normal with mean $g(X_1, ..., X_n)$7 and variance $g(X_1, ..., X_n)$8,

$g(X_1, ..., X_n)$9

where $X_k$0.

Theoretical and Practical Consequences

The theoretical implications are twofold:

1. Optimality in principle: The paper establishes that sharpest tail bounds are, in principle, attainable and precisely characterizable (at least in form), reducing questions about the quality of tail bounds to tractable suprema over explicit classes of distributions/measures.
2. Structural rigor for probabilistic optimization: The methods supply a blueprint for encoding deterministic structure (monotonicity, boundary exposure, shift operations) within probabilistic measure optimization.

Practically, these results inform the design of statistical tests, probabilistic algorithms, or high-dimensional simulation protocols where worst-case tail risks must be quantified as tightly as possible. Moreover, for cases with a finite support or grid, the results are fully algorithmic.

Future Directions

Several open directions are identified or suggested:

• Extension of explicit sharpest bound computations to non-monotonic or more general functions $X_k$1.
• Constructive or computational advances for high-dimensional $X_k$2 with continuous structure, which remains exponentially complex even with grid reduction.
• Connection with optimal transport and measure concentration in more general metric spaces.

Additionally, the paper proposes that future developments could generalize the shift operator methods to treat left, right, and two-sided tail scenarios in a unified framework—currently, each requires separate constructions.

Conclusion

This work provides a rigorous and highly general foundation for the derivation and representation of genuinely optimal tail bounds for functions of tail-bounded random variables. Through domain simplification, shift operator constructions, boundary reductions, and measure-theoretic analysis, the paper advances both the theory and practical applicability of sharp concentration inequalities. These advances deepen our understanding of the probabilistic geometry of extremal measures and lay the groundwork for further developments in non-asymptotic probability theory and risk quantification.