Tail Bounds via Southwest Boundary

Abstract: We derive upper bounds for probabilities of the form $P(g(\mathbf{X})\geq t)$ using the southwest boundary (recently introduced in our previous work) $\partial_{\mathrm{SW}} Q(g<sup>{-1}[t,\infty))$, where $Q$ is a reflection to the first quadrant. Under natural continuity, symmetry, and monotonicity assumptions on $g$, this yields explicit and computable bounds of the form $P(g(\mathbf{X})\ge t)\le ns_t$, where $s_t$ is the unique parameter at which the line $L(s)=(f_1<sup>{-1}(s),\dots,f_n<sup>{-1}(s))$ intersects the southwest boundary. In particular, when $g$ is a homogeneous polynomial of degree $k$ (plus a constant $C$) and all tail bounds on the random variables are identical, the bound proves to the closed-form expression $$P(g(\mathbf{X})\ge t)\leq nf\bigg(\frac{(t-C)^{{1/k}}{(\sum_i|a_i|)^{1/k}}\bigg)}$$ where $a_i$ are the coefficients of the monomials in $g$. We then obtain an explicit tail bound for the trace of a Schur multiplier acting on random matrices with identical tail bounds on the random variables. No assumptions are made about independence or dependence.

• The paper introduces a new framework using the southwest boundary to derive explicit tail bounds for functions of random vectors.
• It provides closed-form inequalities for homogeneous polynomials and Schur multipliers on random matrices without requiring independence.
• The methodology refines classical union bounds by leveraging geometric structure, offering tighter results for dependent variables.

Tail Bounds via Southwest Boundary: A Technical Overview

Introduction

The paper "Tail Bounds via Southwest Boundary" (2604.21113) develops a new framework for deriving probabilistic tail bounds of the form $P(g(\mathbf{X}) \geq t)$, where $g$ is a function on $\mathbb{R}^n$ and $\mathbf{X}$ is a random vector with possibly dependent coordinates. The approach leverages the notion of the southwest boundary of sets in the positive orthant, allowing for explicit, computable upper bounds under minimal assumptions. This framework generalizes and geometrizes union bounds and provides new closed-form inequalities for functions of dependent or independent random variables, including polynomials and traces of Schur multipliers on random matrices.

Methodology: The Southwest Boundary Construction

The central innovation is the use of the southwest boundary, $\partial_{SW}$, which identifies the set of minimal elements of a closed subset $V \subset [0,\infty)^n$ with respect to the coordinatewise partial order. By reflecting events into the positive orthant and focusing on $\partial_{SW} Q(V)$, where $Q$ maps vectors to their coordinatewise absolute values, the problem of bounding $P(g(\mathbf{X}) \geq t)$ reduces to analyzing minimal points on the boundary of the relevant set. This yields a general bound:

$P(g(\mathbf{X}) \geq t) \leq n s_t,$

where $g$0 is the unique value for which the curve $g$1 intersects $g$2, and $g$3 are tail bound functions for $g$4.

A key theoretical statement is that the southwest boundary is precise enough that, for any measurable set $g$5, the maximal probability over all measures with prescribed tail bounds is achieved on the southwest boundary pushforward measure, thus ensuring the optimality (subject to knowledge of the geometry of the boundary).

Closed-Form Results for Homogeneous Polynomials

The construction simplifies significantly when $g$6 is a homogeneous polynomial (plus constant). In this case, if all $g$7 are identical, the bound simplifies to:

$g$8

where $g$9 is a constant, $\mathbb{R}^n$0 is the polynomial degree, and $\mathbb{R}^n$1 are the monomial coefficients. Notably, this result holds regardless of the dependence structure between the components of $\mathbb{R}^n$2. The analytic tractability and lack of an independence assumption distinguish these results from most classical concentration inequalities.

Extension: Schur Multipliers on Random Matrices

The technique yields a new explicit tail inequality for the trace of a Schur multiplier acting on random matrices whose entries are subject to identical tail bounds. If $\mathbb{R}^n$3 is a $\mathbb{R}^n$4-linear Schur multiplier and $\mathbb{R}^n$5 are random matrices with $\mathbb{R}^n$6 for all entries, then:

$\mathbb{R}^n$7

This result generalizes classical tail bounds for matrix-valued random variables, again making no independence assumption and capturing the geometric structure of dependencies through the southwest boundary.

Comparative Analysis with Classical Concentration Inequalities

The paper provides an explicit comparison with Hoeffding-type inequalities and union bounds. For linear functions and sub-gaussian variables:

• The southwest boundary bound yields $\mathbb{R}^n$8, whereas the optimal Hoeffding-type bound is $\mathbb{R}^n$9 ($\mathbf{X}$0 the maximal sub-gaussian norm).
• The new bounds may outperform classical results for specific choices of coefficients or when $\mathbf{X}$1, but have a looser leading constant.
• The new bounds generalize the union bound for homogeneous functions and demonstrate strict improvements in cases such as the determinant of a random matrix, giving bounds that are better by a combinatorial factor ($\mathbf{X}$2) compared to the naive union bound.

Theoretical and Practical Implications

The approach gives the following implications:

• Optimal bounds for dependent variables: The framework does not rely on independence, and recovers tight bounds under only marginal tail control.
• Sharper union bounds: For polynomials and multilinear functions, the framework avoids over-counting arising in the union bound, accounting for overlap in variable occurrence.
• Applications to matrix analysis: New tail inequalities for Schur multipliers and determinants of random matrices offer tools for random matrix theory and multilinear spectral analysis.
• Desiderata for future inequalities: The notion of the southwest boundary could structure future improvements to concentration inequalities, especially in dependent or geometrically structured settings.

Future Directions

Further directions include:

• Extension to non-homogeneous or non-polynomial functions using approximate or local boundary analysis.
• Tightening constant factors and investigating matching lower bounds in the tail regime, especially for high-degree polynomials.
• Application to non-Euclidean or infinite-dimensional settings, exploiting the coordinatewise structure of tail bounds.
• Integration into high-dimensional probability and random matrix theory for functions where classical approaches are not so directly tractable.

Conclusion

The southwest boundary methodology presented in "Tail Bounds via Southwest Boundary" (2604.21113) yields new, explicit, and optimal tail bounds for a broad class of functions of random vectors with prescribed marginal tail behavior. The framework both recovers and improves upon union bound-based estimates, generalizes to non-independent settings, and offers new insight on tail probabilities for functions such as multilinear polynomials and traces of Schur multipliers. This framework enriches the analytical toolbox for concentration of measure in both classical and non-classical settings, and opens avenues for further geometric approaches to probability inequalities.