Tail Bounds via Southwest Boundary

TL;DR

This paper introduces a method to derive explicit upper bounds for tail probabilities of functions of random variables using the southwest boundary, applicable to homogeneous polynomials and matrix traces without independence assumptions.

Contribution

The authors develop a novel boundary-based approach for tail bounds that yields explicit formulas for complex functions like polynomials and matrix traces.

Findings

Derived explicit tail bounds for homogeneous polynomials of random variables.

Provided a closed-form tail bound for the trace of Schur multipliers on random matrices.

Bounded tail probabilities without requiring independence or dependence assumptions.

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) $\partial_{\mathrm{SW}} Q(g^{-1}[t,\infty))$, 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 $L(s)=(f_1^{-1}(s),\dots,f_n^{-1}(s))$ 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.