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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2604.21113 |
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| _version_ | 1866910161741807616 |
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| author | Harrison, Stephen Jordan |
| author_facet | Harrison, Stephen Jordan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21113 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tail Bounds via Southwest Boundary Harrison, Stephen Jordan Probability 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. |
| title | Tail Bounds via Southwest Boundary |
| topic | Probability |
| url | https://arxiv.org/abs/2604.21113 |