Extended Convolution Bounds Insights

• Extended convolution bounds are a unified framework of analytical inequalities that extend classical convolution estimates to control growth, tail risk, and spectral characteristics.
• They are applied to diverse areas including risk aggregation, analytic function theory, high-dimensional probability, convex geometry, spectral analysis, and algorithmic complexity.
• These bounds reveal phase transitions, sharp thresholds, and extremal structures that enable precise performance control in adaptive systems and deep learning.

Extended convolution bounds comprise a broad class of analytical inequalities that generalize and unify the estimation of convolutions in complex, algebraic, geometric, probabilistic, and algorithmic contexts. These bounds extend classical results by tightly controlling growth, tail, or spectral properties under convolution, and reveal phase transitions, sharp thresholds, and structural extremals across a variety of domains such as quantitative risk aggregation, high-dimensional probability, adaptive systems, convex geometry, spectral theory, and computational complexity.

1. Convolution Bounds in Probability and Quantitative Risk

Extended convolution bounds were introduced to sharpen extremal quantile and risk aggregation problems under dependence uncertainty. In the classical Fréchet problem—where the goal is to determine the maximal or minimal value of a functional $\Phi(\nu)$ of $S = X_1 + \dots + X_n$ subject to fixed marginals—recent works (Liu et al., 26 Nov 2025, Blanchet et al., 2020) provide convolution-based inequalities that represent the optimal risk in terms of quantile-based allocations. The central innovation is to express, for functionals such as range-Value-at-Risk (RVaR) or differences of quantiles, exact upper and lower bounds as explicit infima or suprema over simplex-constrained allocations of tail risk (see Table 1 below):

Functional	Aggregate Convolution Bound	Sharpness Conditions
$\sup_\nu RVaR_{\beta,\beta+s}(\nu)$	$$\inf_{(\beta_0,\dots,\beta_n)} \sum_i RVaR_{1-\beta_i-\beta_0, 1-\beta_i}(\mu_i)$$	Monotone tail densities
$\sup_\nu q^+_t(\nu)$	$$\inf_{(\beta_0,\dots,\beta_n)} \sum_i R_{\beta_i, \beta_0}(\mu_i)$$	Mutually exclusive/monotone tails

For RVaR aggregation, the paper "Extended Convolution Bounds on the Fréchet Problem" (Liu et al., 26 Nov 2025) establishes the following sharp inequality for $0$$R_{[r,r+s]}(\nu) \leq \sum_{i=1}^n \left[ \frac{1-r-\beta_i}{s} R_{[r, r+\alpha_i] \cup [r+\alpha_i+\beta_i, 1]}(\mu_i) + \left(1 - \frac{1-r-\beta_i}{s}\right) R_{[r+\alpha_i, r+\alpha_i+\beta_i]}(\mu_i) \right],$$ where $(\alpha_i, \beta_i)$ satisfy sum constraints on the interval size and tail mass. Such bounds are sharp (attained as equalities) when densities are monotonic over the relevant tails.

The duality theory further connects these to primal inf-convolutions and reveals that the structure of extremal copulas, attaining the bounds, can be explicitly characterized—typically as (counter-)comonotonic allocations on tail events (Blanchet et al., 2020). In the risk-sharing context, the minimal aggregate for risk measures averaging quantiles is achieved by comonotonic sharing of large losses and counter-comonotonic splitting of small/gain events (Liu et al., 26 Nov 2025).

2. Convolution Bounds in Analytic Function Theory: Univalence and Growth

Convolution bounds also describe fine-grained phase transitions in the univalent function theory, particularly for the Hadamard product of analytic maps $f_\alpha$ representing convex $2$-gons in the disk (Chuaqui et al., 2023). Given $f_\alpha(z) = [((1+z)/(1-z))^\alpha - 1]/(2\alpha)$ with $0<\alpha<1$, the convolution $f_\alpha * f_\beta$ exhibits sharp trichotomy based on $S = \alpha+\beta$:

• If $S<1$, the convolution is bounded and extends analytically beyond $|z|=1$.
• For $S=1$, the growth is logarithmic: $(f_\alpha*f_\beta)(z) \sim -(\ln(1-|z|))^{-1}$ as $z\to 1$.
• If $S>1$, the growth is polynomial: $(1-|z|)^{S-1} (f_\alpha*f_\beta)(z) \to C$.

These growth rates are precisely characterized using Taylor coefficient asymptotics, recurrence relations, and the geometry of the image domains via the "angle at infinity," which encodes the asymptotic sector opening at infinity for the convex mapping. For $n$-fold convolutions with random $\alpha_j$, the probability that the convolution is unbounded is exactly $1/n!$, tracing to the simplex volume for sums of independent uniforms on $(0,1)$ (Chuaqui et al., 2023).

3. High-Dimensional Probability: Shearer-Type and Stability Inequalities

Extended convolution bounds in high-dimensional probability control mixing, concentration, and stability by bounding the Poincaré constant and related functionals (entropy, Fisher information) under convolutions of probability measures (Courtade, 2018). The central Shearer-type inequality generalizes subadditivity and monotonicity: $$C_P(\mu_1 * \dots * \mu_n) \leq \frac{1}{r} \sum_{S \in \mathcal{C}} C_P(*_{i \in S} \mu_i),$$ where $\mathcal{C}$ is a family of subsets covering indices, and $r$ is the minimum number covering each index. This yields the monotonicity of the Poincaré constant along the CLT convolution chain and dimension-free stability corrections to naive subadditivity, substantially improving classical one-dimensional estimates. The variational projection lemma that underpins these inequalities is the linearization of Shearer's entropy inequality and connects with monotonicity of Fisher information and entropy as well.

4. Extended Convolution Bounds in Convex and Discrete Geometry

In convex geometry, convolution bodies interpolate between Minkowski sums and projection bodies. The $k$-th limiting convolution body $C_k(K, L)$ of convex bodies $K,L\subset\mathbb{R}^n$ admits sharp volume bounds generalizing Rogers–Shephard's and Zhang's inequalities (Alonso-Gutiérrez et al., 2013): $$|C_k(K, L)| \geq \frac{ |K| W_{n-k}(L) + |L| W_{n-k}(K) }{ W_{n-k}(K \cap (-L)) },$$ with equality if and only if $K = -L$ is a simplex. For $k=n$, this recovers the polar projection body and Zhang's reverse Petty-projection inequality. These inequalities follow from delicate layer-cake integrations, Brunn-Minkowski convexity, and Crofton's formula.

5. Convolution Bounds in Spectral, Algorithmic, and Functional Analysis Contexts

Several domains leverage extended convolution bounds in controlling spectral norms, complexity, and analytic behavior:

Spectral Norms in Deep Learning: Recent work (Grishina et al., 18 Sep 2024, Singla et al., 2019) establishes that the spectral norm of the Jacobian of a convolutional layer $T$ in a CNN is optimally bounded by the tensor spectral norm $\|K\|_\sigma$ of the kernel times a tight filter-dependent factor: $$\|K\|_\sigma \leq \|T\|_2 \leq \sqrt{h w} \, \|K\|_\sigma,$$ substantially improving previous "Fantastic Four" matrix-unfolding bounds and enabling efficient, differentiable, provably accurate regularization during training.

Finite Free Convolution and Polynomial Root Bounds: Advanced polynomial convolution inequalities, such as the submodular inequalities for largest roots proved by Leake–Ryder (Leake et al., 2018), extend the Marcus–Spielman–Srivastava root barrier method to all differential operators preserving real-rootedness. The resulting largest-root bounds unify spectral interlacing and discrepancy theory, with explicit counterexamples delineating the sharp boundary for multivariate extensions.

Convolution Powers and Local Limit Theorems: In the context of functions on $\mathbb{Z}$, convolution power sup-norm bounds,

$$C n^{-1/m} \leq \| \varphi^{(n)} \|_\infty \leq C' n^{-1/m},$$

where $m$ is determined by the major-arcs expansion of the Fourier transform, generalize the classical heat kernel local limits and show that attractors can be oscillatory functions (e.g., Airy functions) in the nonpositive, complex, or defective cases (Randles et al., 2012).

Adaptive Control and Signal Processing: In $d$-step-ahead adaptive control, the closed-loop regressor admits a uniform linear-like convolution bound,

$$\|\phi(t)\| \leq c \lambda^{t-t_0} \|\phi(t_0)\| + c \sum_{j=t_0}^{t-1} \lambda^{t-1-j} (|y^*(j)| + |w(j)|),$$

granting exponential stability, $\ell_2$-bounded noise gain, and robustness to plant variation (Miller et al., 2019).

Algorithmic Lower Bounds: The cell-probe complexity of online convolution algorithms is bounded below (and above, by optimal constructions) by $\Theta\big((d/w) \log n\big)$, with the information transfer method rigorously linking time complexity to memory traffic through Toeplitz-matrix rank arguments (Clifford et al., 2011).

6. Interrelations and Structural Insights

A recurring theme in these results is that sharp convolution bounds are often attained by highly structured solutions—comonotonic or anti-comonotonic extremal couplings in risk aggregation, simplex extremality in geometric inequalities, or rank-constrained obstructions in complexity lower bounds. Phase transitions and monotonicity properties (e.g., thresholds for boundedness, switching of extremals due to tail monotonicity, or symmetry breakings in local limit attractors) typify the fine control enabled by these extended bounds. Many fundamental inequalities, previously considered case-specific, are now understood as projections or inf-convolutions in generalized functional or geometrical frameworks.

7. Open Directions and Further Applications

The theory of extended convolution bounds continues to evolve. Current and prospective lines include full submodular Horn-theory for polynomial convolutions (Leake et al., 2018), multidimensional and dynamic extensions for risk measures, distributionally robust optimization using inf-convolutions, and deeper connections to majorization, high-dimensional limit theorems, and functional inequalities in geometric and analytic settings. The explicit characterization of extremal configurations, the design of efficient computation schemes for high-dimensional or online problems, and the identification of new sharp constants across disciplines remain active areas of mathematical research.

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References:

• "Extended Convolution Bounds on the Fréchet Problem: Robust Risk Aggregation and Risk Sharing" (Liu et al., 26 Nov 2025)
• "Convolution Bounds on Quantile Aggregation" (Blanchet et al., 2020)
• "On the convolution of convex 2-gons" (Chuaqui et al., 2023)
• "Bounds on the Poincaré constant for convolution measures" (Courtade, 2018)
• "Volume inequalities for the $i$-th-Convolution bodies" (Alonso-Gutiérrez et al., 2013)
• "Tight and Efficient Upper Bound on Spectral Norm of Convolutional Layers" (Grishina et al., 18 Sep 2024)
• "Fantastic Four: Differentiable Bounds on Singular Values of Convolution Layers" (Singla et al., 2019)
• "On the Further Structure of the Finite Free Convolutions" (Leake et al., 2018)
• "On the convolution powers of complex functions on Z" (Randles et al., 2012)
• "Classical d-Step-Ahead Adaptive Control Revisited: Linear-Like Convolution Bounds and Exponential Stability" (Miller et al., 2019)
• "Tight Cell-Probe Bounds for Online Integer Multiplication and Convolution" (Clifford et al., 2011)
• "A Bounded $p$-norm Approximation of Max-Convolution for Sub-Quadratic Bayesian Inference on Additive Factors" (Pfeuffer et al., 2015)
• "Generalization Bounds for Convolutional Neural Networks" (Lin et al., 2019)