Papers

This paper investigates the problem of statistical inference for a mixture distribution consisting of a discrete and a continuous component, with a particular focus on the class of rational-infinitely divisible distributions. We consider non-parametric estimation of both components of the mixture as well as the quasi-Lévy measure, assuming that the mixture belongs to the class of rational-infinitely divisible distributions. We propose an estimation framework based on band-limited kernels, which are the functions characterized by compactly supported Fourier transform. Under mild assumptions, the proposed estimators are theoretically shown to achieve polynomial (and in some cases even almost parametric) convergence rates. Finally, we demonstrate the numerical performance of the algorithm on simulated examples.

We investigate approximation of random differential equations driven by semimartingales satisfying a singularly perturbed Langevin equation with scaled mixing random force. By a diffusion approximation approach, we explore the limit of the rough path lift of this semimartingale, and a universal limit theorem is applied to identify the limit of random differential equation. A structurally parallel proof also applies to establish an iterated weak invariance principle for the mixing random force, which is itself an independent interesting result. We find that, the limit of both of the second-level processes, have the form of iterated integral of Stratonovich form plus an anti-symmetric part which is proportional to the time increment.

We develop thermodynamic formalism for random Anosov maps and uniformly Hölder random potentials. We assume uniform fibre hyperbolicity given by deterministic invariant cone fields, a one-dimensional stable direction, and a fibrewise mixing condition whose mixing time may depend on the base point. To do so, we construct adapted projective cones for the random Perron–Frobenius cocycle and prove that the cocycle contracts the associated Hilbert projective metrics. This allows us to construct a $\mathbb P$-relative equilibrium state, prove its uniqueness, and establish quenched exponential decay of correlations.

Post-hoc conditioning of pretrained diffusion models can be addressed using Sequential Monte Carlo (SMC) methods. By evolving an interacting particle system, SMC-guided diffusion samplers combine unconditional reverse-diffusion dynamics with sequential reweighting to approximate conditional distributions. Nevertheless, even in the infinite-particle limit, the implemented sampler may differ from the ideal conditional target because of errors in the diffusion model, its numerical implementation, and the guidance mechanism. We characterize how these local errors propagate through forward-smoothing kernels, which jointly account for the reverse dynamics and the remaining conditioning information. This yields non-asymptotic error bounds that capture both finite-particle fluctuations and approximation errors arising from initialization, numerical integration, score approximation, and potential design. In doing so, we extend stability guarantees for diffusion models to the conditional setting. Finally, we apply our framework to several state-of-the-art SMC-guided diffusion algorithms, providing a unified theoretical perspective on their approximation mechanisms and sources of error.

In this paper, we study the asymptotic behaviour for distribution of supercritical branching process in a Markovian environment with linear fractional offspring distributions. A phase transition in the behaviour of the process is discovered, through investigating the strongly and intermediately strongly supercritical cases.

In this article, we first discuss how triplewise independent random variables (rvs) are connected to a complete bipartite graph. Using the connection, we construct a sequence of triplewise independent rvs. We next consider a variance gamma (VG) approximation of sums of such triplewise independent rvs. Using Stein's method and the generalized zero-bias transformation, we obtain our bounds. Related limit theorems are also discussed.

Every square matrix is uniquely the sum of a symmetric matrix and a skew-symmetric matrix. We extend this familiar fact to higher-order tensors: every cubic $k$-tensor is uniquely the sum of an $m$-piecewise symmetric tensor and a $(k-m)$-piecewise skew-symmetric tensor, for each choice of $m\leq k$. We study these tensor spaces via the combinatorics of word descents and compositions. Under Schur–Weyl duality, the two spaces naturally decompose into descent representations of the symmetric group. In fact, we prove a variant of Solomon's classical decomposition of its group algebra that explains this decomposition. Our motivation stems from signature tensors in stochastic analysis, algebraic geometry and data science: more specifically, we show that our tensor space decompositions determine the vanishing ideals of the signatures of piecewise linear paths with a fixed number of segments.

For a nonnegative supermartingale $(M_n)$ with $M_0=1$, let $T_b$ be the first time it reaches a level $b>1$. Ville's inequality gives only the bound $\mathbb P(T_b<\infty)\le 1/b$. We prove the exact identity $\mathbb P(T_b<\infty)=(1-D_b-R_b)/(b+O_b)$, where $O_b$ is the expected overshoot at crossing, $D_b$ is the cumulative predictable supermartingale loss before crossing, and $R_b=\lim_n\mathbb E[M_n{\bf 1}\{T_b>n\}]$ is the residual mass carried by paths that never cross. Thus the slack in Ville's inequality is decomposed completely into overshoot, loss, and survival. The proof is given first in the absorbing case, where non-crossing paths decay to zero and $R_b=0$, and then in full generality using a conservation identity for stopped martingales at extended-valued stopping times. The formula yields Ville's inequality as a corollary, gives a sharp tightness criterion, and is worked out in examples including double-or-absorb processes, multiplicative decay, gambler's ruin, likelihood-ratio martingales, and a bounded martingale with $R_b>0$. Finally, with $b=1/\alpha$, the same identity gives the exact type-I error of a sequential e-value test and identifies when threshold recalibration can safely recover unused significance.

We study the small-noise asymptotics of Euclidean heat regularizations of probability measures supported on manifolds with corners. Near a boundary or corner stratum, the relevant regime is a conical boundary layer in which the observation point approaches the stratum at the same scale as the Gaussian smoothing parameter. After rescaling this layer, the support is replaced to leading order by its inward tangent cone. We prove a two-term expansion for the heat-regularized density in this regime. The leading coefficient is the Gaussian mass of the linearized cone, weighted by the density on the support and by the adapted corner Jacobian; the first correction records the variation of the density, the Jacobian, and the quadratic geometry of the embedding. A localization argument then yields the corresponding expansion for the full heat regularization, with the nonlocal contribution exponentially small. From this density expansion we derive logarithmic asymptotics and uniform expansions for the score, the log-Hessian, and the scale derivative of the score. These formulas show how lower-dimensional support, boundary faces, corners, and curvature are encoded in the singular differential structure of small-noise Gaussian regularizations.

Several positivity questions in stochastic processes, dependence modeling, fractional analysis, and renewal theory reduce to a common recognition task: after normalization, identify the object as a Laplace transform, a potential density, an inverse-flow coefficient, or a finite kernel average, and then read the sign pattern from that representation. We develop this recognition calculus for completely monotone functions, Bernstein functions, special Bernstein functions, and probabilistic realizations through subordinators and mixing measures. The main affirmative results settle three narrowly stated source questions in the conventions used by their source papers. Möhle's Problem 6.3 on the block-counting process of exchangeable coalescents with residual singleton mass (dust) is proved by a finite-simplex ordered-pair kernel certificate. For the Pearse–Bondell power-divergence copula generators, we prove complete monotonicity of the inverse throughout the remaining strict negative range $\lambda\le-1$ identified in their Section 3.8. Together with the special cases already verified in the source paper, this yields Archimedean copulas in every dimension for $\lambda\le-1$. The Bendikov–Cygan monotonicity question for discrete renewal sequences attached to special Bernstein functions is answered by representing the potential kernel as a Gamma average of a nonincreasing density. Supporting representation and boundary results cover Sibisi's Prabhakar–Pollard $Q$-measure, the Mecke–Nagel–Weiss atom at zero, and the cubic branch criterion $a^2\ge3b$.

We study two recently discovered "dimension-free" Monte Carlo sampling algorithms, the multiproposal and multiple-try preconditioned Crank-Nicolson methods (mpCN and MTpCN). These methods were designed to address certain non-parametric (i.e. infinite-dimensional) sampling problems, defined relative to a Gaussian reference measure, by combining proposal and acceptance mechanisms that take non-trivial advantage of parallel computing architectures. We provide the first rigorous analysis of both algorithms, establishing exponential convergence to the target measure through the weak Harris framework, both for a finite number of proposals and in the infinite-proposal limit. The resulting mixing rates are independent of the dimension and uniform in the number of proposals, and apply to targets with bounded, Lipschitz log-likelihoods, without requiring convexity. At the center of the analysis are two new coupling constructions, together with analytical tools of independent interest, yielding Wasserstein contraction estimates, $L^2$ spectral gaps, and associated statistical guarantees (laws of large numbers, central limit theorems, and non-asymptotic concentration bounds) for the corresponding Monte Carlo estimators. These theoretical results are complemented by a numerical study on benchmark problems with complex posterior geometries and high-dimensional structure, comparing mpCN and MTpCN against standard pCN and independent parallel-chain implementations. The experiments indicate that the multiproposal methods can offer a shorter warm-up phase and greater robustness to the choice of tuning parameters as the number of proposals grows.

In static risk measurement, law invariance expresses the principle that the risk of a position should depend only on its distribution, and not on the particular probability space on which it is represented. In a dynamic setting, the same principle leads naturally to adapted law invariance: the risk assessment should depend only on the probabilistic structure of the financial position together with the way information about it is revealed over time. We show that, for time-consistent risk measures, adapted law invariance is equivalent to a recursive one-step conditional-law representation. More precisely, assuming Fatou regularity, the one-step risk evaluations are exactly conditional lifts of static law-invariant risk measures, and the full dynamic risk measure is obtained by backward composition of these one-step maps. Convexity and coherence of the dynamic risk measure are characterized by the corresponding properties of the static one-step risk measures. This identifies adapted law invariance as the dynamic counterpart of ordinary law invariance. It also clarifies the strength of terminal-law invariance, as it appears in the rigidity theorem of Kupper and Schachermayer: it does not distinguish risks with the same distribution but different times of resolution. We further obtain an adapted Kusuoka representation in the coherent case and establish an extension of the Kupper–Schachermayer theorem.

The Dvoretzky–Kiefer–Wolfowitz–Massart inequality gives an upper bound on the probability that the empirical distribution function of a finite sequence of independent random variables deviates from its theoretical value. It is widely used in statistics in tests and in the production of confidence intervals. The original proof by Massart, later reformulated by Dudley, is very long and technical. Recently, Reeve slightly extends Dvoretzky–Kiefer–Wolfowitz–Massart inequality and presents an alternative, much shorter proof, that uses a continuous time martingale. We present a simpler and less technical proof of the original Dvoretzky–Kiefer–Wolfowitz–Massart inequality, based on a discrete-time martingale and Sion's minimax theorem.

We solve two open problems about spatial (or geometric) quantiles and depth. First we show that in infinite dimension, the spatial distribution function and the associated spatial quantiles characterize the underlying distribution, which has been established in Koltchinski (1997) in finite dimension but remained unknown in infinite dimension. Second, and more surprisingly, we show that the spatial depth also fully characterizes probability measures, which has been an open problem even in finite dimension since the introduction of these concepts in Chaudhuri (1996) and Vardi & Zhang (2000). Our results provide theoretical foundations for nonparametric depth-based statistical inference and introduce novel proof techniques to investigate these questions for other depth and quantile concepts.

We prove a canonical limit theorem for the macroscopic Feynman cycles of finite-volume ideal Bose gases. Cycles carry marks in a general Polish space $\mathsf{M}$, encoding spatial, geometric, spectral, or internal data. After removing a deterministic background density $\rho_{\mathrm{bg}}$, the marked macroscopic cycle process converges in the canonical ensemble to a marked Poisson–Kingman bridge of total mass $\rho - \rho_{\mathrm{bg}}$. The bridge is constructed from a marked Poisson point process with intensity $x^{-1}\eta_x(dm)\,dx$, conditioned on total mass~$\rho - \rho_{\mathrm{bg}}$, where the kernel $x \mapsto \eta_x$ and its total-mass profile $\phi(x) = \eta_x(\mathsf{M})$ are determined by the low-energy spectral data visible on the scale $j \sim V_L$. When $\phi$ is constant, the bridge reduces to a Gamma bridge and the ranked cycle lengths follow the Poisson–Dirichlet law. We verify this for the ideal Bose gas in dimension $d > 2$ under periodic, Dirichlet, and Neumann boundary conditions: in all three cases $\phi \equiv 1$ and the ranked lengths converge to $\mathrm{PD}(0,1)$, while the mark kernels distinguish the three models through their winding, killed-bridge, and reflected-bridge geometry. When $\phi$ is not constant, the bridge is no longer Gamma and the ranked lengths are not Poisson–Dirichlet. As a concrete example, a critical double-well potential whose tunnelling splitting satisfies $V_L \Delta_L \to \gamma$ gives $\phi_\gamma(x) = 1 + e^{-\beta\gamma x}$; more generally, a finite-type visible spectrum with $Q$ components yields $\phi(x) = \sum_{r=1}^{Q} \theta_r e^{-\beta\lambda_r x}$. These results identify Poisson–Kingman bridges as the canonical universality class for marked macroscopic Bose cycles, with the visible low-energy spectrum selecting the particular bridge.

We mathematically model the dynamics of the number of migratory fish observed at a fixed location along a river in a random environment. Particularly, as a new approach, we construct a stochastic differential equation that incorporates the influence of environmental factors on the fluctuations in the start and end of migration. The model is a diffusion bridge with a non-Lipschitz diffusion coefficient, called the Cox-Ingersoll-Ross bridge, and has random initial and terminal times arising from time-change, so that the influences of environmental factors can be efficiently incorporated. The well-posedness of the model is first established, which is considered novel and significant in applied mathematics. Second, we estimate the parameters of the model based on the latest multiyear daily data set for the upstream migration of Plecoglossus altivelis altivelis (Ayu) by relying on the hypothesis that water temperature affects the migration of the fish, which has been suggested in existing studies. We also explore the application of the proposed model to the challenging task of analyzing environmental DNA data. This study advances the development of a theory of fish migration that is simple yet can take environmental factors into account.

Random exploration is usually quantified by how fast new space is found, from the range of a single walker to the territory collectively covered by many walkers. In competitive exploration, first arrival secures an exclusive resource, as when foragers compete for food items or agents capture distributed targets. It is then no longer enough to know which sites have been discovered: one must determine, for each discovered site, which searcher reached it first. We introduce the discovery share $X_n$, the fraction of the first $n$ collective discoveries secured by a tagged searcher. For two identical competitors, exchange symmetry fixes $\langle X_n\rangle=1/2$, but the central question is whether this equal split emerges in each long exploration history or only on average, i.e. whether early competitive advantages are erased or persist. Here we show that the answer is controlled by the spectral dimension $d_s$, defined by the large-time decay of the probability that a single searcher is at its starting point after $t$ steps, $p_0(t)\sim t^{-d_s/2}$. Across ordinary diffusion, long-range superdiffusion and subdiffusion induced by crowding or memory, $d_s$ separates persistent randomness in recurrent exploration $(d_s<2)$, anomalously slow non-Gaussian concentration for $2\le d_s<3$, and Gaussian concentration, logarithmically corrected at $d_s=3$, for $d_s\ge3$. For $d_s\ge2$, we derive exact asymptotic variances, including prefactors, and the discovery scale on which competitive imbalances are erased. Two-point correlations of first-discovery labels identify the memory mechanism behind these regimes. The same phase structure persists under changes in geometry, competitor heterogeneity, number of competitors and memory, revealing a general fluctuation theory of first-arrival inequalities.

In this paper, we derive the Onsager–Machlup functional for a class of degenerate stochastic differential equations on $\mathbb{R}^{m+n}$ driven by $n$-dimensional fractional Brownian motion with time-dependent diffusion coefficients, where the Hurst parameter satisfies $H\in(1/4,1)$. The main difficulty arises from the interaction between the degenerate structure and the multidimensional fractional noise, which produces nontrivial coupling terms under small-ball conditioning. By combining the Gaussian correlation inequality with an approximation argument for infinite-dimensional convex sets, we establish a decoupling mechanism that allows these coupling effects to be controlled by unconditional Gaussian expectations. Furthermore, through regularity estimates for the degenerate components and sharp Hölder norm estimates, we eliminate the remaining coupling contributions and obtain an explicit expression for the Onsager–Machlup functional. As applications of the derived functional, we obtain the corresponding constrained Euler–Lagrange equations characterizing the most probable transition paths of the system, and establish a sufficient condition under which the stochastic differential equation preserves the most probable path.

Krajči, Liu, Mikeš, and Moser proved in 2015 that the redundancy of binary Shannon-Fano coding is always below one bit. We sharpen this to a bound depending on the largest source probability $p_1$: an explicit seven-piece envelope $R<f(p_1)$. The envelope equals the exact supremum of $R$ given $p_1$ for every $p_1\ge\tfrac12$ and on a subinterval below $\tfrac13$, and gives the cap $R<\tfrac52-\tfrac56\log_2 5=0.5651$ for $p_1<\tfrac12$. It is the first $p_1$-dependent redundancy bound for Fano codes. The method is more sophisticated than the approach typical for Huffman codes: Fano trees are built top-down by contiguous balanced splits and lack the sibling property. From the $R<1$ theorem the rest follows from the Fano recursion, through a min-corrected affine potential and a no-burial lemma. Every scalar inequality in the proof reduces to a comparison of integer powers.

We construct random hives from deformed GUE minor processes. Starting from two independent diagonally deformed GUE matrices \[ X=\sqrt{n}(wG+uD),\qquad Y=\sqrt{n}(w'G'+u'D'), \] where \(D,D'\) are diagonal and have GUE spectra, we use their minor processes to form a double hive and then apply the octahedron recurrence. Under the matching condition \[ \frac{u}{w^2}=\frac{u'}{(w')^2}, \] we prove that the resulting hive law is close, in relative entropy, to a GUE hive law. More precisely, if \[ a^2=w^2+u^2,\qquad b^2=(w')^2+(u')^2, \] then the produced hive density $q_n$ satisfies \[ D_{\mathrm{KL}}\!\left( q_n\, \middle\|\, \operatorname{Density}\bigl(H_n(a\sqrt n,b\sqrt n,c_{**}\sqrt n)\bigr) \right) = O(n\log n). \] The third scale $c_{**}$ is determined by a limiting tetrahedral optimization problem; equivalently, writing \(\delta=u+u'\), \[ \delta^2 = \frac{ 2c_{**}^4(c_{**}^2-a^2-b^2) }{ (c_{**}^2-a^2+b^2)(c_{**}^2+a^2-b^2) }. \] Thus the construction realizes GUE hive laws, up to subleading relative entropy, throughout the right-angled and obtuse regime. The introduction records two explicit surface-tension approximations and numerical comparisons which motivated the construction.

This second part of the series develops the statistical and applied layer of the theory built in Part I [1]. Unlike current Hilbert diffusion models [2-5], we focus not only on well-posedness of the infinite-dimensional generative dynamics, but on the noncommutative $A$-geometry, explicit entropy constants, and checkable lower bounds. The analytic estimate of Part I reduces stability of the backward evolution to control of a validation error $\mathcal{E}_{val}$. We prove three results. First, we construct a cylindrically weak denoising label for the infinite-dimensional score field, consistent with the exact logarithmic gradient. Second, the local parametric closure is replaced by a global nonparametric score closure, its complexity controlled via the Dudley entropy integral and uniform empirical bounds in $L^2(\nu_*;A)$. Third, for the trace-smoothed nonlinear class we construct minimax bounds by the Le Cam-Assouad method, showing the root statistical rate cannot be improved without additional quadratic structure. The final section applies the theory to anisotropic diffusion in a nanosystem model and checks the constants $c_A,C_A,C_{LSI}^A$ independently of any smallness condition, establishing explicit accuracy orders: the parametric trace-smoothed minimax score risk is $p/M$, while the uniform error of the risk functional is $\sqrt{p/M}$, showing the statistical plateau of order $p/M$ is not a proof artifact. The applied layer closes with an independent analytic benchmark for the isotropic case, a comparison with classical cell homogenization, and an approximation theorem for smooth logarithmic gradients by $A$-adapted spectral networks.

For a finite nonempty poset \(F\), the normalized probabilistic powerdomain \(\Vone(F)\) is an RB-domain exactly when \(F\) is a finite rooted tree. We extend this classification to arbitrary nonempty dcpos from the viewpoint of forbidden structure. The principal-ideal chain condition is expressed by the absence of a lower fork, i.e. a triple \((x,y,t)\) with \(x\leq t\), \(y\leq t\), and \(x\parallel y\). A useful point is that any dcpo $P$ without lower forks is continuous. For normalized valuations the least element remains necessary, and we prove \[ \begin{aligned} \Vone(P)\text{ is RB} \Longleftrightarrow \Vone(P)\text{ is a pointed bc-domain} \Longleftrightarrow P\text{ has a least element and contains no lower fork}. \end{aligned} \] For subprobability and extended valuations, the analogous classifications hold without the pointedness assumption on \(P\).

We study the top singular value for a sum of $m$ independent $n \times n$ random matrices, each of which is a product of $N$ i.i.d. $n\times n$ Gaussian matrices. Our main conceptual observation is that when $m,n,N\rightarrow \infty$, the top singular value coincides with the partition function in a random energy model at the inverse temperature $\beta=\sqrt{2(N-1)/(n\log m)}$, with energies depending on the ratio $N/n$. We provide several non-asymptotic results making this approximation precise.

We prove a Hölder-type regularity estimate for the martingale integrand of a backward stochastic Volterra integral equation (BSVIE). The estimate is formulated in $L^p(\Omega)$ after averaging in $L^2$ over the first time variable, and gives an averaged Hölder estimate of order $1/2$ in the second time variable. Our approach is based on the approximation of the BSVIE by a system of BSDEs. For this system, we establish a uniform regularity estimate for the martingale component using Malliavin calculus, and then pass to the limit to obtain the result for the BSVIE. We allow for a general Malliavin differentiable free term and generator. In particular, neither is assumed to come from a forward stochastic differential equation or to have a specific functional form. We also propose an explicit Euler scheme for the approximating BSDE system and show that the regularity estimate yields a convergence rate for the resulting discrete approximation of the BSVIE.

We study collision properties of simple random walks in unimodular random rooted graphs. This work continues the study initiated in~\cite{HutchcroftPeres2015}: under recurrence and an integrability condition on the root, two independent random walks collide infinitely often a.s. We prove that, under transience and an integrability assumption involving the Green function on the root, two independent random walks collide only finitely often a.s. We apply these results to several random graphs with unbounded degree: the Gilbert graph, the Delaunay graph, the Gabriel graph; and the long-range percolation model. We use these collision properties to characterize stationary measures of the voter model on these graphs.

We solve in semi-explicit form a class of non-Markovian stochastic optimal control problems with path-dependent rewards, using path signatures. We reformulate the control problem as the computation of Laplace transforms of signature functionals thanks to the Boué-Dupuis representation. Exploiting recent signature representations of such transforms on tensor algebras, we determine the value process and the optimal control through an infinite-dimensional system of Riccati equations on the extended tensor algebra. We establish an explicit feedback representation of the optimal control and the value process as an infinite linear combination of the time-extended signature of the controlled process, with time-dependent coefficients. The expansions being intrinsically local, we propose a dynamic recentering algorithm to ensure a global representation over the entire time horizon. We illustrate the approach on genuinely path-dependent, non-linear examples that go beyond the tractable linear-quadratic setting, including the tracking of linear functionals of the signature and signature lifts of Volterra control problems.

In a 1934 Annals of Mathematics paper, Samuel S. Wilks posed the problem of computing joint moments of disjoint principal minors of Wishart random matrices, describing the general case as “extremely complicated.” These moments arise in classical multivariate statistics, including multivariate regression, analysis of variance, generalized correlation coefficients, and likelihood ratio testing. We solve Wilks' problem for nonnegative integer exponents and any finite number of disjoint principal minors using Stein's method, specifically the Wishart Stein characterization recently developed by Bailly et al. (2026). The Wishart Stein identity yields a degree-lowering recursion that terminates after finitely many steps and evaluates the desired moment exactly. Numerical experiments validate our recursive formulas against direct Monte Carlo simulations from the Wishart distribution.

Positive operator-valued measures on a tree admit intrinsic local coordinates coming from the way each cylinder value splits into its children. We show that these local splittings, taken on the range spaces of the cylinder values, recover the measure and at the same time build an intrinsic direct limit dilation whose cylinder projections yield the minimal Naimark dilation. In these coordinates, the commutant of the dilation becomes a martingale calculus on the range spaces. This gives local descriptions of extremality and domination, and it also yields a bounded change-of-measure transform that updates the tree coordinates in a natural way. For self-adjoint range martingales we obtain a quadratic variation formula from the range space isometries, and the associated local variance terms detect the projection-valued case.

For a multidimensional reflected diffusion, determining whether the associated basic adjoint relationship (BAR) uniquely characterizes the stationary distribution is a basic uniqueness problem in the BAR approach. The problem has remained unresolved for more than 35 years since the introduction of the BAR approach. In this paper, we resolve the finite-signed uniqueness problem for stable Harrison–Reiman data with a nonsingular $M$-matrix reflection matrix. The proof uses pathwise differentiability of the reflected diffusion implies feasible directional differentiability of the probabilistic resolvent to show that, at boundary points, its one-sided initial-state derivative factors through the tangent projection and vanishes along active reflection directions. An interior one-sided convolution then yields smooth test functions whose oblique derivatives are uniformly bounded and converge pointwise to zero on each closed face. The interior signed measure is consequently invariant for the reflected semigroup. The proof was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors. We also show that the nonsingular $M$-matrix assumption is structural. In the larger completely-$\mathcal{S}$ class, a nonsingular reflection matrix with a singular proper principal block admits boundary gauges supported on lower-dimensional strata. Under standard exponential ergodicity and a mild one-step regulator bound, these gauges produce nonzero zero-mass signed BAR tuples; indeed the zero-mass interior BAR coordinates contain an infinite-dimensional subspace. A four-parameter three-dimensional family, including an explicit rational example, verifies the obstruction. Thus the finite signed version of the Dai–Dieker question has a positive answer in the Harrison–Reiman $M$-matrix class and a negative answer in a natural completely-$\mathcal{S}$ extension.

By a convergent set is meant a set of stochastic matrices where every infinite product of matrices from every compact subset converges to a rank one matrix. Well-known examples include the set of all scrambling matrices, the set of all stochastic matrices with all diagonal entries positive and a rooted graph, the set of all Sarymsakov matrices, and the set of doubly stochastic matrices with positive diagonal entries and a weakly connected graph. It is known that every infinite product from each compact set of every convergent set converges to its limit exponentially fast, but not much is known about the rate of convergence when not all matrices involved are scrambling matrices. This paper deals with bounding the rate of convergence in convergent sets using submultiplicative seminorms. It is shown that only in some convergent sets all matrices are contractions in the same seminorm, and in particular that this method cannot be used to determine the convergence rate for the class of matrices with positive diagonal entries and a rooted graph. As a second contribution, it is shown that for every compact convergent set and every submultiplicative seminorm, there is a finite number $k$ such that all products of $k$ matrices from the set are contractions in the seminorm. Finally, several open questions are posed for future research.