Papers

We introduce a self-exciting point process with power-law intensity dynamics that admits a finite-dimensional Markovian state representation. The model is constructed to preserve the local jump and slope update structure of power-law Hawkes processes, while replacing global history dependence with a nonlinear Markov chain governing the intensity dynamics. Within a general state-space framework, we establish irreducibility, aperiodicity, and the T-chain property under mild regularity conditions on the inter-arrival time distribution. Under an explicit stability condition, we further prove that the latent state process is positive Harris recurrent, ensuring the existence of a unique invariant distribution. Simulation results based on the local Whittle estimator show that the proposed Markovian intensity model exhibits long-memory behavior near the boundary of the stability region.

Stochastic reaction networks often involve components at low copy number, where individual production and degradation events generate substantial fluctuations. Yan et al.\ conjectured in 2019 that for networks with linear degradation and arbitrary cross-regulatory production rates, feedback cannot suppress the stationary fluctuations for each component below the fluctuations of its constant-rate counterpart. Their formulation allows random burst sizes $K_i$: the unit-birth case has $K_i\equiv1$, the biologically important burst model takes $K_i$ to be geometrically distributed, but more general positive integer-valued burst laws are also permitted. The conjecture was recently proved for unit births, $K_i\equiv1$. We show here that the conjecture is false in general by constructing a two-component network with bounded production rates and burst sizes in $\{1,2\}$ for which both stationary Fano factors lie below their common constant-rate baseline. We then prove the conjecture for positive geometric bursts, the canonical burst model in stochastic gene expression. For arbitrary regulatory architecture and nonlinear cross-regulatory production rates, we prove an exact weighted tradeoff that rules out simultaneous suppression of every component below its geometric-burst baseline. We also prove a complementary structural impossibility result for arbitrary positive integer-valued burst laws with finite second moments: if the activating and inhibiting interactions have a globally consistent sign structure, in the sense that every cycle of the regulatory interaction graph contains an even number of negative interactions, then every component individually satisfies $F_{X_i}\ge B_i$, where $F_{X_i}$ is its stationary Fano factor and $B_i$ is its constant-rate burst baseline.

Many instances of sequential sampling, including audit and inspection scheduling, representative sampling, and treatment assignment, require selections to be distributed evenly without becoming easy to anticipate or exploit. We study a family of sequential sampling rules that adaptively bias sampling probabilities in order to achieve faster convergence of the empirical distribution to a desired target law, while keeping the resulting samples as unpredictable as possible. The resulting self-balancing sampler is simple to implement, arises naturally among a class of Markovian samplers sharing a certain invariance property, and admits a stochastic mirror-descent interpretation. Our main results show that (i) this self-balancing sampler converges at the fastest possible $O(n^{-1})$ rate with explicit dependence on biasing parameters, beating the standard $O(n^{-1/2})$ rate of IID sampling, (ii) it is the unique solution to a natural entropy-regularized optimization problem which balances the convergence rate of the empirical law and the unpredictability of the samples, and (iii) in the weak-biasing regime, the properly centered counts process converges to an Ornstein-Uhlenbeck process in the diffusive limit. Together, these results support a practical framework for reducing repeated selections and long gaps in coverage without making future selections overly predictable.

Imprecise probability generalizes standard probability theory by replacing a single distribution with a convex set of possible distributions. We show that this generalization requires no change to the standard BDD compilation and weighted model counting pipeline used by discrete probabilistic languages. An imprecise coin flip is simply a BDD variable whose weight is left free rather than fixed. We introduce Imp, a Haskell-embedded DSL for imprecise probabilistic programming. A graded monad, indexed by finite sets of named sources of epistemic uncertainty, restores the commutativity that the standard convex powerset monad lacks, and GHC's type system enforces this at compile time. Weighted model counting is parametric in the semiring, so the same compiled BDD supports exact, differentiable, and interval-bounded inference.

Let $X\sim Bin(N,p)$ and let $c=Nr$ be a prescribed centre with $r\ne p$. The finite formula for $E|X-c|$ is classical, and the existence of a complete large-deviation expansion in powers of $N^{-1}$ follows from Timashev's saddle-point expansion for binomial tails. We give the coefficients in closed form for the stop-loss tail produced by the absolute deviation, with the lattice defect of $Nr$ retained explicitly. The construction combines three standard ingredients: Bernoulli polynomials for the logarithm of successive binomial masses, the exponential Bell recursion, and Eulerian polynomials for weighted geometric sums. The $k$-th coefficient has denominator exactly $(1-\rho)^{2k+2}$, where $\rho$ is the exponential tilt; hence the effective expansion parameter is $1/(N(1-\rho)^2)$, the lattice weighted analogue of the parameter in the incomplete gamma expansion. The case $r=\tfrac12$ gives the fold at the origin of a biased $\pm1$ random walk, and its parity split is identified with the lattice defect of the fold.

We develop a real Laplace method for integrals $$ I(z)=\int_N a(x)e^{-z f(x)}\operatorname{dvol}_g(x),\qquad z\to\infty, $$ near a compact, possibly stratified, minimum set of the phase. The basic input is an adapted normal-cell disintegration, defined up to null sets, together with piecewise anisotropic dilations and homogeneous limiting models. The framework simultaneously permits finite $C^2$ stratifications; Borel base refinements and $C^1$ adapted charts; measurable base-dependent cells, relative domains, and incident-stratum assignments; different fiber dimensions and weights on different pieces; a continuous phase and a merely measurable amplitude; and limiting models controlled either by ambient coercivity or directly by cellwise exponential tightness. The cells may converge only in weighted measure, or pointwise under an integrable base majorant. To the best of our knowledge, no existing theorem for real Laplace integrals accommodates this full combination of geometric and analytic assumptions. The tradeoff is that the adapted disintegration and limiting models are supplied hypotheses to be verified in each application.

Let $X(t)$, $t\in K$, be a centred Gaussian process with continuous sample paths on a compact metric space $K$, and let $M=\min_{t\in K}X(t)$. Let $\sigma_*^2$ denote the minimum covariance energy associated with $X$, and assume that $\sigma_*^2>0$. Motivated by the results of \cite{chakrabarty2018asymptotic} for smooth Gaussian processes, we show that, conditionally on $M>u$, the scaled overshoot $u(M-u)$ converges, as $u\to\infty$, to an exponential random variable with mean $\sigma_*^2$. Moreover, every weak subsequential limit of the conditional law of a measurable minimizer of $X$ is an optimal covariance-energy measure. In particular, if this measure is unique, then the conditional law converges weakly to it. The results are illustrated by stationary Gaussian processes, fractional Brownian motion, and fractional Brownian sheet.

Despite more than six decades of research, a tractable closed-form distribution for the typical Poisson–Voronoi cell volume remains unknown beyond one dimension. Building on the classical complementary-theorem structure for the Poisson–Voronoi fundamental region, we develop an explicit configuration-space factorization of the Palm-typical cell volume for a tessellation generated by a stationary Poisson point process of intensity \(\lambda>0\) in \(\mathbb R^d\), \(d\geq1\). Conditional on the number \(k\) of effective facets, the Voronoi flower volume is a \(\operatorname{Gamma}(k,\operatorname{rate}=\lambda)\) scale variable independent of the effective-neighbour configuration normalized to have unit flower volume. Mapping this normalized configuration to its cell-to-flower volume ratio \(A_k\) gives the conditional cell-volume representation as the product of the classical Gamma scale and a bounded geometric shape factor. From this representation, we derive exact mixture formulae, transform and moment identities, and criteria characterizing when the conditional cell-volume laws are Gamma. We also distinguish shape-factor variability within facet-number strata from mixing across strata as two sources of departure from a single Gamma law and recover the universal bound \(0<A_k\leq2^{-d}\). For the lower tail, we reduce critical negative moments of the shape factor to inverse-volume integrals over normalized configuration spaces. Under explicit critical-integrability and higher-facet summability assumptions, the density of the intensity-normalized cell volume has leading order \(y^d\) as \(y\downarrow0\). The framework recovers the one-dimensional distribution, admits explicit planar coordinates, and supports numerical evaluation of the mixture representation in dimensions two through four.

This paper introduces planar web geometry into the study of characterization problems for probability laws arising from independence preservation. Let $F(x,y) = (u, v)$ be a real-analytic diffeomorphism defined on an open domain in $\mathbb{R}^2$. Let $X$ and $Y$ be independent random variables and set $(U, V) = F(X, Y)$. The map $F$ is said to preserve the independence if $U$ and $V$ are independent. This condition is highly restrictive and can therefore be used to characterize the laws for which it holds. This framework provides unified proofs of characterization theorems, including classical results such as the Kac–Bernstein and Lukacs theorems and the Matsumoto–Yor property, as well as recent results for the quadrirational Yang–Baxter maps $H_I^+$, $H_{II}^+$, and $H_{III}^A$. For measures with positive continuous densities, the map preserves independence if and only if their logarithmic densities solve the corresponding inhomogeneous Abelian functional equation. The solution space is identified with an affine space modeled on the space of Abelian relations, whose dimension is at most three according to Bol's rank bound. The same method also produces further examples, including a class generalizing transformations studied by Koudou and Vallois; every map in this class admits at least a two-parameter family of preserved product measures. The independence-preserving property of a map is preserved by coordinatewise one-variable reparametrizations, which also preserve the associated web. Thus the web associated with a map is a natural invariant under the corresponding equivalence relation of independence-preserving maps. Together with the local algebraizability of nonsingular planar $4$-webs of maximal rank, this correspondence suggests a route toward a complete classification of maps preserving the independence of a three-parameter family of measures.

We construct the sine-Gordon measure and its renormalized effective potential in finite volume up to the seventh threshold. The construction relies on a weak stochastic control problem, its associated weak FBSDE, and an analysis of the renormalization flow equation using a multipole Mayer expansion. The analysis is fully inductive, and we include a complete order-by-order analysis in the subcritical regime $\beta^{2}<8\pi$.

We give a derivation of the Gibbs measure for the focusing nonlinear Schrödinger equation (NLS) on the circle with rough cut-off. This extends earlier work by Sohinger and the second author, which proved analogous results for smooth cut-offs. Our proof is based on the perturbative expansion developed by Fröhlich, Knowles, Schlein, and Sohinger (2017), and provides an alternative proof of the recent derivation given in Lü, Nam, and Zhu (2026). To prove convergence of the explicit terms, we employ a Wigner measure approach and an inductive argument to overcome the lack of smoothness for the cut-off. In particular, we give a derivation of the Gibbs measure for the focusing quintic NLS with the optimal cut-off from Oh, Sosoe, and Tolomeo (2022).

Posterior probabilities are bounded martingales, but this fact alone does not identify the experiment that generates them. We show that every time-homogeneous martingale diffusion on $[0,1]$ with strictly positive $C^1$ volatility on the interior and absorbing endpoints, which we call an autonomous win-martingale, can be realized as the Bayesian posterior of an explicit binary sequential experiment observed through a scalar diffusion. Conversely, in a general binary diffusion experiment, the current time and observation are jointly sufficient for the hidden state precisely when the difference between the signal drifts under the two hypotheses satisfies a Riccati equation. Equivalently, this holds when the likelihood ratio is generated by a Doob $h$-transform associated with a positive space-time harmonic function for the null dynamics. Within this class, when that function is spatially harmonic, the posterior is an autonomous martingale. Outside this harmonic case, autonomy forces the drift difference to be constant, recovering the classical two-state Shiryaev-Wonham filter from binary testing. These results give concrete sequential inference interpretations to a broad class of bounded martingale diffusions, including absorbed Brownian motion, the Wright-Fisher diffusion, and the Aldous martingale of the "most exciting game."

Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, put $\beta=\dim_{\mathbb{R}}\mathbb{K}$, and let $G_n^{\mathbb{K}}$ be an $n\times n$ matrix with i.i.d. standard $\mathbb{K}$-Gaussian entries, namely a standard $\mathbb{K}$-Ginibre matrix. We prove that the normalized row-ordered permanent $W_n^{\mathbb{K}}=\operatorname{per}_{\mathbb{K}}G_n^{\mathbb{K}}/\sqrt{n!}$ has a radial density $p_n^{\mathbb{K}}$ satisfying $\|p_n^{\mathbb{K}}\|_\infty=p_n^{\mathbb{K}}(0)\lesssim_\beta n^{(\beta+2)/4}$ and $\sup_{z\in\mathbb{K}}\mathbb{P}(|W_n^{\mathbb{K}}-z|\leq\varepsilon)\lesssim_\beta n^{(\beta+2)/4}\varepsilon^\beta$. In particular, for $\mathbb{K}=\mathbb{C}$, this resolves the Permanent Anticoncentration Conjecture of Aaronson and Arkhipov. The proof compares the squared Gaussian permanent with the squared (Study) determinant in Laplace-transform order.

The atypical behaviour of random walks in time-dependent random environment was recently related to Kardar-Parisi-Zhang (KPZ) growth. While this is now well-understood in spatial dimension $d=1$, further efforts are necessary to better understand these connections in dimensions $d>1$. In this paper, we study this problem numerically for $d=1, 2$ and $3$, focusing on a discrete model with Dirichlet distributed transition probabilities. This model is a generalization of an integrable model in $d=1$, and it has the advantage of admitting an explicit, product-form, stationary measure. We verify that the growth of the variance of the logarithm of point-to-point probabilities, namely from the origin to position $x$ in time $t$, is compatible with KPZ growth in dimension $d=1$ and $d=2$. In spatial dimension $d=3$, we confirm the existence of a phase transition as the angle $\vert x\vert /t$ increases and we obtain a lower bound based on an exact second moment calculation. We find that in the weak disorder phase the point-to-point probability acquires a heavy tailed distribution, and that in the strong disorder phase the cumulants of its logarithm grow with time. Further, we show that for this model, we can compute exactly the sample to sample variance of the thermal average $\overline{ \langle x \rangle^2}$ and that it is related to the extreme diffusion coefficient introduced recently.

In a recent work [Adv. Math. 401 (2022), Paper No. 108318], a central limit theorem was established for the linear combinations of the iterates of a non-rotational inner function fixing the origin. In this paper, we prove the law of iterated logarithm (LIL) in the same setup, with a very mild condition on the coefficients. We also identify the full set of subsequential limit points at the LIL scale. Using the Aleksandrov–Clark decomposition and measure-preserving properties of the inner functions, one can construct a reverse martingale that is close to the linear combinations of inner functions. We prove the LIL for the partial sums of reverse martingale differences under a Feller-type assumption, which then transfers to the linear combinations of iterates of the inner functions.

To derive dimension-free convergence rates of empirical measures for Markov processes on a Banach space, we adopt the sliced Wasserstein distance (SW distance) induced by a probability measure with full support on the unit ball in the dual space. This distance refers to the convergence in finite dimensions, and is topologically equivalent to the classical Wasserstein distance when the Banach space is finite-dimensional. With the SW distance we derive dimension-free convergence rates for the empirical measures of ergodic Markov processes on Banach space, which can be sharp as illustrated by concrete examples. The study provides an efficient way to simulate infinite-dimensional distributions using sample trajectories of Markov processes, avoiding $``$the curse of dimensionality" in the classical Wasserstein distance. The main results apply to a broad class of infinite-dimensional models, and are illustrated by partially dissipative SPDEs in the end of the paper.

We consider supercritical branching random walks (BRW) on countable groups $G$ and we prove that the asymptotic entropy of the empirical distributions of the BRW has a phase transition at $\rho_* = e^{h(\mu)}$, where $h(\mu)$ is the asymptotic entropy of the underlying random walk on $G$ with step distribution $\mu$. Below this value $\rho_*$, the asymptotic empirical entropy of BRW equals the logarithm of the exponential growth rate of the population. Above this value, it is constantly equal to the asymptotic entropy of the underlying random walk. In particular, this answers questions from Kaimanovich-Woess [MR4663513, Section 6.3] about the existence and the behavior of the asymptotic entropy.

This work presents analytical tools for studying functions defined on spaces of measures, together with two applications: mean field game master equations and first order Hamilton-Jacobi-Bellman equations on the set of probability measures. The first part introduces several notions of differentiability for such functions - vertical (or flat) and horizontal derivatives - as well as associated notions of convexity, monotonicity and sub-differentiability. The second part is concerned with master equations, for which uniqueness and stability of monotone solutions are established, and existence is discussed. The third part studies mean field optimal control problems and characterizes their value functions as the unique viscosity solutions of the associated Hamilton-Jacobi-Bellman equations. This work is intended to be accessible to readers unfamiliar with spaces of measures.

We prove a local limit theorem for the quantity $\| g_n \cdots g_1 \|$, where $g_1, g_2, \ldots$ is a sequence of independent and identically distributed random invertible square matrices. The norm $\| \cdot \|$ is arbitrary and we make no proximality assumption.

We study binomial probabilities at a fixed integer distance from the upper mode, in the regime where the fractional part of the mean remains visible. The complete asymptotic expansion is obtained to all orders, uniformly for the success probability in compact subintervals of $(0,1)$ and for bounded shifts from the mode. The main point is structural. A logarithmic tail in the standard local expansion is exactly the binomial size-bias factor, and removing it leaves a pure Appell expansion. A second, symmetric normalisation is governed by an even Appell sequence and recovers the classical Stirling prefactor. The resulting coefficients explain the exact binomial mode rule, recover the central-binomial expansions as a special case, and give closed forms for averaged oscillating coefficients.

We construct weak equilibrium measures and weak capacities for the hypoelliptic third-order Langevin diffusion motivated by an accelerated sampling algorithm (Mou et al. (2021) J. Mach. Learn. Res., 22(42), 1–41). In this process, the Brownian noise acts only in the highest-order auxiliary variable and reaches the physical variables through a step-three Hörmander chain, so the standard uniformly elliptic boundary-flux theory is not directly applicable at characteristic points of phase-space balls. We prove an elliptic-regularization stability theorem for the corresponding hitting laws and then define the weak equilibrium measure and weak capacity. The proof combines the boundary-hitting stability strategy of Lee–Ramil–Seo (2026, arXiv:2503.12610v2) with localized hypoelliptic heat-kernel estimates (Pigato (2022) Stoch. Process. Appl., 145, 117–142) adapted to the third-order chain. We obtain the bounded-domain weak capacity–hitting identity and a Lyapunov drift argument in the spirit of Lee–Ramil–Seo that yields positive Harris recurrence and extends the construction to a whole-space weak equilibrium measure, and whole-space capacity–hitting identity.

In this paper, we study the smallest gaps between successive zeros of nondegenerate smooth stationary centered Gaussian processes on the real line with the assumption that the covariance kernel $\kappa(x)$ and its derivatives decay to 0 as $|x|\to\infty$. We prove that, after rescaling, the smallest gaps converge to a Poisson point process with a specific rate. Moreover, the positions where these smallest gaps occur tend to a uniform distribution. Consequently, we can derive the limiting density for the $k$-th smallest gap.

In this paper, we study the random walk on the symmetric group $\mathfrak{S}_n$ generated by the conjugacy class of $k$-cycles, where $2\le k=o(n/(\log n)^4)$. We prove that the walk exhibits hitting-time mixing: at the first time when every card has been touched, the distribution is already close to equilibrium. For odd $k$, the equilibrium measure is the uniform measure on $\mathfrak{A}_n$. For even $k$, the walk first mixes to the parity mixture determined by the hitting time, and in our range this mixture is asymptotically $U_{\mathfrak{S}_n}$. Our argument combines a refined fixed-time approximation for the random $k$-cycle walk near the cutoff window with an auxiliary marking scheme inspired by Jain-Sawhney's work (arXiv:2410.23944) on random transpositions. The main new feature is a parity-compatible coupling which handles both odd and even $k$-cycles in a unified framework. We also prove a hitting-time mixing result in the opposite regime $k\ge n-o(n^{1/2})$, and formulate a conjecture for all $2\le k\le n-1$.

We consider a stochastic opinion dynamics model on a fully connected social network with $N$ actors interacting by expressing opinions from a set of $M$ opinions. At any time $t\geq 0$, each actor is associated to an $M$-tuple representing the social pressure exerted on this actor for each opinion. The evolution of the matrix containing the social pressure of all actors for all opinions is a Markov jump process. Each actor tends to express opinions according to their social pressure vector and this tendency is modulated by a polarization coefficient. When an actor expresses an opinion $o$, its social pressure for all opinions is reset to zero, while for other actors the social pressure for $o$ increases by 1 and the social pressure for other opinions decreases by $1/(M-1)$. In this setting, we prove fast consensus formation, existence of a unique invariant measure and metastability in a highly polarized network. Moreover, by considering a communication bias parameter, the system exhibits a phase transition described as follows. With a negative communication bias parameter, all actors except one stop expressing in a finite time almost surely. Otherwise, no actor stops expressing opinions.

Let $f_\alpha(x)=(x^q-(1-\alpha))^d-\alpha^d x^z$, where $0<\alpha<1$, $q\ge 2$, $d\ge 2$, $1\le z\le q-1$, and $\gcd(q,z)=1$. These are the non-degenerate full-degree Type II reduced Ito polynomials of order $n=qd$. We give a complete parametrisation, up to permutation similarity, of the stochastic matrices whose characteristic polynomial is $f_\alpha$. The support data are a composition $a_0+\cdots+a_{d-1}=z$ and a cyclic origin $\rho\in\mathbb{Z}_q$. The transfer support in block 0 is an arbitrary nonempty subset of $[\rho,\rho+a_0]_q$, while the support in block $t\ge 1$ is an arbitrary nonempty subset of $[\rho-\sum_{j=1}^t(a_j+1),\rho-\sum_{j=1}^t(a_j+1)+a_t]_q$. The horizontal weights at the selected positions are arbitrary elements of $(0,1)$ whose product in every block is $1-\alpha$. The proof first shows that the order-$n$ Type II arc lies strictly outside $\Theta_{n-1}$, which permits the Dmitriev–Dynkin/Kirkland–Smigoc two-shift reduction. A circular separation theorem then classifies the possible supports, and Coates' formula supplies the characteristic polynomial. This resolves the non-sparse full-degree Type II characteristic-polynomial realiser problem.

We prove that, for every fixed degree $d$ and a uniformly random simple $d$-regular graph $G_{n,d}$ on an even number $n>d$ of vertices, $$\mathbb{E}\operatorname{MaxCut}(G_{n,d})-\mathbb{E}\operatorname{MaxBis}(G_{n,d})=o(n).$$ In other words, requiring the two sides of a cut to have exactly the same size changes the expected optimal cut by only a sublinear number of edges. We prove this first for the configuration model by comparing cuts of each possible cardinality on an $n$-vertex graph with bisections of a related graph on $2n$ vertices. An important technical tool is Huang's interpolation theorem (Huang, 2018); to apply it, we establish a structural property of the change in the optimal cut when a single edge is added. This comparison, together with concentration estimates and Huang's convergence theorem for maximum bisection, shows that maximum cut and maximum bisection have the same limiting density. Conditioning the configuration model on being simple then gives the result for uniformly random simple regular graphs.

We show that, for a metric graph equipped with the Laplacian operator $\Delta=-\frac{d^2}{dx^2}$, the graph trace formula admits a new interpretation in terms of quantum probability and curvature. Our approach is based on a notion of graph curvature inspired by Wilson-lines and holonomy, together with von-Neumann's ergodic theorem.

We study the local density-dependent diffusion $dY_t=-\Xi(p_t(Y_t))\nabla\Phi(Y_t)\,dt+\sqrt2\,dW_t$ and a clipped, randomly shifted histogram particle approximation on $\mathbb{R}^d$. The central difficulty is that the empirical density is evaluated at the particles' locations and re-enters their drift, while the confining force $\nabla\Phi$ may be unbounded. We provide a path-space entropy proof under two verifiable analytic conditions: a uniform pointwise Gaussian envelope for the true density $p_t$, and a Gaussian–polynomial bound for its spatial gradient $\nabla p_t$. The potential is allowed to have a gradient of at most linear growth. The probabilistic input is a weighted exponential occupancy estimate under the independent product law. It is proved by Poissonizing the system at total intensity $N-1$, performing a one-cell leave-one-out estimate bounded via Poisson information, using Gaussian cell summability, and de-Poissonizing. For every fixed time horizon $T$, we obtain $\operatorname{Ent}(P_t^{N,k}|p_t^{\otimes k})\leq C_T k(h^2(1+|\log h|)+(h^{-d}+\log N)/N)$. Consequently, selecting the optimally balanced bandwidth $h\asymp (N\log N)^{-1/(d+2)}$ yields a total variation error of $\Vert P_t^{N,k}-p_t^{\otimes k}\Vert_{\operatorname{TV}}\leq C_T\sqrt{k}\,N^{-1/(d+2)}(\log N)^{d/[2(d+2)]}$ for fixed $k$. This includes the usual Ornstein–Uhlenbeck density and the density-dependent OU model whenever the PDE estimates hold on the considered interval. Furthermore, the histogram estimator offers a scalable approach for particle approximations. Using occupied-cell hashing, one algorithm step evaluates in expected $O(dLN)$ operations under standard constant-time hashing assumptions. For a fixed dimension and number of shifts, this requires expected $O(N)$ time, avoiding the $O(N^2)$ evaluation cost typical of standard kernel density estimators.

In this paper, we investigate the generalized friendship paradox for eigenvectors (alternatively called the eigen friendship paradox and abbreviated hereafter as EFP) in the setting of inhomogeneous Erdős–Rényi random graphs whose edge probabilities are generated by a continuous graphon. We consider the adjacency matrix of the graph and take the entries of the eigenvector corresponding to its largest eigenvalue as the vertex attributes. It was shown in \cite{hazra2026generalized} that the generalized friendship paradox holds in this setting. We study the empirical distribution of the resulting bias values across the vertices and derive its limiting distribution explicitly in terms of the principal eigenvalue and the corresponding eigenfunction of the integral operator whose kernel is the underlying graphon.

We introduce point conditional probability spaces (PCPSs) as primitive building blocks for stochastic choice. This concept goes back to Rényi (1955), who proposed conditional probability spaces (CPSs) as a basis for probability theory. Luce (1959) noted the connection between CPSs and stochastic choice, and Cerreia-Vioglio et al. (2021) have developed the connection further. A PCPS is a CPS each of whose component probability measures concentrates on a singleton selection. We build a four-level PCPS-based hierarchy of families of stochastic choice rules. Level 1 consists of PCPSs, Level 2 is made up of finitely additive "conditionally consistent" mixtures of PCPSs, Level 3 comprises all finitely additive probabilistic mixtures of PCPSs, and Level 4 consists of all finitely additive signed probabilistic mixtures of PCPSs. We construct our hierarchy at a general measure-theoretic level that encompasses infinite choice sets. We also connect each level of our hierarchy to well-known axioms for stochastic choice, namely, the Weak Axiom of Stochastic Revealed Preference, Independence of Irrelevant Alternatives, and no Dutch book. We establish the relationship between total orders and PCPSs and demonstrate a sense in which PCPSs can be a more parsimonious representation of choice.