Blockchain Papers

Hongru Zhao

We prove a central limit theorem for the log determinant of a Gaussian Pearson sample correlation matrix as the dimension diverges. Only two conditions are imposed: the population correlation matrix is positive definite, and the sample degrees of freedom are at least the dimension. Both are necessary for the ordinary log determinant to be finite. To the best of our knowledge, no previous central limit theorem covers this full nonsingular domain. It covers every aspect ratio from dilute growth to the square hard edge. No uniform lower or upper bound is imposed on the eigenvalues of the population correlation matrices: the smallest may approach zero and the largest may diverge. The proof develops a coordinatewise Wiener chaos reduction for the random diagonal normalization and combines it with an exact Wishart transform comparison. Geometrically, the statistic is twice the log volume of a random parallelotope spanned by standardized Gaussian coordinate vectors.

José Ignacio Peinador Sala

Conditional Refutation of Erdős Problem #463 via Arithmetic Quantum Chaos Author: José Ignacio Peinador Sala Overview This repository contains the full manuscript, companion computational notebooks, and formal Lean 4 verification for the paper "Conditional Refutation of Erdős Problem #463 in Hyper‑Slow Growth Regimes via Arithmetic Quantum Chaos". We demonstrate that, under the hypothesis that the survival variance of rough numbers around primorials is controlled by the fractal dimension D2≈0.24338 of the Riemann‑GUE Hamiltonian (Bridge Conjecture), no function f(n)≤log⁡(log⁡n) satisfies Erdős' condition for all sufficiently large n. The proof is constructed by bridging Galois projection operators, power‑law random banded matrices (PRBM), the Altshuler‑Shklovskii effect, and optimal transport (Kantorovich–Rubinstein duality). The ultimate goal of this program is to elevate this conditional result to an unconditional proof by integrating the supersymmetric Non-Linear Sigma Model (NLσM) limit with the most recent 2025 sieve bounds on rough numbers in short intervals. Contents Article: Open pdf One‑Click Reproducibility This project is designed for frictionless, one‑click reproducibility. No compiler installation, no supercomputing cluster. All experiments run on Google Colab with zero local setup — you can audit the physics of the arithmetic vacuum from a browser on your laptop or even your phone. What the notebooks validate You can run the experiments directly in your browser: Notebook Contents What it certifies Main experiments: Open in Colab Experiments 1–4 + Chirikov map Collapse of Nₖ, monotonic decrease of D₂, massive suppression of Σ²(L), sub‑diffusive SFF ramp, classical chaos suppression Lean 4 verification: Open in Colab Lean 4 formal proofs Idempotence of the Galois projector, discrete variance floor lemma, modular classification of primes Experiments (Main Notebook) Collapse of the survival variable Nk – deterministic emptiness of the critical interval for primorials k≥10 (M=5,000 samples). Fractal dimension D2 of pruned Hamiltonians – monotonic decrease under Galois projection (Numba‑accelerated up to N=10,000). Number variance Σ2(L) and Thouless energy – massive spectral suppression (up to 96% below GUE) with the Thouless scale plunging below L=0.5 (M=10,000 realizations). Spectral Form Factor and Finite‑Size Scaling – robust sub‑diffusive ramp (γ→0.61) and convergent D2≈0.106 in the thermodynamic limit (M=100 realizations, N up to 6,000). Chirikov Map (Classical) – Galois projection completely strangulates chaotic transport (D≈0.00 vs D≈11.05), proving universal ergodicity suppression. Formal Verification in Lean 4 Erdős Problem #463 is actively tracked by the mathematical community, including Google DeepMind's formal-conjectures repository. Laying the formal groundwork to resolve this, the notebook Notebooks/erdos_refutation.ipynb compiles and mechanically verifies three foundational lemmas in Lean 4 (v4.29.1, Mathlib4): Discrete Variance Floor Lemma — ∀ x ∈ ℕ, x ≤ x² Galois Projector Idempotence — χ² = χ for the coprimality indicator Modular Classification of Primes — ∀ p > 3 prime, p ≡ 1 ∨ p ≡ 5 (mod 6) These lemmas form the unshakeable logical bedrock of the conditional refutation. 🔭 Philosophical Context "Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." — William Thurston For decades, the distribution of prime numbers and the behaviour of chaotic quantum systems were studied as separate continents of knowledge, occasionally glimpsing each other across a narrow strait —the Hilbert–Pólya conjecture, the Montgomery–Odlyzko law— but never truly merging. This work builds a bridge across that strait. The key insight is that the ring ℤ/6ℤ is not merely a convenient sieve for eliminating multiples of 2 and 3. It is a topological substrate —a discrete analogue of the KO‑dimension in noncommutative geometry— that partitions the integers into resonant channels (𝒞₁ and 𝒞₅) and sterile channels (𝒞₀, 𝒞₂, 𝒞₃, 𝒞₄). When this partition is imposed as a superselection rule on a quantum Hamiltonian, the system does not thermalise. It enters a Non‑Ergodic Extended (NEE) phase where fluctuations are systematically suppressed, variance collapses, and the arithmetic vacuum swallows the survivors. The philosophical lesson is profound: randomness is not the default state of complex systems. The apparent chaos of prime numbers, long regarded as the quintessence of unpredictability, harbours a rigid geometric order. That order can be harnessed —through Galois projection, through PRBM Hamiltonians, through the Altshuler–Shklovskii effect— to prove theorems that have resisted classical sieve methods for half a century. This project also embodies a conviction about how science should be done in the age of artificial intelligence. Every line of code, every formally verified lemma, and every numerical experiment was developed using freely accessible tools. The massive simulations of quantum chaos, which traditionally would demand exclusive access to institutional supercomputers, were executed entirely on Google Colab, democratizing high-performance computing. The formal verification of the mathematical bedrock was achieved using the open-source proof assistant Lean 4. Furthermore, the theoretical framework was built in a genuine symbiosis with DeepSeek, an open-weight AI freely provided to the world. No proprietary models, no paywalled platforms, no computational aristocracy. This work demonstrates that the absolute frontier of mathematical research is now accessible to anyone with a good idea, a standard laptop, and the willingness to engage in dialogue with tools that amplify, rather than replace, human creativity. "The universe is written in the language of mathematics." — Galileo Galilei Perhaps it is written, more precisely, in the language of modular arithmetic. Last Update: May 2026 | Status: Under Peer Review in IOP/LMS Nonlinearity (Ref: NON-110856) | Built with ❤️, 🐍 & 🤖

Akiva Groskin

The Connes–van Suijlekom truncated Weil quadratic form, indexed by a cutoff parameter c that controls the primes p ≤ c entering the operator, produces a ground state whose Fourier–Mellin zeros provably lie on the critical line; whether they converge to the Riemann zeros as c → ∞ is open (Connes 2026; Connes–Consani–Moscovici 2025). We present, to our knowledge, the first independent public implementation of the Connes–van Suijlekom Galerkin matrix at sixteen cutoffs (c = 13 through 67, plus c = 100). Across the in-sample window c = 13 through c = 67 at N = 100, the first-zero absolute error |γ1 − γ1Riemann| shrinks monotonically from ∼2×10−55 to ∼1.5×10−168 — a 113-OOM convergence across fifteen cutoffs. The smallest-positive even-sector eigenvalue λmineven separately reaches ∼10−334 at c = 100, N = 250 (275-OOM span from c = 13). Out-of-sample test at c = 100. On the four-point N-sweep N ∈ {100, 150, 200, 250} at dps = 500, consecutive first-difference ratios 0.837 and 0.836 match to two decimal places. Aitken-Δ2 on the two overlapping triples yields log10|λ∞even| ≈ −536.8 and ≈ −533.7, approaching the Connes 2026 §6.4 heuristic prediction (≈ −530.4) monotonically with N (6.4 and 3.3 OOM gaps out of |x∞| ∼ 530). The same eigenvector recovers γ1, …, γ10 to 307–329 matching digits at N = 250, dps = 500. Under the unitary equivalence with Connes–Consani–Moscovici Lemma 5.1, this is the deepest such Galerkin-truncation recovery in the public Connes–van Suijlekom / Connes–Consani–Moscovici literature, subject to a hypothesis-status caveat: the raw finite-N matrix carries a small block of dps-stable negative-sign eigenvalues, so we report the smallest-positive branch (continuum positivity of QWλ is RH-equivalent and is not assumed at λ = √100). The fit |log10 λmin| ≈ 13.24 c0.634 on c ≤ 67 at N = 100 is shown to be a finite-N rate, falsified at c = 100, N = 200 by 49 OOM in the direction of faster decay. Structural observations include approximate eigenvector c-invariance (overlap ≥ 0.9498 on all 105 cutoff pairs despite eigenvalues differing by 113 OOM), multi-zero convergence universality (all ten detectable zeros within 3.8% of each other), an empirical Galerkin-convergence exponent s(c) ≈ 55 log c − 128, un-rescaled Galerkin bulk-spectrum Poisson statistics (β < 0.05; this is a structural diagnostic of the truncated operator, not a test of Montgomery's conjecture, which applies to locally-rescaled zero spacings), and tight bulk invariants log|det Qc| ≈ −65.6 c + 542 (R2 = 0.997). We make no claim of proof; the contribution is reproducible numerical data and its careful interpretation under the existing CvS / CCM framework. All code, data, and ancillary files are publicly available.

Kyle Liss, Kunhui Luan

We study passive scalar mixing by parallel shear flows in the presence of weak molecular diffusion. We recover the sharp uniform-in-diffusivity mixing rate for shear flows with finitely many critical points, recently proven in [1]. Our approach is based on the stochastic representation formula of the associated advection-diffusion equation and yields two short proofs. The first uses a stochastic integration-by-parts argument and gives optimal mixing under the weakest regularity assumption required in the zero-diffusion case, answering Question II in [1, Section 4]. The second adopts a dynamical systems perspective and provides a proof of shear-induced mixing that, to our knowledge, is new even in the zero-diffusivity setting.

Michele Battagliola, Laura Mattiuz, Alessio Meneghetti

Abstract The Vector Oblivious Linear Evaluation in the Head (VOLEitH) paradigm has proven to be a versatile tool to design zero-knowledge proofs and signatures in post-quantum cryptography. In this paper, we propose three VOLE-friendly modellings for Proofs of Knowledge (PoK) of a solution of an instance of the Linear Code Equivalence Problem (LEP). For the first two schemes, we propose two new reductions from LEP to the Multivariate Quadratic (MQ) problem, that may be of independent interest for the cryptanalysis of LEP. Instead, the last model is obtained by generalizing a recent work by Bettaieb et al. to the context of monomial matrices instead of permutation matrices. While our proposed schemes exhibit larger signature sizes compared to LESS, they improve the computational efficiency, reducing the overall complexity from $$O(n^3)$$ to $$O(n^2\log n )$$ and $$O(n^2\log ^2 n )$$ , where n is the length of the code.

Nicolas Lanchier, Max Mercer, H. M. Yun

This paper is concerned with a natural variant of the contact process modeling the spread of knowledge on the integer lattice. Each site is characterized by its knowledge, measured by a real number ranging from 0 = ignorant to 1 = omniscient. Neighbors interact at rate $λ$, which results in both neighbors attempting to teach each other a fraction $μ$ of their knowledge, and individuals die at rate one, which results in a new individual with no knowledge. Starting with a single omniscient site, our objective is to study whether the total amount of knowledge on the lattice converges to zero (extinction) or remains bounded away from zero (survival). The process dies out when $λ\leq λ_c$ and/or $μ= 0$, where $λ_c$ denotes the critical value of the contact process. In contrast, we prove that, for all $λ&gt; λ_c$, there is a unique phase transition in the direction of $μ$, and for all $μ&gt; 0$, there is a unique phase transition in the direction of $λ$. Our proof of survival relies on block constructions showing more generally convergence of the knowledge to infinity, while our proof of extinction relies on martingale techniques showing more generally an exponential decay of the knowledge.

Nicolò Vallarano, Matteo Bruno, Emiliano Marchese, Giuseppe Di Trapani · 8 authors

Exponential Random Graph Models (ERGMs) have gained increasing popularity over the years. Rooted into statistical physics, the ERGMs framework has been successfully employed for reconstructing networks, detecting statistically significant patterns in graphs, counting networked configurations with given properties. From a technical point of view, the ERGMs workflow is defined by two subsequent optimization steps: the first one concerns the maximization of Shannon entropy and leads to identify the functional form of the ensemble probability distribution that is maximally non-committal with respect to the missing information; the second one concerns the maximization of the likelihood function induced by this probability distribution and leads to its numerical determination. This second step translates into the resolution of a system of $O(N)$ non-linear, coupled equations (with $N$ being the total number of nodes of the network under analysis), a problem that is affected by three main issues, i.e. accuracy, speed and scalability. The present paper aims at addressing these problems by comparing the performance of three algorithms (i.e. Newton's method, a quasi-Newton method and a recently-proposed fixed-point recipe) in solving several ERGMs, defined by binary and weighted constraints in both a directed and an undirected fashion. While Newton's method performs best for relatively little networks, the fixed-point recipe is to be preferred when large configurations are considered, as it ensures convergence to the solution within seconds for networks with hundreds of thousands of nodes (e.g. the Internet, Bitcoin). We attach to the paper a Python code implementing the three aforementioned algorithms on all the ERGMs considered in the present work.

Jonathan Hermon, Justin Salez

We consider the zero-range process with arbitrary bounded monotone rates on the complete graph, in the regime where the number of sites diverges while the density of particles per site converges. We determine the asymptotics of the mixing time from any initial configuration, and establish the cutoff phenomenon. The intuitive picture is that the system separates into a slowly evolving solid phase and a quickly relaxing liquid phase: as time passes, the solid phase dissolves into the liquid phase, and the mixing time is essentially the time at which the system becomes completely liquid. Our proof uses the path coupling technique of Bubley and Dyer, and the analysis of a suitable hydrodynamic limit. To the best of our knowledge, even the order of magnitude of the mixing time was unknown, except in the special case of constant rates.

Di Wang, Xiangyu Guo, Chaowen Guan, Shi Li · 5 authors

Recently, many machine learning and statistical models such as non-linear regressions, the Single Index, Multi-index, Varying Coefficient Index Models and Two-layer Neural Networks can be reduced to or be seen as a special case of a new model which is called the \textit{Stochastic Linear Combination of Non-linear Regressions} model. However, due to the high non-convexity of the problem, there is no previous work study how to estimate the model. In this paper, we provide the first study on how to estimate the model efficiently and scalably. Specifically, we first show that with some mild assumptions, if the variate vector $x$ is multivariate Gaussian, then there is an algorithm whose output vectors have $\ell_2$-norm estimation errors of $O(\sqrt{\frac{p}{n}})$ with high probability, where $p$ is the dimension of $x$ and $n$ is the number of samples. The key idea of the proof is based on an observation motived by the Stein's lemma. Then we extend our result to the case where $x$ is bounded and sub-Gaussian using the zero-bias transformation, which could be seen as a generalization of the classic Stein's lemma. We also show that with some additional assumptions there is an algorithm whose output vectors have $\ell_\infty$-norm estimation errors of $O(\frac{1}{\sqrt{p}}+\sqrt{\frac{p}{n}})$ with high probability. We also provide a concrete example to show that there exists some link function which satisfies the previous assumptions. Finally, for both Gaussian and sub-Gaussian cases we propose a faster sub-sampling based algorithm and show that when the sub-sample sizes are large enough then the estimation errors will not be sacrificed by too much. Experiments for both cases support our theoretical results. To the best of our knowledge, this is the first work that studies and provides theoretical guarantees for the stochastic linear combination of non-linear regressions model.

Andreas Masuhr, Mark Trede

Abstract This paper proposes a Bayesian estimation algorithm to estimate Generalized Partition of Unity Copulas (GPUC), a class of nonparametric copulas recently introduced by [18]. The first approach is a random walk Metropolis-Hastings (RW-MH) algorithm, the second one is a random blocking random walk Metropolis-Hastings algorithm (RBRW-MH). Both approaches are Markov chain Monte Carlo methods and can cope with ˛at priors. We carry out simulation studies to determine and compare the efficiency of the algorithms. We present an empirical illustration where GPUCs are used to nonparametrically describe the dependence of exchange rate changes of the crypto-currencies Bitcoin and Ethereum.

Fadhel Ayed, Marco Battiston, Federico Camerlenghi, Stefano Favaro

Given $n$ samples from a population of individuals belonging to different types with unknown proportions, how do we estimate the probability of discovering a new type at the $(n+1)$-th draw? This is a classical problem in statistics, commonly referred to as the missing mass estimation problem. Recent results by Ohannessian and Dahleh \citet{Oha12} and Mossel and Ohannessian \citet{Mos15} showed: i) the impossibility of estimating (learning) the missing mass without imposing further structural assumptions on the type proportions; ii) the consistency of the Good-Turing estimator for the missing mass under the assumption that the tail of the type proportions decays to zero as a regularly varying function with parameter $α\in(0,1)$. In this paper we rely on tools from Bayesian nonparametrics to provide an alternative, and simpler, proof of the impossibility of a distribution-free estimation of the missing mass. Up to our knowledge, the use of Bayesian ideas to study large sample asymptotics for the missing mass is new, and it could be of independent interest. Still relying on Bayesian nonparametric tools, we then show that under regularly varying type proportions the convergence rate of the Good-Turing estimator is the best rate that any estimator can achieve, up to a slowly varying function, and that minimax rate must be at least $n^{-α/2}$. We conclude with a discussion of our results, and by conjecturing that the Good-Turing estimator is an rate optimal minimax estimator under regularly varying type proportions.

Nicholas A. Cook

We obtain lower tail estimates for the smallest singular value of random\nmatrices with independent but non-identically distributed entries.\nSpecifically, we consider $n\\times n$ matrices with complex entries of the form\n\\[ M = A\\circ X + B = (a_{ij}\\xi_{ij} + b_{ij}) \\] where $X=(\\xi_{ij})$ has iid\ncentered entries of unit variance and $A$ and $B$ are fixed matrices. In our\nmain result we obtain polynomial bounds on the smallest singular value of $M$\nfor the case that $A$ has bounded (possibly zero) entries, and $B= Z\\sqrt{n}$\nwhere $Z$ is a diagonal matrix with entries bounded away from zero. As a\nbyproduct of our methods we can also handle general perturbations $B$ under\nadditional hypotheses on $A$, which translate to connectivity hypotheses on an\nassociated graph. In particular, we extend a result of Rudelson and Zeitouni\nfor Gaussian matrices to allow for general entry distributions satisfying some\nmoment hypotheses. Our proofs make use of tools which (to our knowledge) were\npreviously unexploited in random matrix theory, in particular Szemer\\'edi's\nRegularity Lemma, and a version of the Restricted Invertibility Theorem due to\nSpielman and Srivastava.\n

Oriane Blondel, Patrícia Gonçalves, Marielle Simon

In this paper we prove the convergence to the stochastic Burgers equation\nfrom one-dimensional interacting particle systems, whose dynamics allow the\ndegeneracy of the jump rates. To this aim, we provide a new proof of the second\norder Boltzmann-Gibbs principle introduced in [Gon\\c{c}alves, Jara 2014]. The\nmain technical difficulty is that our models exhibit configurations that do not\nevolve under the dynamics - the blocked configurations - and are locally\nnon-ergodic. Our proof does not impose any knowledge on the spectral gap for\nthe microscopic models. Instead, it relies on the fact that, under the\nequilibrium measure, the probability to find a blocked configuration in a\nfinite box is exponentially small in the size of the box. Then, a dynamical\nmechanism allows to exchange particles even when the jump rate for the direct\nexchange is zero.\n

Oriane Blondel, Patrícia Gonçalves, Marielle Simon

In this paper we prove the convergence to the stochastic Burgers equation from one-dimensional interacting particle systems, whose dynamics allow the degeneracy of the jump rates. To this aim, we provide a new proof of the second order Boltzmann-Gibbs principle introduced in [7]. The main technical difficulty is that our models exhibit configurations that do not evolve under the dynamics - the blocked configurations - and are locally non-ergodic. Our proof does not impose any knowledge on the spectral gap for the microscopic models. Instead, it relies on the fact that, under the equilibrium measure, the probability to find a blocked configuration in a finite box is exponentially small in the size of the box. Then, a dynamical mechanism allows to exchange particles even when the jump rate for the direct exchange is zero.

Alistair Sinclair, Piyush Srivastava, Marc Thurley

No abstract is available for this record.

Hubert Lacoin, François Simenhaus, Fabio Lucio Toninelli

Let be a bounded, smooth enough domain of ℝ 2 . For L &gt; 0 consider the continuous‐time, zero‐temperature heat bath stochastic dynamics for the nearest‐neighbor Ising model on (ℤ/ L ) 2 (the square lattice with lattice spacing 1/ L ) with initial condition such that σ x =−1 if x ∊ and σ x = + 1 otherwise. We prove the following classical conjecture due to H. Spohn: In the diffusive limit where time is rescaled by L 2 and L → ∞, the boundary of the droplet of “‐” spins follows a deterministic anisotropic curve‐shortening flow such that the normal velocity is given by the local curvature times an explicit function of the local slope. Locally, in a suitable reference frame, the evolution of the droplet boundary follows the one‐dimensional heat equation. To our knowledge, this is the first proof of mean‐curvature‐type droplet shrinking for a lattice model with genuine microscopic dynamics. An important ingredient is in our forthcoming work, where the case of convex was solved. The other crucial point in the proof is obtaining precise regularity estimates on the deterministic curve‐shortening flow. This builds on geometric and analytic ideas of Grayson, Gage and Hamilton, Gage and Li, Chou and Zhu, and others.© 2015 Wiley Periodicals, Inc.

Hubert Lacoin

We study the Glauber dynamics for the zero-temperature Ising model in dimension d=4 with "plus" boundary condition.Let T+ be the time needed for an hypercube of size L entirely filled with "minus" spins to become entirely "plus". We prove that T+ is O(L^2(log L)^c) for some constant c, not depending on the dimension. This brings further rigorous justification for the so-called "Lifshitz law" T+ = O(L^2) [5, 3] conjectured on heuristic grounds. The key point of our proof is to use the detail knowledge that we have on the three-dimensional problem: results for fluctuation of monotone interfaces at equilibrium and mixing time for monotone interfaces dynamics extracted from [2], to get the result in higher dimension.

Sandy Klemm

In the present work, a framework is proposed for studying autonomous agents which interact locally yet effect a globally coherent behavior. This problem of locally induced organization is ubiquitous in decentralized multi-robot environments and various micro- and macroscopic biological contexts (e.g., cellular chemotaxis, avian flocking). In analogy with the local equations of motion which arise in various elastic rod and vorticity theories, we pursue this question in a continuum setting where agents are uniquely associated with material points of a virtual filament. The governing dynamics for this filament are chosen so that an established set of control objectives is achieved. The appropriate configuration space of continua is shown to be an infinite dimensional Hilbert Lie group admitting a separable topology. A class of filament models is studied in a Lagrangian formalism on this manifold, leading to a natural curvature feedback law.

Oded Schramm

Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.

Kai Lai Chung

exists for every i (Theorem 9 of [3]).Following Levy the state i is called stable or instantaneous according as gt-as finite or infinite.We refer to [2 ] for the foundations of the theory of Markov chains under consideration.Although knowledge of these foundations will be necessary for a thorough understanding of what follows, we shall strive to make the present paper readable by itself.Let i be a stable state with q,>0; such a state always exists unless P{x(t) =x(0), 0^t<°° }=l [8, p. 375].Suppose that P{x(0)>=*} =1.LetX=X,(w) be the "first sojourn time" in the state i, namely the length of the first tinterval in which x(t, w)=i (seeTheorem 1 of [2]).Then P{\^t} =l-e~">', t^O [3, p. 54].Let j be an arbitrary state (not oo!) and defineIf J9*i, a is the "first entrance time intoj"; ii j = i, a is the "second entrance time into i" (the first being zero by hypothesis).It is easily shown that a is a random variable in the broad sense, namely a measurable w-iunction defined on a measurable w-set whose probability may be less than one.We define its distribution function in the broad sense by Fij(t)=P{a^t}.Now it can be proved that the two random variables X and a-X (which may be zero with positive probability) are independent^).This is a special case of Theorem 5 of [2], but we give a simple proof as follows.Let us first note that the distribution of a -X may be derived as follows.It can be shown that(3) a(w) considered as a point on the i-axis is the limit from the right of points of Sj(w) ={t: x(t, w) =j}; hence we havewhere h = 2~m, ra->• oo.Now for each 5 > 0 define two random variables X,=X,(w) and a, = aa(w) on the set {w: x(s, w)=i} as follows: X"(w) is the supremum of T such that x(t, w)=i, s^t<s + T; as(w) is the infimum of t such that t>\"(w) and x(t, w) =j.Thus X0 and a0 reduce to the previous X (2) The random variables Zi(w), ■ ■ ■ , z"(w), with domains of definition Ai, • ■ • , A", are said to be independent iff P{ flLx A*[z*M Set] }/P{ f|Li A*l = LTt-i P{A*tz*W £c*]}/P(At) for every real Cx, • • • , ck.(3) In fact, the set Sj(w) is dense in itself (see [2, §4, (ii)]).