Statistics

2608 Submissions

[2] viXra:2608.0041 [pdf] submitted on 2026-08-11 02:53:06

Continuous Random Variables

Authors: Teo Banica
Comments: 400 Pages.

This is an introduction to probability, written with a quantum idea in mind, namely that the semicircle law comes first. We first discuss discrete probability, notably with the binomial and hypergeometric laws, positive and negative, and the Poisson and compound Poisson laws. Then we get into the continuous case, with the basics of the theory explained, and with as starting examples the exponential, semicircle and beta distributions. Afterwards, we investigate the central limits and normal variables, both real and complex, and with a look at Rayleigh laws, and hyperspherical laws too. Finally, we discuss a number of more specialized distributions, and more specialized techniques too, and we end with an introduction to random matrices and freeness.
Category: Statistics

[1] viXra:2608.0028 [pdf] submitted on 2026-08-07 05:37:17

Classical vs Path Coupling on a Geometric State Space: Glauber Dynamics for (q)-Colorings of (C_n)

Authors: Iz Tecum
Comments: 26 Pages.

We analyze one coupling of single-site Glauber dynamics for proper (q)-colorings of the cycle (C_n) in two ways.Drift on the Hamming disagreement count and path coupling on Hamming-adjacent pairs give the one-step contractions[mathbb{E}bigl[d_{H}(X_1,Y_1)bigr]lealpha_{mathrm{cl}}(q,n),d_{H}(X_0,Y_0),qquadmathbb{E}bigl[d_{G}(X_1,Y_1)bigr]lealpha_{mathrm{pc}}(q,n),d_{G}(X_0,Y_0),]with (alpha_{mathrm{cl}}(q,n)=1-(q-4)/bigl(n(q-2)bigr)) and (alpha_{mathrm{pc}}(q,n)=1-(q-6)/bigl(n(q-2)bigr)), both attained. The thresholds are(qge5) and (qge7), and the constants separate by (2/bigl(n(q-2)bigr)) uniformly in (n). The gap is geometric.Transposing two colors across an edge produces a pair at Hamming distance (2) that no single recoloring connects,so the path metric (d_{G}) of the color graph charges a created disagreement (+2) rather than (+1); such pairspack (lfloor n/2floor) to a cycle, giving (operatorname{diam} d_{H}=n) and (operatorname{diam} d_{G}=lfloor3n/2floor) for(qge5), hence closed (O(nlog n)) bounds for both methods. The loss is attributable to the sparse edge setrather than to path coupling, since the complete edge set recovers (alpha_{mathrm{cl}}). Exact enumeration on (C_4)confirms every bound and gives worst-start (t_{mathrm{mix}}(0.05)=57,29,22,19,18) for (q=3,dots,7).
Category: Statistics