On random walks, projecting election results and statistical physics

2021 ◽  
Vol 18 (2 Jul-Dec) ◽  
pp. 020202
Author(s):  
J. Valentín Escobar
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Density Of States ◽  
Statistical Physics ◽  
Statistical Tools ◽  
Physics Courses ◽  
The Central Limit Theorem ◽  
Election Results

Several important statistical tools and concepts are covered in upper division undergraduate Statistical Physics courses, including those of random walks and the central limit theorem. However, some of their broad applicability tends to be missed by students as well as the connection between these and other physical concepts. In this work, we apply a 1D random walk to study the evolution of the probability that a candidate will win an election given she holds some lead over her opponent, and connect the result found to the concept of density of states and occupation probabilities. This paper is intended to serve as a guide to the Statistical Physics instructor who wishes to motivate students beyond the boundaries of the official syllabus.

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (4) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U(∊)n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M∊ = max {U(∊)n, n ≧ 0}, v0 = min {n : U(∊)n = M∊}, v1 = max {n : U(∊)n = M∊}. The joint limiting distribution of ∊2σ∊–2v0 and ∊σ∊–2M∊ is determined. It is the same as for ∊2σ∊–2v1 and ∊σ–2∊M∊. The marginal ∊σ–2∊M∊ gives Kingman's heavy traffic theorem. Also lim ∊–1P(M∊ = 0) and lim ∊–1P(M∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U(∊)n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment

2004 ◽  
Vol 41 (01) ◽  
pp. 83-92 ◽  
Author(s):  
Jean Bérard
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Space Time ◽  
Nearest Neighbour ◽  
One Dimensional ◽  
The Central Limit Theorem ◽  
Almost All

The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabeiet al.for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for anarbitrarylevel of randomness.

A Quenched Central Limit Theorem for Reversible Random Walks in a Random Environment on Z

2014 ◽  
Vol 51 (04) ◽  
pp. 1051-1064
Author(s):  
Hoang-Chuong Lam
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Integrability Condition ◽  
The Central Limit Theorem ◽  
The Poisson Equation ◽  
Convergence Of Moments ◽  
Quenched Central Limit Theorem

The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on Z without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple and was introduced by Depauw and Derrien [3]. More precisely, for a given realization ω of the environment, we consider the Poisson equation (P ω - I)g = f, and then use the pointwise ergodic theorem in [8] to treat the limit of solutions and then the central limit theorem will be established by the convergence of moments. In particular, there is an analogue to a Markov process with discrete space and the diffusion in a stationary random environment.

GALTON'S QUINCUNX: RANDOM WALK OR CHAOS?

2007 ◽  
Vol 17 (12) ◽  
pp. 4463-4469 ◽  
Author(s):  
KEVIN JUDD
Keyword(s):  
Dynamical System ◽  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Mechanical Device ◽  
Dynamical Models ◽  
The Central Limit Theorem ◽  
Random Events ◽  
Low Dimensional ◽  
Deterministic Dynamical System

In 1873 Francis Galton had constructed a simple mechanical device where a ball is dropped vertically through a harrow of pins that deflect the ball sideways as it falls. Galton called the device a quincunx, although today it is usually referred to as a Galton board. Statisticians often employ (conceptually, if not physically) the quincunx to illustrate random walks and the central limit theorem. In particular, how a Binomial or Gaussian distribution results from the accumulation of independent random events, that is, the collisions in the case of the quincunx. But how valid is the assumption of "independent random events" made by Galton and countless subsequent statisticians? This paper presents evidence that this assumption is almost certainly not valid and that the quincunx has the richer, more predictable qualities of a low-dimensional deterministic dynamical system. To put this observation into a wider context, the result illustrates that statistical modeling assumptions can obscure more informative dynamics. When such dynamical models are employed they will yield better predictions and forecasts.

scholarly journals A note on the central limit theorem for geodesic random walks

1984 ◽  
Vol 30 (2) ◽  
pp. 169-173 ◽  
Author(s):  
Gilles Blum
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
The Central Limit Theorem

In this article we use a theorem of T.G. Kurtz to prove a central limit theorem for geodesic random walks.

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