scholarly journals Arcsine laws for random walks generated from random permutations with applications to genomics

2021 ◽  
Vol 58 (4) ◽  
pp. 851-867
Author(s):  
Xiao Fang ◽  
Han L. Gan ◽  
Susan Holmes ◽  
Haiyan Huang ◽  
Erol Peköz ◽  
...  
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Simple Random Walk ◽  
Random Permutation ◽  
Asymptotic Distributions ◽  
Maximum Height ◽  
Strong Embedding ◽  
Arcsine Distribution ◽  
Special Cases ◽  
Functional Central Limit Theorems

AbstractA classical result for the simple symmetric random walk with 2n steps is that the number of steps above the origin, the time of the last visit to the origin, and the time of the maximum height all have exactly the same distribution and converge when scaled to the arcsine law. Motivated by applications in genomics, we study the distributions of these statistics for the non-Markovian random walk generated from the ascents and descents of a uniform random permutation and a Mallows(q) permutation and show that they have the same asymptotic distributions as for the simple random walk. We also give an unexpected conjecture, along with numerical evidence and a partial proof in special cases, for the result that the number of steps above the origin by step 2n for the uniform permutation generated walk has exactly the same discrete arcsine distribution as for the simple random walk, even though the other statistics for these walks have very different laws. We also give explicit error bounds to the limit theorems using Stein’s method for the arcsine distribution, as well as functional central limit theorems and a strong embedding of the Mallows(q) permutation which is of independent interest.

CENTRAL LIMIT AND FUNCTIONAL CENTRAL LIMIT THEOREMS FOR HILBERT-VALUED DEPENDENT HETEROGENEOUS ARRAYS WITH APPLICATIONS

1998 ◽  
Vol 14 (2) ◽  
pp. 260-284 ◽  
Author(s):  
Xiaohong Chen ◽  
Halbert White
Keyword(s):  
Central Limit ◽  
Adaptive Learning ◽  
Limit Theorems ◽  
Asymptotic Distributions ◽  
Central Limit Theorems ◽  
Cumulative Distribution ◽  
Density Estimator ◽  
Dependent Observations ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

We obtain new central limit theorems (CLT's) and functional central limit theorems (FCLT's) for Hilbert-valued arrays near epoch dependent on mixing processes, and also new FCLT's for general Hilbert-valued adapted dependent heterogeneous arrays. These theorems are useful in delivering asymptotic distributions for parametric and nonparametric estimators and their functionals in time series econometrics. We give three significant applications for near epoch dependent observations: (1) A new CLT for any plug-in estimator of a cumulative distribution function (c.d.f.) (e.g., an empirical c.d.f., or a c.d.f. estimator based on a kernel density estimator), which can in turn deliver distribution results for many Von Mises functionals; (2) a new limiting distribution result for degenerate U-statistics, which delivers distribution results for Bierens's integrated conditional moment tests; (3) a new functional central limit result for Hilbert-valued stochastic approximation procedures, which delivers distribution results for nonparametric recursive generalized method of moment estimators, including nonparametric adaptive learning models.

Queues in transportation systems, I: a Markovian system

1973 ◽  
Vol 10 (03) ◽  
pp. 630-643
Author(s):  
Michael A. Crane
Keyword(s):  
Limit Theorems ◽  
Transportation Systems ◽  
Mild Conditions ◽  
Special Cases ◽  
Arrival Processes ◽  
Random Fluctuations ◽  
Functional Central Limit Theorems ◽  
Passenger Terminals ◽  
Functional Central Limit ◽  
Markovian System

We study a transportation system consisting of S vehicles of unit capacity and N passenger terminals. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to a terminal j, 1 ≦ j ≦ N, with probability Pij . Customers at each terminal are served as vehicles become available. Each vehicle is dispatched from a terminal when loaded, whereupon it travels to the destination of its passenger, according to a stochastic travel time. It is shown under mild conditions that the system is unstable, due to random fluctuations of independent customer arrival processes. We obtain limit theorems, in certain special cases, for the customer queue size processes. Where a steady-state limit exists, this limit is expressed in terms of the corresponding limit in a related GI/G/S queue. In other cases, functional central limit theorems are obtained for appropriately normalized random functions.

Queues in transportation systems, I: a Markovian system

1973 ◽  
Vol 10 (3) ◽  
pp. 630-643 ◽  
Author(s):  
Michael A. Crane
Keyword(s):  
Limit Theorems ◽  
Transportation Systems ◽  
Mild Conditions ◽  
Special Cases ◽  
Arrival Processes ◽  
Random Fluctuations ◽  
Functional Central Limit Theorems ◽  
Passenger Terminals ◽  
Functional Central Limit ◽  
Markovian System

We study a transportation system consisting of S vehicles of unit capacity and N passenger terminals. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to a terminal j, 1 ≦ j ≦ N, with probability Pij. Customers at each terminal are served as vehicles become available. Each vehicle is dispatched from a terminal when loaded, whereupon it travels to the destination of its passenger, according to a stochastic travel time. It is shown under mild conditions that the system is unstable, due to random fluctuations of independent customer arrival processes. We obtain limit theorems, in certain special cases, for the customer queue size processes. Where a steady-state limit exists, this limit is expressed in terms of the corresponding limit in a related GI/G/S queue. In other cases, functional central limit theorems are obtained for appropriately normalized random functions.

scholarly journals Strong Limit Theorems for a Simple Random Walk on the 2-Dimensional Comb

2009 ◽  
Vol 14 (0) ◽  
pp. 2371-2390 ◽  
Author(s):  
Endre Csáki ◽  
Miklós Csörgö ◽  
Antonia Feldes ◽  
Pál Révész
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Simple Random Walk ◽  
Strong Limit

The size of the region occupied by one type in an invasion process

1976 ◽  
Vol 13 (02) ◽  
pp. 355-356 ◽  
Author(s):  
Aidan Sudbury
Keyword(s):  
Random Walk ◽  
Simple Random Walk ◽  
Rectangular Lattice ◽  
Invasion Process

Particles are situated on a rectangular lattice and proceed to invade each other's territory. When they are equally competitive this creates larger and larger blocks of one type as time goes by. It is shown that the expected size of such blocks is equal to the expected range of a simple random walk.

Queues in transportation systems, II: An independently-dispatched system

1974 ◽  
Vol 11 (01) ◽  
pp. 145-158 ◽  
Author(s):  
Michael A. Crane
Keyword(s):  
Limit Theorems ◽  
Transportation Systems ◽  
Linear Network ◽  
Random Functions ◽  
Arrival Processes ◽  
Vehicle Fleet ◽  
Vehicle Capacity ◽  
Service Groups ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

We consider a transportation system consisting of a linear network of N + 1 terminals served by S vehicles of fixed capacity. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to some terminal j, 0 ≦ j ≦ i − 1, and are served as empty units of vehicle capacity become available at i. The vehicle fleet is partitioned into N service groups, with vehicles in the ith group stopping at terminals i, i − 1,···,0. Travel times between terminals and idle times at terminals are stochastic and are independent of the customer arrival processes. Functional central limit theorems are proved for random functions induced by processes of interest, including customer queue size processes. The results are of most interest in cases where the system is unstable. This occurs whenever, at some terminal, the rate of customer arrivals is at least as great as the rate at which vehicle capacity is made available.

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

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