Four limit theorems for quadratic functionals of Brownian motion and Brownian bridge

2004 ◽  
pp. 75-87 ◽  
Author(s):  
Giovanni Peccati ◽  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Limit Theorems ◽  
Brownian Bridge ◽  
Quadratic Functionals

Super-Brownian Motion and Critical Spatial Stochastic Systems

2004 ◽  
Vol 47 (2) ◽  
pp. 280-297 ◽  
Author(s):  
Ed Perkins
Keyword(s):  
Brownian Motion ◽  
Limit Theorems ◽  
Stochastic Systems ◽  
Contact Process ◽  
Voter Model ◽  
Oriented Percolation ◽  
Short Introduction ◽  
Super Brownian Motion

AbstractThis article is a short introduction to super-Brownian motion. Some of its properties are discussed but our main objective is to describe a number of limit theorems which show super-Brownian motion is a universal limit for rescaled spatial stochastic systems at criticality above a critical dimenson. These systems include the voter model, the contact process and critical oriented percolation.

scholarly journals How linear reinforcement affects Donsker’s theorem for empirical processes

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1173-1192 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Long Range ◽  
Limit Theorems ◽  
Brownian Bridge ◽  
Empirical Processes ◽  
Constant Factor ◽  
Empirical Measure ◽  
Reinforced Random Walks ◽  
Bernoulli Processes

Abstract A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) , $${\hat{U}}_{n+1}$$ U ^ n + 1 is sampled uniformly from $${\hat{U}}_1, \ldots , {\hat{U}}_n$$ U ^ 1 , … , U ^ n , and with complementary probability $$1-p$$ 1 - p , $${\hat{U}}_{n+1}$$ U ^ n + 1 is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when $$p<1/2$$ p < 1 / 2 , and that a further rescaling is needed when $$p>1/2$$ p > 1 / 2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

The first-passage density of the Brownian motion process to a curved boundary

1992 ◽  
Vol 29 (02) ◽  
pp. 291-304 ◽  
Author(s):  
J. Durbin ◽  
D. Williams
Keyword(s):  
Brownian Motion ◽  
Brownian Bridge ◽  
Sample Path ◽  
Practical Method ◽  
Time Interval ◽  
Brownian Motion Process ◽  
First Passage ◽  
Curved Boundary ◽  
Multiple Integrals ◽  
High Degree

An expression for the first-passage density of Brownian motion to a curved boundary is expanded as a series of multiple integrals. Bounds are given for the error due to truncation of the series when the boundary is wholly concave or wholly convex. Extensions to the Brownian bridge and to continuous Gauss–Markov processes are given. The series provides a practical method for calculating the probability that a sample path crosses the boundary in a specified time-interval to a high degree of accuracy. A numerical example is given.

Distributions of the Longest Excursions in a Tied Down Simple Random Walk and in a Brownian Bridge

2007 ◽  
Vol 44 (04) ◽  
pp. 1056-1067 ◽  
Author(s):  
Andreas Lindell ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Weak Convergence ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Brownian Bridge ◽  
Simple Random Walk ◽  
Marginal Distributions

Expressions for the joint distribution of the longest and second longest excursions as well as the marginal distributions of the three longest excursions in the Brownian bridge are obtained. The method, which primarily makes use of the weak convergence of the random walk to the Brownian motion, principally gives the possibility to obtain any desired joint or marginal distribution. Numerical illustrations of the results are also given.

scholarly journals On the time spent above a level by Brownian motion with negative drift

1986 ◽  
Vol 18 (04) ◽  
pp. 1017-1018 ◽  
Author(s):  
J.-P. Imhof
Keyword(s):  
Brownian Motion ◽  
Limit Theorems ◽  
Distribution Functions ◽  
Probability Densities ◽  
Negative Drift

Limit theorems of Berman involve the total time spent by Brownian motion with negative drift above a fixed or exponentially distributed negative level. We give explicitly the probability densities and distribution functions, obtained via an equivalence of laws.

A note on two-level superprocesses

2004 ◽  
Vol 41 (1) ◽  
pp. 202-210
Author(s):  
Wen-Ming Hong
Keyword(s):  
Brownian Motion ◽  
Central Limit ◽  
Random Fields ◽  
Limit Theorems ◽  
Gaussian Random Fields ◽  
Central Limit Theorems ◽  
Gaussian Fields ◽  
Super Brownian Motion ◽  
Random Immigration

We prove some central limit theorems for a two-level super-Brownian motion with random immigration, which lead to limiting Gaussian random fields. The covariances of those Gaussian fields are explicitly characterized.

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