scholarly journals Research Topics in Empirical Processes

2021 ◽  
Author(s):  
◽  
Christopher Ball
Keyword(s):  
Brownian Motion ◽  
Brownian Bridge ◽  
Research Problem ◽  
Empirical Processes ◽  
Mixture Distributions ◽  
Research Topics ◽  
Martingale Approach ◽  
Model Section ◽  
Reflection Principles ◽  
And Function

<p>The first chapter consists of an overview of the theory of empirical processes, covering an introduction to empirical processes in R, uniform empirical processes and function parametric empirical processes in Section 1.1. Section 1.2 contains an overview of the theory related to the law of the iterated logarithm for Brownian motion and the modulus of continuity for Brownian motion. Section 1.3 contains the theory of the limiting processes for the empirical process, most importantly Brownian motion, Brownian bridge and the connections and relationships between them, with distributions of selected statistics of Brownian motion and Brownian bridge derived from reflection principles. Section 1.4 contains an overview of the theory required to prove central limit results for the empirical processes, covering the theory of the space C and Donsker’s theorem.  The second chapter covers research topics, starting with Fourier analysis of mixture distributions and associated theory in Section 2.1. Section 2.2 covers findings in a research problem about non-linear autoregressive processes. Section 2.3 introduces a martingale approach to testing a regression model. Section 2.4 links the theory of ranks and sequential ranks to the theory of empirical processes.</p>

scholarly journals Research Topics in Empirical Processes

2021 ◽  
Author(s):  
◽  
Christopher Ball
Keyword(s):  
Brownian Motion ◽  
Brownian Bridge ◽  
Research Problem ◽  
Empirical Processes ◽  
Mixture Distributions ◽  
Research Topics ◽  
Martingale Approach ◽  
Model Section ◽  
Reflection Principles ◽  
And Function

<p>The first chapter consists of an overview of the theory of empirical processes, covering an introduction to empirical processes in R, uniform empirical processes and function parametric empirical processes in Section 1.1. Section 1.2 contains an overview of the theory related to the law of the iterated logarithm for Brownian motion and the modulus of continuity for Brownian motion. Section 1.3 contains the theory of the limiting processes for the empirical process, most importantly Brownian motion, Brownian bridge and the connections and relationships between them, with distributions of selected statistics of Brownian motion and Brownian bridge derived from reflection principles. Section 1.4 contains an overview of the theory required to prove central limit results for the empirical processes, covering the theory of the space C and Donsker’s theorem.  The second chapter covers research topics, starting with Fourier analysis of mixture distributions and associated theory in Section 2.1. Section 2.2 covers findings in a research problem about non-linear autoregressive processes. Section 2.3 introduces a martingale approach to testing a regression model. Section 2.4 links the theory of ranks and sequential ranks to the theory of empirical processes.</p>

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.

Brownian motion on fractals and function spaces

1996 ◽  
Vol 222 (3) ◽  
pp. 495-504 ◽  
Author(s):  
Alf Jonsson
Keyword(s):  
Brownian Motion ◽  
Function Spaces ◽  
And Function

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

2007 ◽  
Vol 44 (4) ◽  
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 Brownian local times

1995 ◽  
Vol 8 (3) ◽  
pp. 209-232 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Brownian Motion ◽  
Brownian Bridge ◽  
Distribution Functions ◽  
Local Times ◽  
Explicit Formulas ◽  
Brownian Excursion ◽  
Reflecting Brownian Motion ◽  
Brownian Meander

In this paper explicit formulas are given for the distribution functions and the moments of the local times of the Brownian motion, the reflecting Brownian motion, the Brownian meander, the Brownian bridge, the reflecting Brownian bridge and the Brownian excursion.

scholarly journals Enhancing Critical Writing towards Undergraduate Students in Conducting Research Proposal

2020 ◽  
Author(s):  
Moh. Yamin ◽  
Oikurema Purwati
Keyword(s):  
Critical Thinking ◽  
Undergraduate Students ◽  
Research Method ◽  
Research Problem ◽  
Research Proposal ◽  
Research Topics ◽  
Daily Lives ◽  
Critical Writing ◽  
Conducting Research ◽  
Descriptive Method

Critical thinking in writing for learners is needed to able to build a sense of crisis towards any issues. This capacity is required to create a framework based on reflecting, exploring, and solving some practical problems in their learning, work, and daily lives. This article aims to enhance critical writing towards undergraduate students in conducting research. The research method used is descriptive method with qualitative approach. The data used to analyze is gotten from many research articles in international journals discussing critical writing to conduct a research proposal. The research discussion and result state that learners’ capacity to be familiar with critical thinking writing is essential. It deals with brainstorming the problems of research topics, reflecting, and exploring the issues of research topics. Students should also have the capacity to write the research background, research problem, review of related literature, and research method based on the guidelines of the research proposal. The writing skill relating to the competence of organization, elaborating content, paying attention mechanics, language use, and vocabulary become the leading and supporting capital to be successful in conducting the research proposal.

scholarly journals The value of the high, low and close in the estimation of Brownian motion

2020 ◽  
Author(s):  
Kurt Riedel
Keyword(s):  
Brownian Motion ◽  
Conditional Expectation ◽  
Brownian Bridge ◽  
Conditional Variance ◽  
Conditional Density ◽  
Computational Results ◽  
Final Value ◽  
Additional Information ◽  
High Maximum ◽  
Extremal Properties

AbstractThe conditional density of Brownian motion is considered given the max, $$B(t|\max )$$ B ( t | max ) , as well as those with additional information: $$B(t|close, \max )$$ B ( t | c l o s e , max ) , $$B(t|close, \max , \min )$$ B ( t | c l o s e , max , min ) where the close is the final value: $$B(t=1)=c$$ B ( t = 1 ) = c and $$t \in [0,1]$$ t ∈ [ 0 , 1 ] . The conditional expectation and conditional variance of Brownian motion are evaluated subject to one or more of the statistics: the close (final value), the high (maximum), the low (minimum). Computational results displaying both the expectation and variance in time are presented and compared with the theoretical values. We tabulate the time averaged variance of Brownian motion conditional on knowing various extremal properties of the motion. The final table shows that knowing the high is more useful than knowing the final value among other results. Knowing the open, high, low and close reduces the time averaged variance to $$42\%$$ 42 % of the value of knowing only the open and close (Brownian bridge).

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