scholarly journals A Metropolis-class sampler for targets with non-convex support

2021 ◽  
Vol 31 (6) ◽  
Author(s):  
John Moriarty ◽  
Jure Vogrinc ◽  
Alessandro Zocca
Keyword(s):  
Random Walk ◽  
Law Of Large Numbers ◽  
Rare Event ◽  
General Purpose ◽  
Strong Law ◽  
Numerical Examples ◽  
Large Numbers ◽  
Random Walk Metropolis ◽  
Improved Performance ◽  
Random Walk Metropolis Algorithm

AbstractWe aim to improve upon the exploration of the general-purpose random walk Metropolis algorithm when the target has non-convex support $$A\subset {\mathbb {R}}^d$$ A ⊂ R d , by reusing proposals in $$A^c$$ A c which would otherwise be rejected. The algorithm is Metropolis-class and under standard conditions the chain satisfies a strong law of large numbers and central limit theorem. Theoretical and numerical evidence of improved performance relative to random walk Metropolis are provided. Issues of implementation are discussed and numerical examples, including applications to global optimisation and rare event sampling, are presented.

scholarly journals Strong Law of Large Numbers for a Function of the Local Times of a Transient Random Walk in $${\mathbb {Z}}^d$$

2019 ◽  
Vol 33 (4) ◽  
pp. 2315-2336
Author(s):  
Inna M. Asymont ◽  
Dmitry Korshunov
Keyword(s):  
Random Walk ◽  
Law Of Large Numbers ◽  
Acta Math ◽  
Local Times ◽  
List Type ◽  
Strong Law ◽  
Statistics And Probability ◽  
Large Numbers ◽  
Berkeley Symposium ◽  
Transient Random Walk

Abstract For an arbitrary transient random walk $$(S_n)_{n\ge 0}$$ ( S n ) n ≥ 0 in $${\mathbb {Z}}^d$$ Z d , $$d\ge 1$$ d ≥ 1 , we prove a strong law of large numbers for the spatial sum $$\sum _{x\in {\mathbb {Z}}^d}f(l(n,x))$$ ∑ x ∈ Z d f ( l ( n , x ) ) of a function f of the local times $$l(n,x)=\sum _{i=0}^n{\mathbb {I}}\{S_i=x\}$$ l ( n , x ) = ∑ i = 0 n I { S i = x } . Particular cases are the number of visited sites [first considered by Dvoretzky and Erdős (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function $$f(i)={\mathbb {I}}\{i\ge 1\}$$ f ( i ) = I { i ≥ 1 } ; $$\alpha $$ α -fold self-intersections of the random walk [studied by Becker and König (J Theor Probab 22:365–374, 2009)], which corresponds to $$f(i)=i^\alpha $$ f ( i ) = i α ; sites visited by the random walk exactly j times [considered by Erdős and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where $$f(i)={\mathbb {I}}\{i=j\}$$ f ( i ) = I { i = j } .

scholarly journals Excursions of a Random Walk Related to the Strong Law of Large Numbers

1998 ◽  
Vol 28 (2) ◽  
pp. 595-606
Author(s):  
Travis Lee ◽  
Max Minzner ◽  
Evan Fisher
Keyword(s):  
Random Walk ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Large Numbers

scholarly journals Tauberian theorems for summability transforms

2005 ◽  
Vol 2005 (1) ◽  
pp. 55-66
Author(s):  
Rohitha Goonatilake
Keyword(s):  
Random Walk ◽  
Rate Of Convergence ◽  
Law Of Large Numbers ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Strong Law ◽  
Random Walk Method ◽  
Large Numbers ◽  
The Law

The convolution summability method is introduced as a generalization of the random-walk method. In this paper, two well-known summability analogs concerning the strong law of large numbers (SLLN) and the law of the single logarithm (LSL), that gives the rate of convergence in SLLN for the random-walk method, are extended to this generalized method.

scholarly journals On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xiaochen Ma ◽  
Qunying Wu
Keyword(s):  
Probability Space ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Sublinear Expectation ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
End Random Variables

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

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