Definition contraction mapping Let (X,d)(X,d) be a complete metric space. Then a map T:X→XT:X→X is called a contraction mapping on XX if there exists c∈[0,1)c∈[0,1) such that d(T(x),T(y))≤cd(x,y)d(T(x),T(y))≤cd(x,y) for all x,y∈X.x,y∈X. Definition fixed point If T:X→XT:X→X, then a point x∈Xx∈X such that T(x)=xT(x)=x is called a fixed point of T.T. Theorem Contraction mapping theorem (also know..
Definition Let ΨΨ be an increasing convex function from [0,∞)[0,∞) onto [0,∞)[0,∞) and XX be a random variable. The Orlicz norm ‖X‖Ψ∥X∥Ψ is defined by ‖X‖Ψ=inf{c>0:EΨ(|X|c)≤1},∥X∥Ψ=inf{c>0:EΨ(|X|c)≤1}, where the infimum over the empty set is ∞. Examples of ΨΨ(x)=exp(xp)−1≡Ψp(x) for $p \geq..
Preliminaries and notation Linear and antilinear maps By definition, an antilinear map (also called a conjugate-linear map) f:H→Y is a map between vector spaces that is additive : f(x+y)=f(x)+f(y)for all x,y∈H, and antilinear (also called conjugate-linear or conjugate-homogeneous) : $$ f(cx) = \bar{c}f(x) ~~~ \text{for all } x ∈ \mathcal{H} \..
Definition Let (T,d) be a semi-metric space. For ε>0, an ε-net of T is a subset Tε of T such that for every t∈T there exists a tε∈Tε with d(t,tε)≤ε. The ε-covering number N(T,d,ε) of T is the infimum of the cardinality of ε-nets of T, that is, $$ N(T, d, \..
Definition Given two metric spaces (X,dX) and (Y,dY), where dX denotes the metric on the set X and dY is the metric on set Y, a function f:X→Y is called Lipschitz continuous if there exists a real constant K≥0 such that, for all x1 and x2 in X, dY(f(x1),f(x2))≤K⋅dX(x1,x2). In particular, a real-valued function $f : \mathbb{R} → \mathbb{..
본 내용은 아래의 노트 (Lecture notes in empirical process theory, Kengo Kato, 2019)를 참조하여 작성하였습니다. https://drive.google.com/file/d/0B7C_CufYq6j6QU5rblF2Yl85d3c/view?resourcekey=0-ItZa4Z1yrAGhUa7scVo_aw empirical_process_v3.pdf drive.google.com The main object is to study probability estimates of the random quantity ‖Pn−P‖F:=supf∈F|Pnf−Pf|, and lim..
본 내용은 아래의 노트 (Lecture notes in empirical process theory, Kengo Kato, 2019)를 참조하여 작성하였습니다. https://drive.google.com/file/d/0B7C_CufYq6j6QU5rblF2Yl85d3c/view?resourcekey=0-ItZa4Z1yrAGhUa7scVo_aw empirical_process_v3.pdf drive.google.com A sample space Ω is the set of all possible outcomes of a random element. A collection of subsets of a sample space Ω denoted by A. A col..
본 내용은 아래의 책을 참조하여 작성하였습니다 https://www.researchgate.net/publication/224839687_Hierarchical_Modeling_and_Analysis_of_Spatial_Data 3.1 Formal modeling theory for spatial processes ... 3.1.1 Some basic stochastic process theory for spatial processes ...
본 내용은 아래의 책을 참조하여 작성하였습니다 https://www.researchgate.net/publication/224839687_Hierarchical_Modeling_and_Analysis_of_Spatial_Data Chap 1의 내용에서 알 수 있듯, 해당 주제의 fundamental concept은 stochastic process {Y(s):s∈D}이며 여기서 D는 r-dimensional Euclidean space이다. r=1이면 단순한 time series이다. 이 spatial model에서는 주로 r>2이며 이러한 경우를 spatial process라 부른다. 2.1 Elements of point-referenced modeling 2...
본 내용은 아래의 책을 참조하여 작성하였습니다 https://www.researchgate.net/publication/224839687_Hierarchical_Modeling_and_Analysis_of_Spatial_Data 1.1 Introduction to spatial data and models Spatial data는 크게 3가지 type으로 구분된다. point-referenced data, where Y(s) is a random vector at a location s∈Rr, where s varies continuously over D, a fixed subset of Rr that contains an r-dimen..