On limit theorems for random fields

A complete separable metric space of functions defined on the positive quadrant of the plane is constructed. The characteristic property of these functions is that at every point x there exist two lines intersecting at this point such that limits limy→x f (y) exist when y approaches x along any path not intersecting these lines. A criterion of compactness of subsets of this space is obtained.


Introduction
The theory of weak convergence in the function space D[0, 1] (Billingsley [3]) has been extended by Bickel and Wichura [2], Neuhaus [6], and Straf [7] to D[0, 1] q , by Lindvall [5] to D[0, ∞), and by Ivanoff [4] to D[0, ∞) q . In case q = 2 the spaces D[0, 1] 2 and D[0, ∞) 2 consist of functions with two parameters having so called quadrant limits. Discontinuity points of these functions form lines parallel to the coordinate axes. Here we construct a more general space of functions with two parameters having so called net limits. Discontinuity points of these functions form lines not necessarily parallel to coordinate axes. The restriction of this space to the space of functions defined on the unit square is a particular case of the space introduced by Banys and Surgailis [1].
The functions under consideration have the property that at each point x in the domain of definition there exist two lines intersecting at x such that limits lim y→x f (y) exist when y approaches x along any path not intersecting these lines.

Definitions
Let R 2 be the two-dimensional Euclidean space with the norm x = x 2 1 + x 2 2 of an element x = (x 1 , x 2 ) ∈ R 2 . We shall denote by U and ∂U respectively the closure and the boundary of a set U ⊂ R 2 and by B δ (x) the open circle with radius δ and center x. DEFINITION 1. A union = ϕ 1 ∪ ϕ 2 of line segments ϕ 1 and ϕ 2 passing through a point x ∈ R 2 is called an elementary net at x if one of the following two conditions holds: (i) is a line segment itself and x is an interior point of ; (ii) ϕ 1 and ϕ 2 are not aligned with a straight line.
, 0 x 1 , x 2 1} be the unit sqare. Denote by ∂ i X, i = 1, . . . , 4, the sides of X and by ∂ 0 X the set of its corner points. DEFINITION 2. A finite union = ∪ n i=1 ϕ i (n 4) of line segments lying in X is called a net if ∂X ⊂ and for every x ∈ there exists a circle B = B(x) such that ∩ B is an elementary net at x. If ∩ B(x) is formed by two line segments not aligned with a straight line, then x is called a node point of the net .
where χ A is indicator function, = {A 1 , . . . , A m } ∈ X and, moreover, Given The class of simple functions will be denoted by D 0 .

The space D[0, 1] 2
Denote by D(X) the uniform clousure of D 0 in the space of all bounded functions from X to R 1 . Let f ∈ D(X) and be an elementary net at x ∈ X. The elementary net divides any sufficiently small circle B(x) into disjoint components B 1 , . . . , B n (2 n 4).
We say that f has -limits at x if there exist limits Put and limits lim y→x f (y) exist when y approaches x along the lines forming the elementary net , and these limits are in L(x, f ). Therefore, ( ) = min x sin γ x , where the minimum extends over all node points x of , and γ x is the smallest positive angle between two line segments ϕ 1 , ϕ 2 ⊂ intersecting at the node point x. For an arbitrary net and the boundary ∂X the following relations hold: With a net , we associate a modulus of its "smoothness" defined by . . , n, and w δ (f ) = inf{ω f ( ), ∈ X , κ(∂ ) > δ}, where the minimum extends over all the partitions ∈ X satisfying κ(∂ ) > δ. To introduce a metric on D(X) we begin with a group of one to one mappings of X onto itself. A one to one mapping λ of X onto itself is called a C 1 -diffeomorphism if it is continuous together with its inverse λ −1 and the partial derivatives Let be the class of all C 1 -diffeomorphisms such that λ(x) = x, x ∈ ∂ 0 X, and λ(∂ i X) = ∂ i (X), i = 1, . . . , 4. Denote by C λ the matrix of the partial derivtives of λ, i.e., Given matrix A = [a i,j ], its norm is defined by |A| = 2 max i,j |a i,j |. Let I be the identity matrix. Define function |· |: → R 1 by is a complete separable metric space.
Now we formulate three statements on the compatness and tightness in D which are analogous to Theorems 1-3 in [1].

The space D[0, ∞) 2
Now we construct a similar space of functions defined on R 2 + = [0, ∞) 2 extending the techniques used by Lindvall [5]. Let The space D c [0, 1) 2 is closed in D[0, 1) 2 and, therefore, is a complete separable metric space. Let R 2 ) and define the mapping : The mapping is a bijection, and D c (R 2 is again a complete and separable. For k = 1, 2, . . . put g k (x) = g k (x 1 )g k (x 2 )), where x = (x 1 , x 2 ) ∈ R 2 + , and g k are defined by g k (t) = 1 for t k, g k (t) = k + 1 − t for k < t < k + 1, and g k (t) = 0 for t k + 1.
Define the mappings c k : by c k (f ) = f × g k , and the mapping : Then f n → f if and only if there exist a sequence λ n ∞ n=1 , λ n ∈ (R 2 + ), such that for all a > 0 the following relations hold: The theorem can be proved using similar arguments like in the proof of Theorem 3.1 in [4].

THEOREM 6.
If T is dense in R 2 + , then F T coincides with Borel σ -algebra of subsets of D(R 2 + ).
For a = (a 1 , a 2 ) ∈ R 2 , denote X a = {(x 1 , x 2 ): 0 x i a i , i = 1, 2}. Let D(X a ) be the space of functions from X a to R 1 defined analogously to the space D(X). Define the mapping r a : D(R 2 + ) → D(X a ) by r a f (x) = f (x), x ∈ X a . For a probability measure P on D(R 2 + ), let T P be the set of those a ∈ R 2 + for which P f : r a f ∈ D(X a ) and r a is continuous at f = 1.
THEOREM 7. Let P , P 1 , P 2 , . . . be probability measures on D(R 2 + ). Then the sequence P n , n = 1, 2, . . . converges weakly to P if and only if for all a ∈ T P the sequence P n r −1 a , n = 1, 2, . . . converges weakly to P r −1 a .
The proof is analogous to the proof of Theorem 3 in [5].