A Kolmogorov–Chentsov Type Theorem on General Metric Spaces with Applications to Limit Theorems for Banach-Valued Processes

This paper deals with moduli of continuity for paths of random processes indexed by a general metric space Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} with values in a general metric space X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {X}}}$$\end{document}. Adapting the moment condition on the increments from the classical Kolmogorov–Chentsov theorem, the obtained result on the modulus of continuity allows for Hölder-continuous modifications if the metric space X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {X}}}$$\end{document} is complete. This result is universal in the sense that its applicability depends only on the geometry of the space Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document}. In particular, it is always applicable if Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} is a bounded subset of a Euclidean space or a relatively compact subset of a connected Riemannian manifold. The derivation is based on refined chaining techniques developed by Talagrand. As a consequence of the main result, a criterion is presented to guarantee uniform tightness of random processes with continuous paths. This is applied to find central limit theorems for Banach-valued random processes.

(1.1) understand a theorem that, under an appropriate moment condition on the distance d X (X θ , X ϑ ) for θ, ϑ ∈ Θ, yields existence of a continuous or Hölder-continuous modification (cf. [4]). We establish the following general result.

Remark 1.1
Technical assumption (1.2) is always satisfied when X is a separable metric space because, in this case, B(X 2 ) = B(X ) ⊗ B(X ). In general, we only have the inclusion B(X 2 ) ⊇ B(X ) ⊗ B(X ), and the assumption is needed to ensure measurability of d X (X θ , X ϑ ).
We consider Theorem 1.1 as our main "building block". In the literature, Kolmogorov-Chentsov type theorems are sometimes formulated in a localized form. A localized version of Theorem 1.1 where Θ is not necessarily totally bounded is presented in Section 2.

Remark 1.2
The key assumption on the geometry of the parametric space Θ is (1.1), where the value of t is important, as we need to have q > t in (1.3). 1 We remark that, if Θ is a bounded subset of R m with the Euclidean metric d m,2 = d Θ , then (1.1) is always satisfied with t = m. 2 More generally, a relatively compact subset Θ of an m-dimensional connected Riemannian manifold always satisfies (1.1) with t = m (we provide more detail in Section 3).
In the classical formulation of the Kolmogorov-Chentsov theorem it is assumed that X is a Banach space and Θ = [0, 1] m for some m ∈ N (see [22,Theorem I.2.1]), and the proof relies on the fact that the dyadic rationals are dense in [0, 1]. Since that time there appeared many other versions of the Kolmogorov-Chentsov theorem that essentially allow to treat more general sets Θ. We mention [19,Theorem 2.1], [6,Theorem 3.9], [10,Lemma 2.19], [12,Proposition 3.9] for several recent formulations where Θ is a subset of R m . Some versions of the Kolmogorov-Chentsov theorem only guarantee that sup(d X (X θ , X ϑ )/d Θ (θ, ϑ) β ) < ∞ a.s. (i.e., it is not claimed that the expectation of the p-th power of that quantity is finite). However, some applications such as the ones discussed in Sections 4 and 5 below require that the expectation is finite. As another example of this kind we mention that the proof of Theorem 6.1 in [2] would not work without finiteness of such an expectation (see formula (106) in [2]).
In the aforementioned references, X is (a closed subset of) a Banach space and all X θ are assumed to be in L p (with p from (1.3)), and the proof involves a certain extension result for Banach-valued Hölder-continuous mappings. That extension result allows to pass from rectangular regions in R m to general subsets Θ ⊆ R m . In our situation when X is only a metric space and we do not assume E[d X (a, X θ ) p ] < ∞ for all θ and some a ∈ X (or the like) such a method of the proof cannot work, so we use essentially different ideas to prove Theorem 1.1.
Another approach, used in [21,Theorem 2.9] (also see [17,Corollary 4.3]), is worth mentioning. In that reference, the existence of a locally Hölder-continuous modification is proved for X = R under assumptions of a different kind. In particular, the assumption on Θ is that it is a dyadically separable metric space. The latter is a requirement of a different type than (1.1) on the geometry of Θ, which allows to pursue the arguments initially elaborated for rectangular regions in R m in more general situations. The setup in [21] is quite different from ours, and the relation between the approaches still has to be worked out. Notice, however, that in the finite-dimensional situation Θ ⊆ R m , the other approach imposes some restrictions on possible sets Θ (see [21,Theorem 4.1]), while our approach allows for arbitrary sets Θ ⊆ R m (see Proposition 2.1 and Remark 2.1 below).
We thus summarize the previous discussion by noting that we obtain inequality (1.4), essentially, only under requirement (1.1) on the geometry of the metric space Θ, which is satisfied for bounded subsets of R m (with t = m) and allows to go beyond R m . It is also worth noting that the right-hand side of (1.4) is the same for all countable subsets Θ ⊆ Θ, and that (1.4) is the right way to formulate the result in the case when d X is incomplete (and thus a continuous modification may fail to exist).
In order to discuss applications of Theorem 1.1, we formulate the following immediate 3) and t from (1.1)), and let L(Θ, C, t, M, p, q, β) be any constant satisfying (1.4). Then, for every at most countable subset Θ ⊆ Θ and arbitrary δ > 0, (1.5) Notice that, like in Theorem 1.1, inequality (1.5) holds universally, i.e., independently of the random process satisfying (1.2) and (1.3). This will turn out to be useful when analyzing weak convergence of X -valued random processes (see Sections 4 and 5).
The crucial step for the proof of Theorem 1.1 is provided by the following auxiliary result. It is interesting in its own right.
Although Proposition 2.1 follows from Theorem 1.1 via standard arguments, we present a proof to make the paper self-contained.
Proof of Proposition 2.1 Fix any n ∈ N. The set Θ n from Property (P) is totally bounded. Therefore, we can find open subsets Θ n,1 , . . . , Θ n,rn of Θ with diameters less than ρ n such that where Θ n,i = Θ n ∩ Θ n,i . By (2.2) we can apply Theorem 1.1 on each Θ n,i . Hence each (X θ ) θ∈Θ n,i has a modification (X • Every relatively compact Θ ⊆ M satisfies (1.1) with t = m (Proposition 3.1); • Every Θ ⊆ M satisfies Property (P) with t n = m, n ∈ N (Corollary 3.1).
Let (M, g) be any connected m-dimensional Riemannian manifold as defined in [8]. This means that M denotes an m-dimensional C ∞ -manifold endowed with the Riemannian metric g. By definition g is a mapping which associates to each point p ∈ M an inner product g p on the tangential space T p M at p such that for C ∞ -vector fields V, W on an open subset G of M the mapping is differentiable of class C ∞ . Furthermore, let for p, q denote by C pq the set of all C ∞ -curves in M joining p to q. The length L(c) of a curve c ∈ C pq defined on the closed interval I c of R is where c ′ (t) stands for the velocity of c at t. Since M is connected, the sets C pq are always nonvoid (see [8, p. 146]), and the mapping is a metric on M (see [   Proof Since M is a C ∞ -manifold, we can find an open covering {Θ n } n∈N of M consisting of relatively compact subsets of M and satisfying Θ n ⊆ Θ n+1 for n ∈ N (see, e.g., [7, (16.1.4)]). By Proposition 3.1 this sequence of subsets satisfies (2.1) w.r.t. d g with t n = m for n ∈ N (and the constants C n indeed depend on n). Hence every Θ ⊆ M satisfies Property (P) with t n = m, n ∈ N, w.r.t. d g , as we can choose 4 Θ n := Θ ∩ Θ n , n ∈ N. ✷ In the rest of this section we prove Proposition 3.1. The proof is based on a couple of auxiliary results.
Lemma 3.1 Let Θ be a nonvoid compact subset of M and assume Θ ⊆ G, where G is an open subset of M allowing a chart u : G → R m which satisfies that u(Θ) is convex. Then there is some δ > 0 such that N(Θ, d g , η) ≤ N u(Θ), d m,2 , η/δ for η > 0.
Proof Let {e 1 , . . . , e m } stand for the standard basis on R m . For any C ∞ -mapping g : U → R on some open subset U of R m we shall use notation d x g to denote the differential of g at x ∈ U. Let us introduce for p ∈ G the set C ∞ M (p) of all real-valued C ∞ -mappings on some open neigbourhood of p. By definition, the tangential space T p M of M at p consists of real-valued mappings on C ∞ M (p). The chart u provides the following basis of (see [8, p.8]). Moreover, ∂ ∂u m p defines some C ∞ -vector field (see [8, 25f.]).
Next, let for x ∈ u(G) denote by d x u −1 the differential of u −1 at x which is a linear mapping Then, with S m−1 denoting the Euclidean sphere in R m , we may conclude from the defining properties of the Riemannian metric g that the mapping is continuous with strictly positive outcomes. Moreover, its domain is a compact subset of R m × R m so that it attains its maximum δ which is a positive number. Now, let p, q ∈ Θ with p = q. Since u(Θ) is assumed to be convex, the mapping Since g u −1 (c(t)) is an inner product on T M and d c(t) u −1 is linear for every t ∈ [0, 1], we obtain where · m,2 stands for the Euclidean norm on R m . Hence by definition of the inner metric d g we end up with d g (p, q) ≤ L(c) ≤ δ u(p) − u(q) m,2 .
Since δ does not depend on p, q, we now easily derive the claim of Lemma 3.1. ✷ In the next step, using Lemma 3.1, we prove the result of Proposition 3.1 first for compact subsets of M.
Lemma 3.2 Let Θ ⊆ M be nonvoid and compact. Then there exists a nonvoid compact subset K m of R m as well as r ∈ N and δ > 0 such that Proof For any p ∈ Θ we may find a chart u p , defined on an open subset G up of M, and some ε p > 0 such that p ∈ G up and For any i ∈ {1, . . . , r} the set Θ i meets the requirements of Lemma 3.1. Hence we may find δ 1 , . . . , δ r > 0 such that The set is a compact subset of R m . Then setting, δ := 4 max{δ 1 , . . . , δ r }, we end up with This completes the proof. ✷ Finally, we are ready to prove Proposition 3.1.
Proof of Proposition 3.1 (i) Let Θ be a nonvoid relatively compact subset of M. The topological closure Θ is compact, and N(Θ, d g , η) ≤ N(Θ, d g , η/2) holds for every η > 0. Therefore, the first claim immediately follows from Lemma 3.2.
(ii) If d g is complete, then by the Hopf-Rinow theorem (see, e.g., [8, Theorem 7.2.8]) every d gbounded subset of M is already relatively compact. Therefore, the second claim follows from the first one. ✷

Tightness for sequences of random processes
Let (Θ, d Θ ) be a compact metric space and (X , d X ) a complete metric space. We denote by C(Θ, X ) the space of all continuous mappings from Θ into X endowed with uniform metric d ∞ w.r.t. the metric d X and the induced Borel σ-algebra B C(Θ, X ) . Some of the results we are going to present simplify in the case when C(Θ, X ) is separable (hence Polish, as it is complete). For some discussions below we recall that, as Θ is compact, C(Θ, X ) is separable if and only if X is separable (see [1,Lemma 3.99]). We, however, stress at this point that we never assume X (equivalently, C(Θ, X )) to be separable.
Let us fix any sequence (X n ) n∈N of Borel random elements X n : Ω → C(Θ, X ) on some probability space (Ω, F , P). We show how Corollary 1.1 leads to a sufficient condition for uniform tightness in C(Θ, X ).
Assume that X n (·, θ) n∈N is a uniformly tight sequence of random elements in (X , B(X )), for all θ ∈ Θ ′ , and that there exist M, p > 0 and q > t such that Then (X n ) n∈N is a uniformly tight sequence of Borel random elements in C(Θ, X ).
We recall that (1.1) need not be assumed if Θ is a compact subset of R m endowed with the Euclidean metric. In this case, it is enough only to require q > m in (4.1) (see Remark 1.2).
Remark 4.1 Notice that (1.2) is satisfied for all processes X n because they are assumed to be Borel random elements in C(Θ, X ) in this section and the projection map Remark 4.2 Observe that, if X is separable, then the statements (A) X n : Ω → C(Θ, X ) is a Borel random element, i.e., a random element in C(Θ, X ), B(C(Θ, X )) ; and (B) X n = (X n (·, θ)) θ∈Θ is an X -valued process (i.e., for all θ ∈ Θ, X n (·, θ) is a random element in (X , B(X ))) with continuous paths are equivalent (see [15,Lemma 14.1]). Thus, whenever X is a Polish space, in Proposition 4.1 (and in what follows) we essentially work with sequences of continuous X -valued processes. In general, when (A) and (B) no longer coincide, the right choice is always (A), i.e., always to consider Borel random elements in C(Θ, X ), as the concept of tightness (in C(Θ, X )) discussed in Proposition 4.1 requires the Borel σ-algebra (in C(Θ, X )).
Proof of Proposition 4.1 We take an arbitrary β ∈]0, (q − t)/p[. By compactness of Θ there exists some at most countable dense subset Θ of Θ. Corollary 1.1 together with the continuity of the processes X n yields, for all δ > 0 and n ∈ N, Using the Markov inequality, we conclude that, for every ε > 0, Now the criterion for uniform tightness in C(Θ, X ) presented in Theorem A.1 applies and completes the proof. ✷ We observe that essentially the same condition achieves rather different aims in Theorem 1.1 and in Proposition 4.1. In Theorem 1.1, condition (1.3) ensures existence of a continuous modification for the process X (when X is complete, which is assumed in Section 4), while in Proposition 4.1, condition (4.1) implies the uniform tightness in C(Θ, X ) for the sequence (X n ). (Notice that (4.1) is nothing else but (1.3) required for all X n uniformly in n.) It is, therefore, tempting to try to shift continuity of the processes into the conclusion of Proposition 4.1. And, indeed, this easily follows from the discussions above, although at the cost of requiring X to be separable.
We consider a sequence (X n ) n∈N of X -valued processes X n = (X n (·, θ)) θ∈Θ . Let Θ ′ ⊆ Θ be dense in Θ. Assume that (X n (·, θ)) n∈N is a uniformly tight sequence of random elements in (X , B(X )), for all θ ∈ Θ ′ , and that there exist M, p > 0 and q > t such that Then each process X n admits a modification X n = (X n (·, θ)) θ∈Θ that has continuous paths θ → X n (ω, θ) for all ω ∈ Ω, the processes X n , n ∈ N, are Borel random elements in C(Θ, X ), and the sequence (X n ) n∈N is uniformly tight in C(Θ, X ).
Proof Theorem 1.1 ensures the existence of the continuous modifications X n , n ∈ N. As X is separable, then, due to the equivalence between (A) and (B) in Remark 4.2, each X n is a Borel random element in C(Θ, X ). The uniform tightness of the sequence (X n ) n∈N now follows from Proposition 4.1. ✷ Remark 4.3 If in Corollary 4.1 we additionally require that each process X n is separable (the definition is recalled below), then we obtain that each process X n is itself continuous almost surely, so that we obtain the uniform tightness for the sequence (X n ) n∈N itself. 5 This immediately follows from Lemma 4.1 below. For when this remark can be useful, we observe that, in some situations, we are given processes that are a priori separable (e.g., càdlàg X -valued processes in the case Θ = [0, 1]).
It remains to justify the previous remark. Recall that an X -valued process (Y (·, θ)) θ∈Θ on some (Ω, F , P) is called separable 6 if there exist an at most countable subset Θ 0 ⊆ Θ dense in Θ and an event Ω 0 ∈ F with P(Ω 0 ) = 1 such that for every open subset G of Θ, and any closed subset D of X the following equality holds true [9]). 5 Formally, we need to identify each process X n with almost all continuous paths with an indistinguishable process with all continuous paths, in order to view X n as a Borel random element in C(Θ, X ). 6 In the sense of Doob. Lemma 4.1 Let Y = Y (·, θ) θ∈Θ be a separable X -valued process that admits a continuous modification. Then Y = Y (·, θ) θ∈Θ is itself continuous almost surely, and hence there is an indistinguishable from Y process Y such that all its paths are continuous.
It is worth noting that, contrary to the general setting in Section 4, for this lemma the metric space X does not need to be complete.
For k ∈ N set G k := {ϑ ∈ Θ | d Θ (θ, ϑ) < 1/k}, and let D k denote the closure of the set , as n → ∞, due to definition of Ω 1 . Moreover, we may select by compactness of Θ a subsequence (ϑ k i(n) ) n∈N of (ϑ k n ) n∈N which converges to some ϑ k ∈ Θ. Then, by continuity As d Θ (θ, ϑ k ) ≤ 1/k, the sequence (ϑ k ) k∈N converges to θ. Hence, drawing on the continuity of Y again, we end up with This completes the proof. ✷

Central limit theorems for Banach-valued random processes
Let (Θ, d Θ ) be a compact metric space, and let (X , · X ) be a Banach space. We shall denote by C(Θ, X ) the space of all continuous mappings from Θ into X . It will be endowed with sup-norm · ∞ w.r.t. · X , and the induced Borel σ-algebra B C(Θ, X ) .
Consider any i.i.d. sequence (X i ) i∈N of Bochner-integrable Borel random elements in C(Θ, X ) on some probability space (Ω, F , P). We want to investigate weak convergence of the sequence (S n ) n∈N consisting of Borel random elements in C(Θ, X ) defined by where E B [X i ] denotes the Bochner-integral of X i . We start with the following observation.
(i) The following statements are equivalent: a) The sequence (S n ) n∈N is uniformly tight; b) The sequence (S n ) n∈N converges weakly to some centered Gaussian random element in C(Θ, X ), B(C(Θ, X )) .
(ii) If the equivalent statements in part (i) are satisfied, then the limiting law in b) is tight.
We remark that, as every Borel probability measure in a Polish space is tight, statement (ii) in Proposition 5.1 has a message only when C(Θ, X ) (equivalently, X ) is non-separable.
Proof As the Borel random element X 1 is Bochner-integrable, it is almost surely separablyvalued. Then we can find a closed separable linear subspace C of C(Θ, X ) such that P({X 1 ∈ C}) = 1 (note that C is itself a Polish space and C ∈ B(C(Θ, X ))). It follows that E B [X 1 ] ∈ C. This yields P({X 1 − E B [X 1 ] ∈ C}) = 1, hence P({S n ∈ C}) = 1 for all n ∈ N. In view of the portmanteau lemma this yields that every weak limit point of the laws of S n , n ∈ N, is concentrated on C (in particular, is tight), thus establishing part (ii). Moreover, the implication b) ⇒ a) in part (i) now follows from Prokhorov's theorem, which applies due to the fact that all measures are concentrated on a Polish space.
We turn to the implication a) ⇒ b) in part (i). By Prokhorov's theorem, the uniformly tight sequence (S n ) n∈N is relatively weakly sequentially compact. It remains to prove uniqueness of a limit point and its Gaussianity. To this end, let r ∈ N and Λ j : C(Θ, X ) → R, j = 1, . . . , r, be continuous linear functionals. Classical multivariate central limit theorem applies to the sequence . . , r ( Λ j denotes the operator norm of Λ j ) and yields weak convergence to a centered Gaussian law in R r . This identifies every weak limit point of the laws of S n , n ∈ N, as a Gaussian measure and uniquely determines every weak limit point on the σ-algebra E generated by continuous linear functionals C(Θ, X ) → R. Notice that E ⊆ B(C(Θ, X )), and the inclusion can be strict (when C(Θ, X ) is non-separable). However, restricted to C both σ-algebras coincide: C ∩ E = C ∩ B(C(Θ, X )) (see [24, Theorem I.2.1]). Recalling that every weak limit point is concentrated on C completes the proof. ✷ For application of Proposition 5.1 we can utilize our criterion in Proposition 4.1 and obtain the following result.
Corollary 5.1 Let Θ satisfy condition (1.1) with constants C, t > 0, and let X 1 ∞ be square integrable. Assume that there is a dense subset Θ ′ ⊆ Θ such that S n (·, θ) n∈N is a uniformly tight sequence of random elements in X , for all θ ∈ Θ ′ , (5.1) and that there exist M, p > 0 as well as q > t with Then the sequence (S n ) n∈N converges weakly to a tight centered Gaussian random element in C(Θ, X ), B C(Θ, X ) .
We want to discuss the requirements of Corollary 5.1 for special choices of the Banach space X . Let us start with type 2 -Banach spaces. To recall, the Banach space X is called a type 2 -Banach space if there is a constant C > 0 such that, for all n ∈ N and X -valued independent centered Borel random elements W 1 , . . . , W n such that W i X are square integrable, we have the following inequality • X is a finite-dimensional vector space, • X is an L p -space on some σ-finite measure space (X, A, ν) with L p -norm · p for p ∈ [2, ∞[ (see [18,Section 9.2]).
If X is a type 2 -Banach space, then conditions (5.1) and (5.2) can be simplified in the following way.
2) The sequence (S n ) n∈N satisfies condition (5.2) with p = 2 whenever there exist M > 0 and q > t such that In particular, under (5.3), the sequence (S n ) n∈N converges weakly to a tight centered Gaussian random element in C(Θ, X ), B C(Θ, X ) .
It is worth noting that, even in the separable case, we cannot get this result from the general central limit theorem in type 2 -Banach spaces (see, e.g., [18,Theorem 10.5]) because, in Proposition 5.2, it is only the space X and not C(Θ, X ) that has type 2.
Proof Consider for θ ∈ Θ the continuous linear operator π θ : C Θ, X → X defined by π θ (f ) := f (θ). Then by Bochner-integrability of the Borel random element X 1 in C(Θ, X ), we may conclude that the Borel random element X 1 (·, θ) = π θ •X 1 of X is Bochner-integrable with Bochner-integral E B X 1 (·, θ) = π θ E B X 1 . In particular, it is almost surely separably-valued. Hence the Borel random element X 1 (·, θ) − E B X 1 (·, θ) is almost surely separably-valued too. This means that X 1 (·, θ)−E B X 1 (·, θ) is concentrated on some separable closed subset of X . Due to completeness of · ∞ this implies that X 1 (·, θ) − E B X 1 (·, θ) is a Radon Borel random element of X (see [24, p. 29, Corollary]). Now, statement 1) follows from the general central limit theorem in type 2 -Banach spaces (see [11,Theorem 3.6]  Concerning statement 2), by the above definition of type 2 -Banach spaces, we can find some constant C > 0 such that We now observe that where in the last step we use Jensen's inequality. This completes the proof. ✷ Let us turn to cotype 2 -Banach spaces. The Banach space X is called a cotype 2 -Banach space if there is a constant C > 0 such that, for all n ∈ N and X -valued independent centered Borel random elements W 1 , . . . , W n such that W i X are square integrable, we have the following inequality (see, e.g., [5]). For a further preparation, let us also recall that a centered tight Borel random element W in X is called pre-gaussian if there is some centered tight Gaussian random element holds for every pair L 1 , L 2 of continuous linear forms on X . If X is cotype 2 -Banach space we can obtain the following criterion for property (5.1).

Proposition 5.3
Let X be a cotype 2 -Banach space, let Θ satisfy condition (1.1) with constants C, t > 0, and let X 1 ∞ be square integrable. Assume that there is a dense subset Θ ′ ⊆ Θ such that Then the sequence (S n ) n∈N satisfies property (5.1) (with this Θ ′ ).

Proofs
Let us retake general assumptions and notations from Section 1. One key of our proofs is the following auxiliary technical result which extends Lemma B.2.7 in [23]. For a finite set B we shall use notation card(B) to denote its cardinality.
Lemma 6.1 Let Θ be some nonvoid finite subset of Θ, and let A ≥ 1 as well as r ∈ N such that A r ≥ card(Θ). Then for c > 0 there exists some U ⊆ Θ × Θ satisfying Proof According to the proof of Lemma B.2.7 in [23] we may find a sequence (V l ) l∈N of subsets of Θ, a sequence (θ l ) l∈N in Θ as well as a sequence (r l ) ∈N in {1, . . . , r} such that the following properties are satisfied We shall show that the set First of all so that U fulfills (6.1).
Lemma 6.2 Let Θ ⊆ Θ be finite with at least two elements. Let n 0 be the largest element in Z such that ∆(Θ) ≤ 2 −n 0 , and let Then n 0 < n 1 , and the following statements are valid.

This implies by Minkowski's inequality
Next, set for abbreviation Then we obtain in view of (1.3) along with (6.8) and (6.9) By (6.15) we end up with This shows (6.12) of statement 4). For the remaining part of the proof we additionally assume that property (1.1) is satisfied with constants C > 0, t ∈]0, q[. Then we have Note that 2 −(k+1) < ∆(Θ) ≤ ∆(Θ) holds for every k ∈ {n 0 , . . . , n 1 − 1} due to choice of n 0 . Now, (6.13) can be derived easily by routine calculations using geometric summation formulas. This concludes the proof. ✷ In this case the statement of Lemma 1.1 is trivial.
This yields the first claim of Proposition 6.1. The second claim is a direct consequence of the expressions in (6.25) and (6.27). ✷ Proof of Theorem 1.1 We first fix any β ∈]0, (q − t)/p[. Let the constant L(Θ, C, t, M, p, q, β) be chosen according to Proposition 6.1, and let us consider any at most countable subset Θ of Θ which consists of at least two elements θ, ϑ. We may select some sequence (Θ k ) k∈Θ of nonvoid finite subsets of Θ with at least two elements satisfying and thus by monotone convergence theorem along with Proposition 6.1 For the remaining part of the proof let us assume that d X is complete, and let Θ be some at most countable subset of Θ which is dense w.r.t. d Θ . As a further consequence of (6.28) we have P(A) = 1, where This implies that on A the random process (X θ ) θ∈Θ has Hölder-continuous paths of order β. By completeness of d X we may define a new random process (X θ ) θ∈Θ via wherex ∈ X is arbitrary. Clearly, this process has Hölder-continuous paths of order β. Furthermore, it can be shown by standard arguments that this random process satisfies (1.2). We now show that it is a modification of (X θ ) θ∈Θ . For this purpose let us fix any θ ∈ Θ, and let (ϑ k ) k∈N be a sequence from Θ which converges to θ w.r.t. d Θ . By construction of (X θ ) θ∈Θ we may invoke inequality (1.3) to conclude In particular, on the one hand the sequence d X (X θ , X ϑ k ) k∈N converges in probability to 0. On the other hand by definition of (X θ ) θ∈Θ , the sequence d X (X θ , X ϑ k ) k∈N converges in probability to 0. Then if l ∈ N 0 ≤ lim sup and thus P d X (X θ , X θ ) > 0 = lim Theorem A.1 Let (X n ) n∈N be a sequence of Borel random elements X n : Ω → C(Θ, X ) on some probability space (Ω, F , P). Let Θ ′ ⊆ Θ be dense in Θ. The sequence (X n ) n∈N is uniformly tight if and only if X n (·, θ) n∈N is a uniformly tight sequence of random elements in X , for all θ ∈ Θ ′ , (A.1) and, for every ε > 0, lim We remark that since, for any δ > 0, the mapping w(·, δ) is continuous, w(X n , δ) is a random variable for every n ∈ N (in particular, the probability in (A.2) is well-defined).
Proof The result is well-known for X = R (see [26,Theorem 1.5.7]), and a similiar one is shown in the proof of Theorem 14.5 from [15] in the case of separable and complete d X . We shall use arguments from the proof of Theorem 14.5 in [15].
Firstly, let us assume that (X n ) n∈N is a uniformly tight sequence in C(Θ, X ), and let ε, η ∈]0, 1[. Then there exists some compact subset K ⊆ C(Θ, X ) such that sup n∈N P({X n ∈ C(Θ, X ) \ K}) ≤ η. (A.3) By a general version of the Arzela-Ascoli theorem (see [20,Theorem 47.1]) the set K is equicontinuous which means that we may find for any θ ∈ Θ some δ θ > 0 such that d X f (ϑ), f (θ) < ε/2 for every f ∈ K whenever d θ (ϑ, θ) < δ θ . Since Θ is compact we may apply Lebesgue's number lemma (see [20,Lemma 27.5]) to the open cover of Θ consisting of the open metric balls B δ θ (θ) around θ with radius δ θ . In this way we may select some δ > 0 such that w(f, δ) < ε holds for every f ∈ K. Then, for all n ∈ N, P({w(X n , δ) ≥ ε}) ≤ P({X n ∈ C(Θ, X ) \ K}) ≤ η, For the if part, let (X n ) n∈N fulfill conditions (A.1) and (A.2). Fix any γ ∈]0, 1[. Since Θ is compact, the mappings X n (ω, ·) are uniformly continuous for ω ∈ Ω and n ∈ N. Hence, for ω ∈ Ω and n ∈ N, we have w(X n , δ) → 0, as δ → 0. Combining this observation with condition (A.2), we may find for any k ∈ N some δ k > 0 such that sup n∈N P w(X n , δ k ) > 2 −k ≤ 2 −k−1 γ. (A.6) Since Θ is compact, the metric on Θ ′ is separable. In addition Θ ′ is dense. Then, there is some sequence (θ k ) k∈N in Θ ′ which is a dense subset of Θ. Hence in view of (A.1) we may find for every k ∈ N some compact subset K k of X such that sup n∈N P X n (·, θ k ) ∈ K k } ≤ 2 −k−1 γ. Hence it is left to show that B is a relatively compact subset of C(Θ, X ).
For an arbitrary ε > 0 and for every θ ∈ Θ, choose some k ∈ N such that 2 −k < ε. Then by construction B, we obtain w(f, δ k ) < ε for f ∈ B. In particular, d X f (ϑ), f (θ) < ε for every f ∈ B and any ϑ ∈ Θ with d Θ (ϑ, θ) < δ k . Thus we have shown that B is some equicontinuous subset of C(Θ, X ). Therefore by a general version of the Arzela-Ascoli theorem [20,Theorem 47.1] it remains to show that the set {f (θ) | f ∈ B} is a relatively compact subset of X for any θ ∈ Θ. This means to show that this set is totally bounded w.r.t. d X due to completeness of d X .
Let us fix any θ ∈ Θ. Choose, for an arbitrary ε > 0, some k 0 ∈ N such that 2 −k 0 < ε/2. Since {θ k | k ∈ N} is dense, we may find some k 1 ∈ N such that d Θ (θ k 1 , θ) < δ k 0 . This implies by the construction of the set B that f (θ k 1 ) ∈ K k 1 and d X f (θ k 1 ), f (θ) < ε/2 for every f ∈ B. (A.8) The set K k 1 is assumed to be compact, in particular, it is totally bounded w.r.t. d X , so that there exist m ∈ N and x 1 , . . . , x m ∈ X satisfying where, for r > 0, B r (x i ) denotes the open d X -metric ball around x i with radius r. Then we may conclude from (A.8) This shows that the set {f (θ) | f ∈ B} is totally bounded w.r.t. d X , which completes the proof. ✷