1 Introduction and statement of the main results
For the basic probabilistic concepts and results, we refer the reader to any standard work on probability theory such as [
1].
Let
S be a Polish space, that is, a separable completely metrizable topological space, and
\(\mathcal{P}(S)\) the collection of Borel probability measures on
S, equipped with the weak topology
\(\tau_{w}\), that is, the weakest topology for which each map
$$ \mathcal{P}(S) \rightarrow\mathbb{R} : P \mapsto \int f \, dP $$
with bounded and continuous
\(f : S \rightarrow\mathbb{R}\) is continuous. The space
\(\mathcal{P}(S)\) is known to be Polish.
We call a collection \(\Gamma\subset\mathcal{P}(S)\) uniformly tight iff for each \(\epsilon> 0\), there exists a compact set \(K \subset S\) such that \(P(S \setminus K) < \epsilon\) for all \(P \in\Gamma\).
The following celebrated result interrelates the \(\tau_{w}\)-relative compactness with uniform tightness.
Fix
\(N \in\mathbb{N}_{0}\) and let
\(\mathcal{C}\) be the space of continuous maps
x of the compact interval
\([0,1]\) into Euclidean
N-space
\(\mathbb{R}^{N}\) equipped with the uniform topology
\(\tau _{\infty}\), that is, the topology derived from the uniform norm
$$ \|x\|_{\infty}= \sup_{t \in[0,1]} \bigl\vert x(t)\bigr\vert , $$
where
\(\vert \cdot \vert \) stands for the Euclidean norm. The space
\(\mathcal{C}\) is also known to be Polish.
Recall that a set \(\mathcal{X} \subset\mathcal{C}\) is said to be uniformly bounded iff there exists a constant \(M > 0\) such that \(\vert x(t)\vert \leq M\) for all \(x \in\mathcal{X}\) and \(t \in [0,1 ]\), and uniformly equicontinuous iff for each \(\epsilon> 0\), there exists \(\delta> 0\) such that \(\vert x(s) - x(t)\vert < \epsilon\) for all \(x \in\mathcal{X}\) and \(s,t \in [0,1 ]\) with \(\vert s - t\vert < \delta\).
In this setting, the following theorem is a classical result [
2].
Let \(\Omega= (\Omega,\mathcal{F},\mathbb{P})\) be a fixed probability space. Throughout, a continuous stochastic process (c.s.p.) is a Borel-measurable map of Ω into \(\mathcal{C}\), and we consider on the set of c.s.p.s the weak topology \(\tau_{w}\), that is, the topology with open sets \(\{\xi\text{ c.s.p.} \mid\mathbb{P}_{\xi}\in\mathcal{G}\}\), where \(\mathbb{P}_{\xi}\) is the probability distribution of ξ, and \(\mathcal{G}\) is a \(\tau_{w}\)-open set in \(\mathcal{P} (\mathcal{C} )\).
A collection Ξ of c.s.p.s is said to be stochastically uniformly bounded iff for each \(\epsilon> 0\), there exists \(M > 0\) such that \(\mathbb{P} (\|\xi\|_{\infty}> M ) < \epsilon\) for all \(\xi\in\Xi\), and stochastically uniformly equicontinuous iff for all \(\epsilon, \epsilon^{\prime}> 0\), there exists \(\delta> 0\) such that \(\mathbb{P} (\sup_{\vert s - t\vert < \delta} \vert \xi (s) - \xi(t)\vert \geq\epsilon ) < \epsilon^{\prime}\) for all \(\xi\in\Xi\), the supremum taken over all \(s,t \in[0,1]\) for which \(\vert s - t\vert < \delta\).
It is not hard to see that combining Theorem
1.1 and Theorem
1.2 yields the following stochastic version of Theorem
1.2, which plays a crucial role in the development of functional central limit theory.
In a complete metric space
\((X,d)\), the Hausdorff measure of noncompactness of a set
\(A \subset X\) [
3,
4] is given by
$$ \mu_{\mathrm{H},d}(A) = \inf_{F} \sup_{x \in A} \inf_{y \in F} d(x,y), $$
the first infimum running through all finite sets
\(F \subset X\). It is well known that
A is
d-bounded if and only if
\(\mu_{ \mathrm{H},d}(A) < \infty\), and
d-relatively compact if and only if
\(\mu_{{\mathrm{H}},d}(A) = 0\).
Fix a complete metric
d metrizing the topology of the Polish space
S. The Prokhorov distance with parameter
\(\lambda\in\mathbb {R}^{+}_{0}\) between probability measures
\(P,Q \in\mathcal{P}(S)\) [
5] is defined as the infimum of all numbers
\(\alpha\in\mathbb {R}^{+}_{0}\) such that
$$ P (A ) \leq Q \bigl(A^{(\lambda\alpha)} \bigr) + \alpha $$
for all Borel sets
\(A \subset S\), where
$$ A^{(\epsilon)} = \Bigl\{ x \in S \bigm|\inf_{a \in A} d(a,x) \leq \epsilon \Bigr\} . $$
This distance is denoted by
\(\rho_{\lambda}(P,Q)\). It defines a complete metric on
\(\mathcal{P}(S)\) and induces the weak topology
\(\tau_{w}\). It is also known that
\(\rho_{\lambda_{1}} \leq\rho _{\lambda_{2}}\) if
\(\lambda_{1} \geq\lambda_{2}\) and that
$$ \sup_{\lambda\in\mathbb{R}^{+}_{0}} \rho_{\lambda}(P,Q) = \sup _{A} \bigl\vert P(A) - Q(A)\bigr\vert , $$
the supremum being taken over all Borel sets
\(A \subset S\).
For a collection
\(\Gamma\subset\mathcal{P}(S)\), we define the measure of nonuniform tightness as
$$ \mu_{\mathrm{ut}}(\Gamma) = \sup_{\epsilon> 0} \inf _{Y} \sup_{P \in\Gamma} P \biggl(S \bigm\backslash \bigcup_{y \in Y} B(y,\epsilon ) \biggr), $$
where the infimum runs through all finite sets
\(Y \subset S\), and
$$ B(y,\epsilon) = \bigl\{ x \in S \mid d(y,x) < \epsilon \bigr\} . $$
It is clear that
\(\mu_{\mathrm{ut}}(\Gamma) = 0\) if Γ is uniformly tight. The converse holds as well. Indeed, suppose that
\(\mu_{\mathrm{ut}}(\Gamma) = 0\) and fix
\(\epsilon> 0\). Then, for each
\(n \in\mathbb{N}_{0}\), choose a finite set
\(Y_{n} \subset S\) such that
$$ P \biggl(S \bigm\backslash \bigcup_{y \in Y_{n}} B(y,1/n) \biggr) < \epsilon/2^{n} $$
for all
\(P \in\Gamma\). Put
$$ K = \bigcap_{n \in\mathbb{N}_{0}} \bigcup _{y \in Y_{n}} B^{\star}(y,1/n) $$
with
\(B^{\star}(y,1/n)\) the closure of
\(B(y,1/n)\). Then
K is a compact set such that
\(P(S \setminus K) < \epsilon\) for all
\(P \in\Gamma\). We conclude that Γ is uniformly tight. The measure
\(\mu_{\mathrm {ut}}\) is slightly weaker than the weak measure of tightness studied in [
6].
By the previous considerations we know that a set
\(\Gamma\subset \mathcal{P}(S)\) is
\(\tau_{w}\)-relatively compact if and only if
\(\mu _{\mathrm{H},\rho_{\lambda}}(\Gamma) = 0\) for each
\(\lambda \in\mathbb{R}^{+}_{0}\), and uniformly tight if and only if
\(\mu_{\mathrm {ut}}(\Gamma) = 0\). Therefore, Theorem
1.4, our first main result, which provides a quantitative relation between the numbers
\(\mu_{\mathrm{H},\rho_{\lambda}}(\Gamma)\) and
\(\mu _{\mathrm{ut}}(\Gamma)\), is a strict generalization of Theorem
1.1. The proof is given in Section
2.
From now on, we consider on the space
\(\mathcal{C}\) the uniform metric derived from the uniform norm, and for a set
\(\mathcal{X} \subset \mathcal{C}\), we let
\(\mu_{\mathrm{H},\infty}(\mathcal{X})\) stand for the Hausdorff measure of noncompactness; more precisely,
$$ \mu_{\mathrm{H},\infty}(\mathcal{X}) = \inf_{\mathcal{F}} \sup _{x \in\mathcal{X}} \inf_{y \in\mathcal{F}} \|x - y\|_{\infty}, $$
the infimum taken over all finite sets
\(\mathcal{F} \subset\mathcal {C}\). Clearly,
\(\mathcal{X}\) is
\(\tau_{\infty}\)-relatively compact if and only if
\(\mu_{\mathrm{H},\infty}(\mathcal{X}) = 0\).
The measure of nonuniform equicontinuity of
\(\mathcal{X} \subset \mathcal{C}\) is defined by
$$ \mu_{\mathrm{uec}}(\mathcal{X}) = \inf_{\delta> 0} \sup _{x \in\mathcal{X}} \sup_{\vert s - t\vert < \delta} \bigl\vert x(s) - x(t) \bigr\vert , $$
the second supremum running through all
\(s,t \in[0,1]\) with
\(\vert s - t\vert < \delta\). We readily see that
\(\mathcal{X}\) is uniformly equicontinuous if and only if
\(\mu_{\mathrm{uec}}(\mathcal {X}) = 0\). In [
3], it was shown that
\(\mu_{\mathrm{ {uec}}}\) is a measure of noncompactness on the space
\(\mathcal{C}\) (Theorem 11.2).
Theorem
1.5, our second main result, entails that the measures
\(\mu_{\mathrm{H},\infty}\) and
\(\mu_{\mathrm {uec}}\) are Lipschitz equivalent on the collection of uniformly bounded subsets of
\(\mathcal{C}\), and thus it strictly generalizes Theorem
1.2. The proof, which hinges upon a classical result of Jung on the Chebyshev radius, is given in Section
3.
We transport the parameterized Prokhorov metric from
\(\mathcal {P}(\mathcal{C})\) to the collection of c.s.p.s via their probability distributions. Thus, for c.s.p.s
ξ and
η,
$$ \rho_{\lambda}(\xi,\eta) = \rho_{\lambda}(\mathbb{P}_{\xi},\mathbb{P}_{\eta}). $$
Note that a set of c.s.p.s Ξ is
\(\tau_{\omega}\)-relatively compact if and only if
\(\mu_{\mathrm{H},\rho_{\lambda}}(\Xi) = 0\) for all
\(\lambda\in\mathbb{R}^{+}_{0}\).
For a set of c.s.p.s Ξ, the measure of nonstochastic uniform boundedness is given by
$$ \mu_{\mathrm{sub}}(\Xi) = \inf_{M \in\mathbb{R}^{+}_{0}} \sup_{\xi\in\Xi} \mathbb{P}\bigl(\Vert \xi \Vert _{\infty}> M\bigr), $$
and the measure of nonstochastic uniform equicontinuity by
$$ \mu_{\mathrm{suec}}(\Xi) = \sup_{\epsilon> 0} \inf_{\delta> 0} \sup_{\xi\in\Xi} \mathbb{P} \Bigl(\sup_{\vert s - t\vert < \delta} \bigl\vert \xi(s) - \xi(t)\bigr\vert \geq\epsilon \Bigr), $$
where the third supremum is taken over all
\(s,t \in[0,1]\) with
\(\vert s - t\vert < \delta\). It is easily seen that Ξ is stochastically uniformly bounded if and only if
\(\mu_{\mathrm{sub}}(\Xi) = 0\), and stochastically uniformly equicontinuous if and only if
\(\mu_{\mathrm{suec}}(\Xi) = 0\). The measure
\(\mu _{\mathrm{suec}}\) was studied in [
6].
In Section
4, we prove that combining Theorem
1.4 and Theorem
1.5 leads to Theorem
1.6, our third main result, which gives upper and lower bounds for
\(\sup_{\lambda\in\mathbb{R}^{+}_{0}} \mu_{\mathrm{H},\rho _{\lambda}}\) in terms of
\(\mu_{\mathrm{sub}}\) and
\(\mu_{\mathrm{suec}}\). Theorem
1.6 strictly generalizes Theorem
1.3.
Competing interests
The author declares that there are no competing interests.
Ben Berckmoes is post doctoral fellow at the Fund for Scientific Research of Flanders (FWO).