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I have some problems with the definition of $\textit{weak convergence of stochastic processes}$. To ask my question, we start with two well-known definitions corresponding to measures and random variables.

$\textbf{Definition 1 (weak convergence of measures).}$ Suppose $\left(\mu_n\right)_{n =1}^{\infty}$ is a sequence of probability measures. Then we say that $\mu_n$ converges weakly (or in distribution) to a probability measure $\mu$ and write $\mu_n \xrightarrow{\mathcal{D}} \mu$, if $$ \forall \, \varphi \in \mathcal{C}_b(\mathbb{R}) \colon \int_{\Omega} \varphi \, \text{d} \mu_n \xrightarrow{n \to \infty} \int_{\Omega}\varphi \, \text{d} \mu, $$ where $\mathcal{C}_b(\mathbb{R})$ denotes the set of continuous and bounded functions.


$\textbf{Definition 2 (weak convergence of random variables).}$ Suppose $\left( \Omega, \mathcal{F}, \mathbb{P}\right)$ is a probability space and $\left(f_n\right)_{n =1}^{\infty}$ is a sequence of random variables. Then we say that $f_n$ converges weakly (or in distribution) to a random variable $f$ and write $f_n \xrightarrow{\mathcal{D}} f$, if $$ \forall \, \varphi \in \mathcal{C}_b(\mathbb{R}) \colon \ \mathbb{E} \varphi(f_n)\xrightarrow{n \to \infty} \mathbb{E} \varphi(f) \Longleftrightarrow \int_{\Omega} \varphi(f_n(\omega)) \, \text{d} \mathbb{P}(\omega) \xrightarrow{n \to \infty} \int_{\Omega} \varphi(f(\omega)) \, \text{d} \mathbb{P}(\omega) \\ \Longleftrightarrow \int_{\Omega} \varphi \, \text{d} \mathbb{P}_{f_n} \xrightarrow{n \to \infty} \int_{\Omega}\varphi \, \text{d} \mathbb{P}_f.$$

These two definitions were clear to me. The following definition is the one I am asking about.

$\textbf{Definition 3 (weak convergence of stochastic processes).}$ Suppose $f$ and $\left(f_n\right)_{n \in \mathbb{N}}$ are random variables with values in $\mathcal{C}([0,\infty))$. Then we say that the stochastic processes $\left(f_n\right)_{n \in \mathbb{N}}$ converge to the stochastic process $f$, if the corresponding laws converge weakly in the topological space $\mathbb{S}$ consiting of $\mathcal{C}([0,\infty))$ equipped with the topology of uniform convergence on comapct subsets of $\mathcal{C}([0,\infty))$. That is, if $\varphi(f_n) \xrightarrow{\mathcal{D}} \varphi(f)$ whenever $\varphi \colon \mathcal{C}([0,\infty)) \rightarrow \mathbb{R}$ Borel measurable.

Now my question is, how can I understand Definition $3$ in the sense of Definition $2$? Can I just replace $\varphi \in \mathcal{C}_b(\mathbb{R})$ by $\varphi \in \mathcal{C}([0,\infty))$? What is meant with "equipped with the topology of uniform convergence on comapct subsets of $\mathcal{C}([0,\infty))$"?

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    $\begingroup$ The point is that Definition 2 makes sense if you replace $\mathbb{R}$ by any other topological space, such as $C([0,\infty))$ with the mentioned topology (do you understand what this topology is?). So you should replace $C_b(\mathbb{R})$ with $C_b(C([0,\infty)))$. $\endgroup$ – Nate Eldredge Apr 20 '15 at 16:25
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I think this is easier to understand in a somewhat broader context. You have a sequence of stochastic processes $f_n$. We think of this in three equivalent ways. We have sets $\Omega,T,A$, with $T$ the time set and $A$ the set of values of the process. We can think of each $f_n$ with any of the signatures:

  • $\Omega \times T \to A$ (a function of the randomization and time, symmetrically)

  • $T \to (\Omega \to A)$ (a time-parametrized family of random variables)

  • $\Omega \to (T \to A)$ (a random function of time).

In discussing weak convergence, we think in terms of the third signature. Specifically, we first give (a restriction of) the space of functions $T \to A$ some notion of convergence, through some topology. For continuous functions on a compact interval, this is naturally given by the metric of uniform convergence. For continuous functions on an unbounded domain, the metric of uniform convergence no longer makes sense, so we extend it with the topology of uniform convergence on compact subsets. That is, a sequence of continuous functions $f_n$ on $X$ converges in the topology of uniform convergence on compact subsets if the restriction of $f_n$ to $K$ converges uniformly for each compact $K \subset X $.

With this topology, we get the corresponding Borel $\sigma$-algebra. Then we say $f_n$ converges weakly to $f$ if for each Borel measurable $\varphi : (T \to A) \to \mathbb{R}$, $\varphi(f_n) \to \varphi(f)$ in the sense of distribution. So really, definition 3 is the natural extension of definition 2, we just have a different space and topology on the codomain of the functions.

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  • $\begingroup$ Did you mean $\mathbb{E} \varphi(f_n)\xrightarrow{n \to \infty} \mathbb{E} \varphi(f)$ with $\varphi(f_n)\rightarrow \varphi(f)$? $\endgroup$ – user214418 Apr 20 '15 at 16:12
  • $\begingroup$ @user214418 I edited. $\endgroup$ – Ian Apr 20 '15 at 16:23

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