# What does weak convergence mean for a stochastic process?

I am reading a paper in which stochastical processes $$\{\mathcal{H}_T(u)\}_{u\in[0,1]}$$ and $$\{\mathcal{H}(u)\}_{u\in[0,1]}$$ on [0,1] with $$u$$ as a time-index occur.

There is a theorem which states that $$\mathcal H_T$$ weakly converges to $$\mathcal{H}$$ as $$T\rightarrow\infty$$ in $$l_\infty([0,1])$$ (Corollary 5.2, p. 12).

Now I have 2 questions:

1. What does $$l_\infty([0,1])$$ mean in this case? According to wikipedia, $$l_\infty$$ is the space containing bounded sequences; however, here we aren't dealing with sequences but with a time-continuous process.

2. What does weak convergence mean for a process? Does it mean pointwise convergence in distribution (i.e. $$\mathcal{H}_T(u) \overset{d}{\longrightarrow}\mathcal{H}(u)$$ as $$T\rightarrow\infty$$ for all $$u\in[0,1]$$)?

• There is no "Corollary 5.2" on p. 12 of linked paper. Commented Aug 13, 2020 at 1:27

1. $$l_\infty([0,1])$$ is a function space that contains the sample paths of the process. Its elements are real-valued functions on $$[0,1]$$, not sequences. In your case, "$$l_\infty([0,1])$$" is not standard notation for a standard function space (nor does it appear to on p12 of the linked paper).
2. A stochastic process induces a probability measure on the function space that contains its sample paths. Weak convergence defines convergence of probability measures on complete metric spaces. Weak convergence for stochastic processes is a generalization of convergence in distribution for random variables---where the metric space is $$\mathbb{R}^n$$. Weak convergence is not the same as convergence in distribution at a given time $$u$$.