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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 $u\in[0,1]$)?

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  • $\begingroup$ There is no "Corollary 5.2" on p. 12 of linked paper. $\endgroup$
    – Michael
    Commented Aug 13, 2020 at 1:27

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  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 $u$.

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