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

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

1 Answer 1

  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 $u$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.