# multi-time limit of a maximum of random walks

Suppose one has $$N$$ iid random walks $$X^{(1)}_t,\ldots,X^{(N)}_t$$ in discrete or continuous time $$t$$, let us say for example Poisson jump processes, and consider the stochastic process $$Y^{(N)}_t = \text{max}(X^{(1)}_t,\ldots,X^{(N)}_t)$$. My question is, broadly, what interesting scaling limits this process $$Y^{(N)}_t$$ may have under any reasonable scaling of $$N$$ and $$t$$. For any fixed $$t$$, as $$N \to \infty$$ I believe that with suitable shift and normalization $$Y^{(N)}_T$$ should converge to one of the extreme value distributions, see the Fisher-Tippett-Gnedenko theorem (note this shouldn't be true for all random walks, e.g. if the $$X^{(i)}$$ are simple random walks then $$Y^{(N)}_t$$ converges to $$t$$, but it seems that for at least some random walks whose fixed-time marginals have unbounded support one ought to see extreme value distributions). However, I am interested in whether there is any meaningfull way to extract a full stochastic process from such a limit of $$Y^{(N)}_T$$ beyond just single-time results. This is an intentionally vague/broad question and I've had trouble finding literature on multi-time distributions of maxima of random walks, so if applicable an answer of the form "here is the question you should have asked" or "limits like this won't exist under any reasonable interpretation of your question, here's why" would also be helpful.

• In 1-d discrete time the process of successive maxima is the ladder height process. Feller vol 2 is a good source.
– mike
Mar 4, 2021 at 8:10
• @mike thank you, I took a look at Feller, but this seems to be a different question than mine. I'm asking about $max(X^{(1)}_T,\ldots,X^{(N)}_T)$ for $N$ random walks, while the ladder process describes $max_{1 \leq t \leq T} X_t$ for a single random walk. Mar 4, 2021 at 19:38

First, you might as well assume that the mean of your walks is $$0$$, because that is just a deterministic shift. If not, as you point out, the maximum may be washed out by the deterministic contribution - unless you scale $$t$$ and $$N$$ appropriately. In what follows I will assume mean $$0$$.
The natural decorrelation length of the random walks is $$t$$ (that is, given the walk at time $$t$$, it takes about $$t$$ additional steps to begin forgetting the value). So, the natural scaling I guess would be $$Z_t^{(N)}=Y_{e^t}^{(N)}$$, where the correlation between $$Z_t^{(N)}$$ and $$Z_{t+1}^{(N)}$$ would be non trivial.
In fact, assume your walks have Gaussian marginals, so the variance of $$X_t$$ is $$t$$. Then, as you point out, $$Z_t^{(N)}$$ scales like $$\sqrt{ e^t \log N}$$. So, the natural process to look at is, IMHO, $$\hat Z_t^{(N)}= \sqrt{\log N}[Z_t^{(N)}- C \sqrt{e^t\log N}]/\sqrt{e^t}$$ (the constant $$C$$ can be computed). I would expect that process to have a non-trivial limit.