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Let X be a continuous stochastic process. I know that (t>s)
P(|X(t) - X(s)|>δ) < |t-s|/δ

Is it possible to say anything (e.g. estimate the decay of the tail) about
Y=sup_{s \in [0,1]} |X(s)|?

I suspect that the answer is no (it would be an easy question if we have |t-s|^{1+a}, for any a>0).

I wonder what are "standard" methods of estimating the suprema of continuous stochastic processes.

So far in my problem I tried to use the Garsia-Rodemich-Rumsey inequality.

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The answer is, indeed, "No" because there is an unbounded with probability $1$ stochastic process that satisfies the given inequality, namely, $X(t)=0.1\log|t-w|$ where $w$ is equidistributed on $[0,1]$. Truncating it at high level $L$, we get a continuous process such that $E|X(t)|$ is uniformly bounded but the supremum is identically $L$. Taking a suitable mixture of such truncations, we see that the tails may decay arbitrarily slowly.

There is essentially only one universal method to gets bounds for the supremum from the bounds for the increments, which is to consider $\delta_ k$-nets with diminishing $\delta_ k>0$ and bound the supremum by the convergent series of suprema taken over finite sets (differences between points from 2 successive nets). Clever choice of the nets may be crucial for the success but not in this case. Whatever you can get from the standard dyadic nets here is the best you can say.

I'm also tempted to ask whether you, indeed, need the estimate for the supremum (which cannot be made without extra assumptions) rather than for some $L^p$ norm.

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  • $\begingroup$ Thanks for the answer. I like the example as it is really simple. Do you have any good references of this "delta" technology? I used once the methods used in Billingsley's "Convergence of probability measures" (new edition) which was quite cumbersome. As to your question, I am not sure if I understand. Do you mean estimation $L^p$ norm of supremum i.e. E|Y|^p? If yes, then this would be enough for me. $\endgroup$
    – Piotr Miłoś
    Commented Nov 3, 2009 at 16:05
  • $\begingroup$ No, I mean the estimates for $Z_ p=\int_ 0^1 |X(t)-m|^p\,dt$ where $m$ is, say, the median of $X(t)$ on $[0,1]$ (we have to normalize somehow to avoid irrelevant big constant summands in $X(t)$). $\endgroup$
    – fedja
    Commented Nov 3, 2009 at 17:26
  • $\begingroup$ I would like to establish tightness of a family of processes in so-called Jakubowski S1 topology (weaker then standard sup toplogy). The first condition is to have tightness of the suprema of proceeses ... $\endgroup$
    – Piotr Miłoś
    Commented Nov 16, 2009 at 13:10

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