Bounding $l^0$ norm of random quantity

Question

There are many techniques in high dimensional probability for bounding quantities of the form

$$ \mathbf{E}( \sup_{s \in S} X_s ) $$

where $\{ X_s \}$ are a family of random variables which are not independent. In my research, I have run into a problem in the complete opposite direction i.e. bounding quantities of the form

$$ \mathbf{E}( \# \{ s \in S : X_s \neq 0 \}| ). $$

If we view the first equation as a bound on an $l^\infty$ norm of the family $\{ X_s \}$, then the second equation can be viewed as a bound on the ''$l^0$ norm'' of the family $\{ X_s \}$. What kind of techniques are there to bound quantities of this form? Is there perhaps a kind of `duality' result that enables us to study one result in terms of the other?

Answer 1

Not so much in terms of high dimensional probability (though this is definitely a high dimensional probability question). I suspect however that you meant a slightly different question, which I answer below.

If $S$ is a subset of $R$ and $X$ is say a Gaussian process then the Kac-Rice formula allows you to compute $E\{\#s: X_s=0\})$ (the number with $\neq 0$ will be typically infinity if the marginal has a density). Dito if $S$ is a subset of $R^k$ and $X$ is a $k$-dimensional vector. Kac-Rice is not limited to the Gaussian setup, but the latter simplifies the computation.

Note that in the one dimensional case, if $X$ is very irregular (e.g., Brownian motion) then the expectation above is $\infty$.

If you really meant what you asked, then in case $P(X_s=0)<1$ and $S$ is one dimensional of positive Lebesgue measure, the answer is $\infty$ by Fubini.