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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. 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? For simplicity, I only need to study such problems where the random variables $\{ X_s \}$ are integer valued and where $S$ is a finite set (though with the $\{ X_s \}$ depending on one another in some way that determines a kind of geometry on $S$ as in certain chaining arguments).

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?

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. 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? For simplicity, I only need to study such problems where the random variables $\{ X_s \}$ are integer valued and where $S$ is a finite set (though with the $\{ X_s \}$ depending on one another in some way that determines a kind of geometry on $S$ as in certain chaining type approaches).

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. 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? For simplicity, I only need to study such problems where the random variables $\{ X_s \}$ are integer valued and where $S$ is a finite set (though with the $\{ X_s \}$ depending on one another in some way that determines a kind of geometry on $S$ as in certain chaining arguments).

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. 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?

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. 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? For simplicity, I only need to study such problems where the random variables $\{ X_s \}$ are integer valued and where $S$ is a finite set (though with the $\{ X_s \}$ depending on one another in some way that determines a kind of geometry on $S$ as in certain chaining type approaches).