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Let $mathcal{H}$ be a separable Hilbert space. Let $v$ be a random variable taking values in $mathcal{H}$ such that $P(v perp h) < 1$ for all $h in mathcal{H}.$ Suppose we sample an infinite sequence $v_1, v_2, ldots.$ Is it the case that, almost surely, the closed span of $v_1, v_2, ldots$ is all of $mathcal{H}?$

(This may turn out to be a simplified version of J. E. Pascoe's answer).

The support of (the distribution of) $v$, that we denote by $operatorname{supp} v$, is the set of vectors $h in mathcal{H}$ such that $P(v in B(h, varepsilon)) > 0$ for every $varepsilon > 0$. We list some properties of this set.

1. The set $operatorname{supp} v$ is the complement of the union of all open sets $B$ such that $P(v in B) = 0$. Thus, the support is a closed set.




2. Since $mathcal{H}$ is a separable metric space, it has a countable topological base $mathcal{B}$, and $operatorname{supp} v$ is the complement of the union of all $B in mathcal{B}$ such that $P(v in B) = 0$. By countable additivity, it follows that $P(v in operatorname{supp} v) = 1$ (the support is a set of full measure).




3. With probability one, the closure of the random set $V = {v_1, v_2, ldots}$ contains $operatorname{supp} v$. Indeed, let ${h_1, h_2, ldots}$ be a countable, dense subset of $operatorname{supp} v$. For every $i, n = 1, 2, ldots$ we have $P(v in B(h_i, tfrac{1}{n})) > 0$, and thus, by Borel–Cantelli, $P(V cap B(h_i, tfrac{1}{n}) = varnothing) = 0$. It follows that $h_i in overline{V}$ for every $i = 1, 2, ldots$, and consequently $operatorname{supp} v subseteq overline{V}$.




4. For every $h in mathcal{H}$, we have $P(h perp v) < 1$, and therefore $h$ is not orthogonal to $operatorname{supp} v$. It follows that the closed span of $operatorname{supp} v$ is $mathcal{H}$.

It remains to note that the closed span of $V$ is the same as the closed span of the closure of $V$, which with probability one contains the closed span of $operatorname{supp} v$, which we have shown to be equal to $mathcal{H}$.

(Item 1 is valid for any topological space; items 2 and 3 work in an arbitrary separable metric space.)

Another Try

We say a $mathcal{H}$-valued random variable $h$ is a random vector if $P(h perp g)<1$ for all $gin mathcal{H}.$

If $h_1, h_2, ldots$ is a sequence independent identically distributed of random vectors,
then, almost surely, the closed span of the $h_i$ is equal to $mathcal{H}.$

First we will need a lemma.

Lemma 1
Let $h$ be a random vector.
There is a countable subset $A$ of $mathcal{H}$ such that the closed span of the elements of $A$ is equal to $mathcal{H}$
and for every point $ain A,$ $P(hin U)>0$ for any neighborhood $U$ of $a.$

Proof
For any subset $A$ such that for every point $ain A,$ $P(hin U)>0$ for any neighborhood $U$ of $a,$ and
the closed span of the elements of $A$ is not equal to $mathcal{H},$
we will show that we can grow $A$ by a single element which is not in closed span of the elements of $A.$
We can only do this a countable number of times because the Hilbert space dimension of $mathcal{H}$ is countable.
(Otherwise, via Gram-Schmidt, we could construct an uncountable orthonormal set by transfinite induction.)

Choose $g$ such that $g perp a$ for all $ain A.$ Now, $P(h perp g)<1.$ So there must be a point $b$ such that
$P(hin U) >0$ for every neighborhood of $b$ and $b$ is not perpendicular to $g,$ therefore, $b$ is not in the span of the elements of $A.$ QED

Suppose $h_1, h_2, ldots$ is a sequence independent identically distributed of random vectors.
Let $A$ be as in Lemma 1. Index $A$ a a sequence $a_n.$
Let $B_{m,n}$ be a ball of radius $1/m$ centered at $a_n$
Almost surely, the sequence $h_i$ must visit $B_{m,n}$ infinitely often,
as $P(h_iin B_{m,n})>0$. Therefore $A$ is a subset of the closure of the values of the sequence. (We have essentially the fact that a random function $f:mathbb{N}rightarrow mathbb{N}^2$ is surjective with infinite multiplicity.)