Let $P$ be a probability distribution over $\mathbb{R}^d$. Note that we DO NOT assume that $P$ is continuous w.r.t. Lebesgue measure. For example, P can be a multinomial distribution over a finite and discrete set of points.
Let $D = \{x_1,...,x_n\}\sim P$ be n i.i.d. data sampled from $P$. Can we upperbound $ \mathbb{P}_{x\sim P}\{x \notin span(D)\}? $
In other words, given n points and their linear span $span(D) = \left\{\sum_{i=1}^n a_i x_i| a_i\in\mathbb{R}, \forall i\right\}$, I want to bound the probability that a new point falls out of $span(D)$, as a function of $n$.