proofs of stochastic boundedness

I'm looking at some statistical literature and trying to compare the results given there in probabilistic big-Oh notation with statements I'm more familiar with.

In particular, I'm trying to interpret statements of the form $$\|\Sigma_{n(p)} - \Sigma \| = O_P\left( \frac{\log p}{n(p)}\right).$$ As far as i can tell from the rather terse wikipedia page on $o_p$ notation, this means that there is some constant $C$ that is independent of $p$ for which $$\lim_{p \rightarrow \infty} \mathbb{P}\left( \|\Sigma_{n(p)} - \Sigma \| > C \frac{\log p}{n(p)}\right) = 0.$$

Is that correct?

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1 Answer

The problem is that there are no universally agreed upon notations. Maybe this will help.

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Thanks for the reference. I think I'll also work through a couple of the proofs because I'm interested to see whether they prove their results by getting explicit tail bounds that hold for finite p (and then just state the theorem in a weaker way because of convention) or by using arguments that only hold asymptotically. – Alex Gittens Nov 3 '11 at 14:19