What does $\tilde{O}_P(N^\alpha)$ mean?

Question

What does $\tilde{O}_P(N^\alpha)$ mean? It appears in an estimation error mention in this paper, in the second paragraph on page 3.

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What does big O subscript P mean in a probability context?

Answer 1

Pulled from: http://www2.math.uu.se/~svante/papers/sjN6.pdf (definition D5).

A probabilistic version of $\mathcal{O}$ that is frequently used is the following:

$X_n = O_\text{P}(\alpha_n)$ if for every $\epsilon > 0$ there exists constants $C_{\epsilon}$ and $n_{\epsilon}$ such that $\mathbb{P}(|X_n| \leq C_{\epsilon}\alpha_n) > 1 − \epsilon$ for every $n \geq n_{\epsilon}$. In other words, $X_n/\alpha_n$ is bounded, up to an exceptional event of arbitrarily small (but fixed) positive probability. This is also known as $X_n/\alpha_n$ being bounded in probability.

So combining this and D.W.'s answer we get:

$X_n \in \tilde{O}_{\text{P}}(\alpha_n)$ means for every $\epsilon > 0$ there exists constants $C_\epsilon$ and $n_{\epsilon}$ such that $\mathbb{P}(|X_n| \leq C_{\epsilon}\alpha_n n^{\gamma}) > 1 - \epsilon$ for every $n \geq n_{\epsilon}$ and for all $\gamma > 0$.

An looser terms $X_n \in \tilde{O}_{\text{P}}(\alpha_n)$ means $X_n$ is bounded by $\alpha_n$ within a polynomial factor with high probability.

Answer 2

It depends on context; often, saying $f(n) \in \tilde{O}(g(n))$ means $f(n) \in O(g(n) n^{\epsilon})$ for all $\epsilon>0$. For example, $n^2 \log n \in \tilde{O}(n^2)$.