- random-number-generator probability
- Updated Fri, 20 May 2022 20:18:16 GMT

I would like to know what $O(v(n))$ really means in detailed and simple words please. I found it everywhere in the literature I am reviewing but I cannot find what the intuition of it (especially if it means the Big O Notation, then what it has to do with the Probability here; also what is $v$, it is never defined except in another paper as a constant -but maybe not related directly to v in this formula-).

Many Thanks.

But this notation is defined (informally) in the first paper.

The notation $O(\nu(n))$ is used for any function, $f(n)$, that vanishes faster than the inverse of any polynomial, that is for every polynomial, $\mathrm{poly} (n)$, and $n$ large enough, $f(n) \leq 1/\mathrm{poly}(n)$

Therefore, what it means is no probabilistic polynomial time (PPT) algorithm $A$ can guess the next bit at inverse polynomially decreasing error rate.

Given any PPT algorithm $A$ this error probability *decays* super-polynomially, for example at a rate $\exp(-\log^2 n)$ which goes to zero faster than any polynomial.

**Claim:** $\exp(-\log^2 n)$ is less than $n^{-c}$ for any constant $c$ for $n$ large enough.

*Proof:* Look at the reciprocal. $$\exp(\log^2 n) > n^c$$ if and only if
$$\log^2 n > c \log n$$ which will clearly happen as soon as $$\frac{\log^2 n}{\log n}>c$$ i.e., as soon as $\log n>c$.

Note that $\exp(\log^{1+\epsilon} n)$ is superpolynomial for all $\epsilon>0.$

**Edit:** The superpolynomial convergence is essentially what is referred to as *negligibility*.

- +0 – Thanks and +1. So a successful predictor should predict with a probability greater than or equal $1-O((n))$, How is the latter evaluated numerically. That's from where should I derive the $O((n))$ numerically? A simple example of a simple algorithm will be very much appreciated. — Sep 01, 2020 at 09:57
- +0 – Is O(v(n)) the error rate? And how is it derived? Practically from the experiment/simulation? Thanks. — Sep 01, 2020 at 10:38
- +2 – It is implicitly negligibility. Adding this into answer will increase the awareness. — Sep 01, 2020 at 18:17

External links referenced by this document: