what does small value of some order mean?

Question

I came across the expression that in an asymptotic sense $\lambda$ will be small, it will be of order $\sqrt{\frac{log p }{n}}$.

Does it mean $\lambda = O(\sqrt{\frac{log p}{n}})$ or something else?

Answer 1

The definition for $f\sim g$, i.e., $f$ is asymptotic to $g$ is as follows: $$g(n)\sim f(n) \iff \lim _{n\to \infty }{\frac {g(n)}{f(n)}}\,=\,1.$$ Then we have $$ g(n)\sim f(n)\iff (g(n)-f(n))\in o(f(n)).$$ So $\lambda(n)\sim \frac{\log(p)}{n}$ would mean that $$ \lim _{n\to \infty }{\frac {\lambda(n)}{\sqrt{\frac{\log(p)}{n}}}}\,=\,1. $$