Suppose for a statistic $X$ of a size-n sample, $E|X-EX|\le f(n)$ for some decreasing function $f(n)$. Can we say $X$ concentrates to its mean at a rate of $f(n)$?
I understand concentration rate is often defined by how the tail probability concentrates. But in this mean concentration case, can we still call $f(n)$ the concentration rate? If not, what is the proper way to call this? Thanks.