Let $X:(\Omega,\Sigma)\rightarrow (\mathbb{R}^d;\mathbb{B}(\mathbb{R}^d))$ be a Borel-measurable random vector. What are some general classes of such random vector for which one can give a "lower concentration inequality" of the form:
$$
\mathbb{P}(\|X\|^2>\lambda) \geq \mbox{(insert non-trivial lower bound)}
$$
where $\lambda>0$.



Anti-concentration inequalities: lower bound on realized second moment
Suppose e.g. that $X=X\_1+\dots+X\_n$, where the $X\_i$' are independent zero-mean random vectors with $\|X\_i\|\le1$ for all $i$. Then the Hoeffding--Azuma inequality (see e.g. [Wikipedia](https://en.wikipedia.org/wiki/Azuma%27s_inequality#A_general_form_of_Azuma's_inequality)) yields
$$P(\|X\|>u)\ge1- e^{-(E\|X\|-u)^2/(2n)}$$
for $u\le E\|X\|$. 


A number of lower bounds on $P(\|X\|\ge u)$ were obtained by de Acosta A. and Samur J.D. (Infinitely divisible probability measures and the converse Kolmogorov inequality in Banach spaces. Studia Math. 1979. V. 66, 143--160). For instance, a special case, for $p=2$, of their Corollary 3.1 on p. 151 is the following: 
$$P(\|X\|>u)\ge \frac14\,\Big(1-\frac{(u+1)^2+u^2/2}{E\|X\|^2}\Big)$$
for $u>0$, where $X$ is just as above. 


Also, in the proof of their Lemma 2.3, de Acosta and Samur showed that, if the $X\_i$'s are also symmetric, then 
$$P(\|X\|>u)\ge\frac12\,P(\max\_i\|X\_i\|>u)
\ge\frac12\,\Big(1-\exp\Big\{-\sum\_i P(\|X\_i\|>u)\Big\}\Big)$$
for $u>0$. 


