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YCor
YCor
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Anti-Concentration Inequalitiesconcentration inequalities: "Lower-Boundlower bound on Realizedrealized second moment"moment

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

Anti-Concentration Inequalities: "Lower-Bound on Realized second moment"

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

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 $\lambda>0$.

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ABIM
ABIM
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Concentration Anti-Concentration Inequalities: Lower bound"Lower-Bound on Realized second moment"

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: $$ \mathbb{P}(\|X\|>\lambda) \geq \mbox{"insert non-trivial lower-bound"} $$$$ \mathbb{P}(\|X\|^2>\lambda) \geq \mbox{"insert non-trivial lower-bound"} $$ where $\lambda>0$.

Concentration Inequalities: Lower bound

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: $$ \mathbb{P}(\|X\|>\lambda) \geq \mbox{"insert non-trivial lower-bound"} $$ where $\lambda>0$.

Anti-Concentration Inequalities: "Lower-Bound on Realized second moment"

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 $\lambda>0$.

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ABIM
ABIM
  • 5.4k
  • 3
  • 19
  • 41

Concentration Inequalities: Lower bound

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: $$ \mathbb{P}(\|X\|>\lambda) \geq \mbox{"insert non-trivial lower-bound"} $$ where $\lambda>0$.