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Chen
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Consider a random vector X with the coordinate distribution is uniformly distributed in the set $\{\sqrt{n}e_i : i = 1,..., n\}$, where $e_i$ denotes the n-element set of the canonical basis vectors in $R^n$. Show that $ \parallel X \parallel_{ \psi 2}\asymp \sqrt{\frac{n}{{ log n}}}.$

By the definition of the sub-gaussian norm of a random vector, $\parallel X\parallel_{ \psi 2}=\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2} $$\parallel X\parallel_{ \psi_2}=\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2} $, for all $x ∈ R^n$.

I tried to consider $\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2}=\sup_{x∈S^{n−1}} \parallel \sum_{i=1}^n x_iX_i\parallel_{\psi 2}$$\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2}=\sup_{x∈S^{n−1}} \parallel \sum_{i=1}^n x_iX_i\parallel_{\psi_2}$, while I am not sure how to precede next? Thank you!

Consider a random vector X with the coordinate distribution is uniformly distributed in the set $\{\sqrt{n}e_i : i = 1,..., n\}$, where $e_i$ denotes the n-element set of the canonical basis vectors in $R^n$. Show that $ \parallel X \parallel_{ \psi 2}\asymp \sqrt{\frac{n}{{ log n}}}.$

By the definition of the sub-gaussian norm of a random vector, $\parallel X\parallel_{ \psi 2}=\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2} $, for all $x ∈ R^n$.

I tried to consider $\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2}=\sup_{x∈S^{n−1}} \parallel \sum_{i=1}^n x_iX_i\parallel_{\psi 2}$, while I am not sure how to precede next? Thank you!

Consider a random vector X with the coordinate distribution is uniformly distributed in the set $\{\sqrt{n}e_i : i = 1,..., n\}$, where $e_i$ denotes the n-element set of the canonical basis vectors in $R^n$. Show that $ \parallel X \parallel_{ \psi 2}\asymp \sqrt{\frac{n}{{ log n}}}.$

By the definition of the sub-gaussian norm of a random vector, $\parallel X\parallel_{ \psi_2}=\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2} $, for all $x ∈ R^n$.

I tried to consider $\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2}=\sup_{x∈S^{n−1}} \parallel \sum_{i=1}^n x_iX_i\parallel_{\psi_2}$, while I am not sure how to precede next? Thank you!

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Chen
Chen
  • 23
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Consider a random vector X with the coordinate distribution is uniformly distributed in the set $\{\sqrt{n}e_i : i = 1,..., n\}$, where $e_i$ denotes the n-element set of the canonical basis vectors in $R^n$. Show that $ \parallel X \parallel_{ \psi 2}\asymp \sqrt{\frac{n}{{ log n}}}.$

By the definition of the sub-gaussian norm of a random vector, $\parallel X\parallel_{ \psi 2}=\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2} $, for all $x ∈ R^n$.

I tried to consider $\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2}= \parallel \sum_{i=1}^n x_iX_i\parallel_{\psi 2}$$\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2}=\sup_{x∈S^{n−1}} \parallel \sum_{i=1}^n x_iX_i\parallel_{\psi 2}$, while I am not sure how to precede next? Thank you!

Consider a random vector X with the coordinate distribution is uniformly distributed in the set $\{\sqrt{n}e_i : i = 1,..., n\}$, where $e_i$ denotes the n-element set of the canonical basis vectors in $R^n$. Show that $ \parallel X \parallel_{ \psi 2}\asymp \sqrt{\frac{n}{{ log n}}}.$

By the definition of the sub-gaussian norm of a random vector, $\parallel X\parallel_{ \psi 2}=\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2} $, for all $x ∈ R^n$.

I tried to consider $\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2}= \parallel \sum_{i=1}^n x_iX_i\parallel_{\psi 2}$, while I am not sure how to precede next? Thank you!

Consider a random vector X with the coordinate distribution is uniformly distributed in the set $\{\sqrt{n}e_i : i = 1,..., n\}$, where $e_i$ denotes the n-element set of the canonical basis vectors in $R^n$. Show that $ \parallel X \parallel_{ \psi 2}\asymp \sqrt{\frac{n}{{ log n}}}.$

By the definition of the sub-gaussian norm of a random vector, $\parallel X\parallel_{ \psi 2}=\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2} $, for all $x ∈ R^n$.

I tried to consider $\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2}=\sup_{x∈S^{n−1}} \parallel \sum_{i=1}^n x_iX_i\parallel_{\psi 2}$, while I am not sure how to precede next? Thank you!

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Chen
Chen
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Show the coordinate distribution has a very large sub-gaussian norm

Consider a random vector X with the coordinate distribution is uniformly distributed in the set $\{\sqrt{n}e_i : i = 1,..., n\}$, where $e_i$ denotes the n-element set of the canonical basis vectors in $R^n$. Show that $ \parallel X \parallel_{ \psi 2}\asymp \sqrt{\frac{n}{{ log n}}}.$

By the definition of the sub-gaussian norm of a random vector, $\parallel X\parallel_{ \psi 2}=\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2} $, for all $x ∈ R^n$.

I tried to consider $\sup_{x∈S^{n−1}}\parallel <X, x>\parallel_{\psi 2}= \parallel \sum_{i=1}^n x_iX_i\parallel_{\psi 2}$, while I am not sure how to precede next? Thank you!