Order Statistics and High Dimension Geometry
$begingroup$
Suppose that I have iid random variables $mathrm U_n sim mathrm U(0,1)$. Then, for $mathrm Y_m$ defined as,
$$mathrm Y_m = min_{n in [1,m]} mathrm U_n$$
it is easy to compute $mathbb{E}[mathrm Y_m] = frac1{m+1}$.
Now consider a compact convex region $mathcal{C}$ in higher dimensions $mathbb{R}^k$ and let $mathbf p_n$ be fixed points in $mathcal C$ where $mathbf p_n$ may lie on the boundary of $mathcal C$. Let $mathrm D_n$ denote iid random variables in $mathcal C$ such that $mathbb P_{mathrm D_n}(S) geq kappa frac{mathrm{vol}(S)}{mathrm{vol}(mathcal C)}~forall Ssubseteq mathcal C$ for some
fixed $kappa < 1$. Thus $mathrm D_n$ denotes a "skewed" uniform distribution of points in $mathcal C$.
Suppose that $mathrm U_n$ now corresponds to the distance of $mathrm D_n$ from $mathbf p_n$ ie. $mathrm U_n triangleq ||mathrm D_n - mathbf p_n||$. Can someone point me in the right direction to find upper bounds on $mathbb E[mathrm Y_n]$ in such cases ( or direct me to references ) ?
probability euclidean-geometry order-statistics expected-value
asked Dec 18 '18 at 14:11
UnadulteratedImaginationUnadulteratedImagination
422213
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add a comment |
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Suppose that I have iid random variables $mathrm U_n sim mathrm U(0,1)$. Then, for $mathrm Y_m$ defined as,
$$mathrm Y_m = min_{n in [1,m]} mathrm U_n$$
it is easy to compute $mathbb{E}[mathrm Y_m] = frac1{m+1}$.
Now consider a compact convex region $mathcal{C}$ in higher dimensions $mathbb{R}^k$ and let $mathbf p_n$ be fixed points in $mathcal C$ where $mathbf p_n$ may lie on the boundary of $mathcal C$. Let $mathrm D_n$ denote iid random variables in $mathcal C$ such that $mathbb P_{mathrm D_n}(S) geq kappa frac{mathrm{vol}(S)}{mathrm{vol}(mathcal C)}~forall Ssubseteq mathcal C$ for some
fixed $kappa < 1$. Thus $mathrm D_n$ denotes a "skewed" uniform distribution of points in $mathcal C$.
Suppose that $mathrm U_n$ now corresponds to the distance of $mathrm D_n$ from $mathbf p_n$ ie. $mathrm U_n triangleq ||mathrm D_n - mathbf p_n||$. Can someone point me in the right direction to find upper bounds on $mathbb E[mathrm Y_n]$ in such cases ( or direct me to references ) ?
probability euclidean-geometry order-statistics expected-value
asked Dec 18 '18 at 14:11
UnadulteratedImaginationUnadulteratedImagination
422213
$endgroup$
add a comment |
$begingroup$
Suppose that I have iid random variables $mathrm U_n sim mathrm U(0,1)$. Then, for $mathrm Y_m$ defined as,
$$mathrm Y_m = min_{n in [1,m]} mathrm U_n$$
it is easy to compute $mathbb{E}[mathrm Y_m] = frac1{m+1}$.
Now consider a compact convex region $mathcal{C}$ in higher dimensions $mathbb{R}^k$ and let $mathbf p_n$ be fixed points in $mathcal C$ where $mathbf p_n$ may lie on the boundary of $mathcal C$. Let $mathrm D_n$ denote iid random variables in $mathcal C$ such that $mathbb P_{mathrm D_n}(S) geq kappa frac{mathrm{vol}(S)}{mathrm{vol}(mathcal C)}~forall Ssubseteq mathcal C$ for some
fixed $kappa < 1$. Thus $mathrm D_n$ denotes a "skewed" uniform distribution of points in $mathcal C$.
Suppose that $mathrm U_n$ now corresponds to the distance of $mathrm D_n$ from $mathbf p_n$ ie. $mathrm U_n triangleq ||mathrm D_n - mathbf p_n||$. Can someone point me in the right direction to find upper bounds on $mathbb E[mathrm Y_n]$ in such cases ( or direct me to references ) ?
probability euclidean-geometry order-statistics expected-value
asked Dec 18 '18 at 14:11
UnadulteratedImaginationUnadulteratedImagination
422213
$endgroup$
Suppose that I have iid random variables $mathrm U_n sim mathrm U(0,1)$. Then, for $mathrm Y_m$ defined as,
$$mathrm Y_m = min_{n in [1,m]} mathrm U_n$$
it is easy to compute $mathbb{E}[mathrm Y_m] = frac1{m+1}$.
Now consider a compact convex region $mathcal{C}$ in higher dimensions $mathbb{R}^k$ and let $mathbf p_n$ be fixed points in $mathcal C$ where $mathbf p_n$ may lie on the boundary of $mathcal C$. Let $mathrm D_n$ denote iid random variables in $mathcal C$ such that $mathbb P_{mathrm D_n}(S) geq kappa frac{mathrm{vol}(S)}{mathrm{vol}(mathcal C)}~forall Ssubseteq mathcal C$ for some
fixed $kappa < 1$. Thus $mathrm D_n$ denotes a "skewed" uniform distribution of points in $mathcal C$.
Suppose that $mathrm U_n$ now corresponds to the distance of $mathrm D_n$ from $mathbf p_n$ ie. $mathrm U_n triangleq ||mathrm D_n - mathbf p_n||$. Can someone point me in the right direction to find upper bounds on $mathbb E[mathrm Y_n]$ in such cases ( or direct me to references ) ?
probability euclidean-geometry order-statistics expected-value
probability euclidean-geometry order-statistics expected-value
asked Dec 18 '18 at 14:11
UnadulteratedImaginationUnadulteratedImagination
422213
asked Dec 18 '18 at 14:11
UnadulteratedImaginationUnadulteratedImagination
422213
asked Dec 18 '18 at 14:11
UnadulteratedImaginationUnadulteratedImagination
422213
asked Dec 18 '18 at 14:11
UnadulteratedImaginationUnadulteratedImagination
422213
asked Dec 18 '18 at 14:11
UnadulteratedImaginationUnadulteratedImagination
422213
422213
add a comment |
add a comment |
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