exercise:8dbb0c4f40: Difference between revisions
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Let <math>d\geqslant3</math> and <math>n</math> be such that <math>2\ln(n)\leqslant d</math> holds. Show that | |||
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holds, when <math>x^{\scriptscriptstyle(1)},\dots,x^{\scriptscriptstyle(n)}</math> are drawn uniformly at random from the <math>d</math>--dimensional unit ball. | holds, when <math>x^{\scriptscriptstyle(1)},\dots,x^{\scriptscriptstyle(n)}</math> are drawn uniformly at random from the <math>d</math>--dimensional unit ball. | ||
''Hint:'' Use that <math>1\leqslant d^2/(d-2\ln n)^2</math> holds and apply then the Theorem of Total Probability. | ''Hint:'' Use that <math>1\leqslant d^2/(d-2\ln n)^2</math> holds and apply then the Theorem of Total Probability. | ||
Revision as of 01:48, 1 June 2024
[math]
\newcommand{\smallfrac}[2]{\frac{#1}{#2}}
\newcommand{\medfrac}[2]{\frac{#1}{#2}}
\newcommand{\textfrac}[2]{\frac{#1}{#2}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\e}{\operatorname{e}}
\newcommand{\B}{\operatorname{B}}
\newcommand{\Bbar}{\overline{\operatorname{B}}}
\newcommand{\pr}{\operatorname{pr}}
\newcommand{\dd}{\operatorname{d}\hspace{-1pt}}
\newcommand{\E}{\operatorname{E}}
\newcommand{\V}{\operatorname{V}}
\newcommand{\Cov}{\operatorname{Cov}}
\newcommand{\Bigsum}[2]{\mathop{\textstyle\sum}_{#1}^{#2}}
\newcommand{\ran}{\operatorname{ran}}
\newcommand{\card}{\#}
\newcommand{\mathds}{\mathbb}[/math]
Let [math]d\geqslant3[/math] and [math]n[/math] be such that [math]2\ln(n)\leqslant d[/math] holds. Show that
[[math]]
\P\bigl[\bigl|\bigl\langle{}\medfrac{x^{\scriptscriptstyle(j)}}{\|x^{\scriptscriptstyle(j)}\|},\medfrac{x^{\scriptscriptstyle(k)}}{\|x^{\scriptscriptstyle(k)}\|}\bigr\rangle{}\bigr|\leqslant\medfrac{\sqrt{6\ln n}}{\sqrt{d-1}} \text{ for all }j\not=k\bigr]\geqslant 1-\medfrac{1}{n}
[[/math]]
holds, when [math]x^{\scriptscriptstyle(1)},\dots,x^{\scriptscriptstyle(n)}[/math] are drawn uniformly at random from the [math]d[/math]--dimensional unit ball.
Hint: Use that [math]1\leqslant d^2/(d-2\ln n)^2[/math] holds and apply then the Theorem of Total Probability.