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=Ranked poset=
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A [[partially ordered set]] $(X,{\le})$ with a ''rank function'' $r  : X \rightarrow \mathbf{N}$ such that $r(m) = 0$ for some minimal element $m \in X$, and $r(q) = r(p)+1$ when $q$ [[Covering element|covers]] $p$: that is, $p. < q$ and there is no $x$ with $p < x < q$. 
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A ''graded poset'' is one in which every minimal element has rank $0$ and every maximal element has the same rank.
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==References==
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* K. Engel, Sperner Theory, Encyclopedia of Mathematics and its Applications '''65''', Cambridge (1997) ISBN 0-521-45206-6  {{ZBL|0868.05001}}
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=Cage=
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A trivalent graph with given [[girth]] and minimum number of vertices.  A $k$-''cage'' is a cage of girth $k$. 
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==References==
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* W.T. Tutte, "Connectivity in graphs", Mathematical Expositions '''15''', University of Toronto Press (1966) {{ZBL|0146.45603}}
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=Girth=
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One of the [[Graph, numerical characteristics of a|numerical characteristics of a graph]]: the size of the smallest cycle in an undirected [[graph]] which is not a [[forest]].
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==References==
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* W.T. Tutte, "Connectivity in graphs", Mathematical Expositions '''15''', University of Toronto Press (1966) {{ZBL|0146.45603}}
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=Moment generating function=
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= Hoeffding inequality =
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A simple inequality giving a bound on the probability of deviation of a sum of independent random variables from a given value.
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Let $X_1,\ldots,X_n$ be independent [[Subgaussian distribution|sub-Gaussian random variable]]s with means $\mu_i$ and sub-Gaussian parameters $\sigma_i$.  Then
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$$
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\mathsf{P}\left\{{\sum_{i=1}^n (X_i-\mu_i) > t }\right\} < \exp\left({-\frac{t^2}{2\sum_{i=1}^n \sigma_i^2} }\right) \ .
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$$
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In particular, if the $X_i$ are bounded, $|X_i|<b$ then they are sub-Gaussian with parameter $b$ and the
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inequality takes the form
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$$
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\mathsf{P}\left\{{\sum_{i=1}^n (X_i-\mu_i) > t }\right\} < \exp\left({-\frac{t^2}{2 n b^2} }\right) \ .
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$$
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==References==
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* Martin J. Wainwright, "High-dimensional Statistics", Cambridge (2019) ISBN 978-1-108-49802-9
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=Subgaussian distribution=
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A probability distribution with tails comparable to those of the Gaussian distribution.  A random variable $X$ with mean $\mu$ is
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sub-Gaussian if there is a positive constant $\sigma$ such that
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$$
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\mathsf{E}[\exp(\lambda(X-\mu))] \le \exp(\lambda^2\sigma^2/2)
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$$
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for all real $\lambda$.
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Clearly any normal random variable $N(\mu,\sigma^2)$ is sub-Gaussian with parameter $\sigma$.
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A random variable is sub-Gaussian if and only if it has a moment generating function satisfying
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$$
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\mathsf{E}[\exp(\lambda X)] \le \exp(\lambda^2\sigma^2/2)
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$$
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for all real $\lambda$.
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==References==
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* Martin J. Wainwright, "High-dimensional Statistics", Cambridge (2019) ISBN 978-1-108-49802-9
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=Markov inequality in probability theory=
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A simple inequality giving a bound on the probability of deviation of a random variable with finite expectation from a given value.
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Let $X(\omega)$ be a random variable.  Then
 +
$$
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\mathsf{P}\{|X| > \epsilon\} \le \frac{\mathsf{E}X}{\epsilon} \ .
 +
$$
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The inequality follows from comparing the indicator function of the set $\{\omega : |X(\omega)| > \epsilon\}$ with the random variable $X/\epsilon$ and then taking expectations.
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Application of Markov's inequality to the random variable $Y = (X-\mu)^2$ yields the [[Chebyshev inequality in probability theory]].  More generally, if $X$ has a central moment of order $k$ about $\mu$, then
 +
$$
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\mathsf{P}\{|X-\mu| > \epsilon\} \le \frac{\mathsf{E}[|X-\mu|^k]}{\epsilon^k} \ .
 +
$$
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If $X-\mu$ has a [[moment generating function]] in the neighbourhood $[-b,b]$ of zero, then for $0 \le \lambda \le b$
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$$
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\mathsf{P}\{|X-\mu| > \epsilon\} = \mathsf{P}\{\exp(\lambda|X-\mu|) > \exp(\lambda\epsilon)\} \le \frac{\mathsf{E}[\exp(\lambda|X-\mu|)]}{\exp(\lambda\epsilon)}
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$$
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so that we obtain the '''Chernoff inequality'''
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$$
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\log \mathsf{P}\{|X-\mu| > \epsilon\} \le \inf_{0\le\lambda\le b} \log \mathsf{E}[\exp(\lambda|X-\mu|)] -  \lambda\epsilon \ .
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$$
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 +
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==References==
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* Geoffrey Grimmett, David Stirzaker, "Probability and Random Processes" (4th ed), Oxford (2020) ISBN 0-19-884759-9
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* Martin J. Wainwright, "High-dimensional Statistics", Cambridge (2019) ISBN 978-1-1-8-49802-9
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=Steinitz exchange lemma=
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A lemma on [[Linear independence|linearly independent]] sets and [[spanning set]]s in a vector space from which it is possible to deduce that dimension is well-defined.
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Let $X = \{x_1,\ldots,x_m\}$ be a linearly independent set and $Y = \{y_1,\ldots,y_n\}$ a spanning set in a vector space $V$.  Then $m \le n$ and $Y$ can be re-ordered so that $\{x_1,\ldots,x_m,y_{m+1},\ldots,y_n$ is also a spanning set.
 +
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As a corollary, if $X$ and $Y$ are both bases (linearly independent and spanning) for $V$, then $n = m$.  Hence all bases for $V$ have the same size, and this shows that the dimension of $V$ is well-defined.
 +
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==References==
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* P. M. Cohn, "Classic Algebra" Wiley (2000) ISBN 047187731X
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=Hopf constant=
 
=Hopf constant=
  

Revision as of 19:35, 27 February 2021

Ranked poset

A partially ordered set $(X,{\le})$ with a rank function $r : X \rightarrow \mathbf{N}$ such that $r(m) = 0$ for some minimal element $m \in X$, and $r(q) = r(p)+1$ when $q$ covers $p$: that is, $p. < q$ and there is no $x$ with $p < x < q$.

A graded poset is one in which every minimal element has rank $0$ and every maximal element has the same rank.

References

  • K. Engel, Sperner Theory, Encyclopedia of Mathematics and its Applications 65, Cambridge (1997) ISBN 0-521-45206-6 Zbl 0868.05001

Cage

A trivalent graph with given girth and minimum number of vertices. A $k$-cage is a cage of girth $k$.

References

  • W.T. Tutte, "Connectivity in graphs", Mathematical Expositions 15, University of Toronto Press (1966) Zbl 0146.45603

Girth

One of the numerical characteristics of a graph: the size of the smallest cycle in an undirected graph which is not a forest.

References

  • W.T. Tutte, "Connectivity in graphs", Mathematical Expositions 15, University of Toronto Press (1966) Zbl 0146.45603

Moment generating function

Hoeffding inequality

A simple inequality giving a bound on the probability of deviation of a sum of independent random variables from a given value. Let $X_1,\ldots,X_n$ be independent sub-Gaussian random variables with means $\mu_i$ and sub-Gaussian parameters $\sigma_i$. Then $$ \mathsf{P}\left\{{\sum_{i=1}^n (X_i-\mu_i) > t }\right\} < \exp\left({-\frac{t^2}{2\sum_{i=1}^n \sigma_i^2} }\right) \ . $$ In particular, if the $X_i$ are bounded, $|X_i|<b$ then they are sub-Gaussian with parameter $b$ and the inequality takes the form $$ \mathsf{P}\left\{{\sum_{i=1}^n (X_i-\mu_i) > t }\right\} < \exp\left({-\frac{t^2}{2 n b^2} }\right) \ . $$

References

  • Martin J. Wainwright, "High-dimensional Statistics", Cambridge (2019) ISBN 978-1-108-49802-9

Subgaussian distribution

A probability distribution with tails comparable to those of the Gaussian distribution. A random variable $X$ with mean $\mu$ is sub-Gaussian if there is a positive constant $\sigma$ such that $$ \mathsf{E}[\exp(\lambda(X-\mu))] \le \exp(\lambda^2\sigma^2/2) $$ for all real $\lambda$.

Clearly any normal random variable $N(\mu,\sigma^2)$ is sub-Gaussian with parameter $\sigma$.

A random variable is sub-Gaussian if and only if it has a moment generating function satisfying $$ \mathsf{E}[\exp(\lambda X)] \le \exp(\lambda^2\sigma^2/2) $$ for all real $\lambda$.

References

  • Martin J. Wainwright, "High-dimensional Statistics", Cambridge (2019) ISBN 978-1-108-49802-9

Markov inequality in probability theory

A simple inequality giving a bound on the probability of deviation of a random variable with finite expectation from a given value. Let $X(\omega)$ be a random variable. Then $$ \mathsf{P}\{|X| > \epsilon\} \le \frac{\mathsf{E}X}{\epsilon} \ . $$

The inequality follows from comparing the indicator function of the set $\{\omega : |X(\omega)| > \epsilon\}$ with the random variable $X/\epsilon$ and then taking expectations.

Application of Markov's inequality to the random variable $Y = (X-\mu)^2$ yields the Chebyshev inequality in probability theory. More generally, if $X$ has a central moment of order $k$ about $\mu$, then $$ \mathsf{P}\{|X-\mu| > \epsilon\} \le \frac{\mathsf{E}[|X-\mu|^k]}{\epsilon^k} \ . $$

If $X-\mu$ has a moment generating function in the neighbourhood $[-b,b]$ of zero, then for $0 \le \lambda \le b$ $$ \mathsf{P}\{|X-\mu| > \epsilon\} = \mathsf{P}\{\exp(\lambda|X-\mu|) > \exp(\lambda\epsilon)\} \le \frac{\mathsf{E}[\exp(\lambda|X-\mu|)]}{\exp(\lambda\epsilon)} $$ so that we obtain the Chernoff inequality $$ \log \mathsf{P}\{|X-\mu| > \epsilon\} \le \inf_{0\le\lambda\le b} \log \mathsf{E}[\exp(\lambda|X-\mu|)] - \lambda\epsilon \ . $$


References

  • Geoffrey Grimmett, David Stirzaker, "Probability and Random Processes" (4th ed), Oxford (2020) ISBN 0-19-884759-9
  • Martin J. Wainwright, "High-dimensional Statistics", Cambridge (2019) ISBN 978-1-1-8-49802-9


Steinitz exchange lemma

A lemma on linearly independent sets and spanning sets in a vector space from which it is possible to deduce that dimension is well-defined.

Let $X = \{x_1,\ldots,x_m\}$ be a linearly independent set and $Y = \{y_1,\ldots,y_n\}$ a spanning set in a vector space $V$. Then $m \le n$ and $Y$ can be re-ordered so that $\{x_1,\ldots,x_m,y_{m+1},\ldots,y_n$ is also a spanning set.

As a corollary, if $X$ and $Y$ are both bases (linearly independent and spanning) for $V$, then $n = m$. Hence all bases for $V$ have the same size, and this shows that the dimension of $V$ is well-defined.

References

  • P. M. Cohn, "Classic Algebra" Wiley (2000) ISBN 047187731X

Hopf constant

A constant $q_\infty$ associated with the Milne problem in radiative transfer theory. It may be defined as $$ q_\infty = \frac{1}{\pi} \int_0^{\pi/2} \left({\frac{3}{\sin^2\theta} - \frac{1}{1-\theta\cot\theta}} \right) d\theta \ . $$ Its numerical value is $$ q_\infty = 0.7104460895 \dots\ . $$

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press (2003) ISBN 0-521-81805-2 Zbl 1054.00001
How to Cite This Entry:
Richard Pinch/sandbox-15. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-15&oldid=51650