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Notation and Conventions

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Part of the Springer Series in Operations Research and Financial Engineering book series (ORFE)

Abstract

Vectors are denoted by bold letters, uppercase for random vectors and lowercase for nonrandom vectors. For example, x = (x(1), ...., x(d)) ∈ ℝd. Operations between vectors should be interpreted componentwise, so that for two vectors x and z,
$$ \begin{gathered} x < z means x^{(i)} < z^{(i)} , i = 1,...,d, x \leqslant z means x^{(i)} \leqslant z^{(i)} , i = 1,...,d, \hfill \\ x = z means x^{(i)} = z^{(i)} , i = 1,...,d, zx = (z^{(1)} x^{(1)} ,...,z^{(d)} x^{(d)} ), \hfill \\ x \vee z = (x^{(1)} \vee z^{(1)} ,...,x^{(d)} \vee z^{(d)} ), \frac{x} {z} = \left( {\frac{{x^{(1)} }} {{z^{(1)} }},...,\frac{{x^{(d)} }} {{z^{(d)} }}} \right), \hfill \\ \end{gathered} $$
and so on. Also, define
$$ \begin{gathered} 0 = (0,...,0), 1 = (1,...,1), \hfill \\ e_i = (0,...,1,...,0), e_i^{ - 1} = (\infty ,...,1,...,\infty ), \hfill \\ \end{gathered} $$
where in ei and e i −1 , the “1” occurs in the ith spot. For a real number c, write cx = (cx(1), ..., cx(d)), as usual. We denote the rectangles (or the higher-dimensional intervals) by
$$ \left[ {a,b} \right] = \left\{ {x \in \mathbb{R}^d :a \leqslant x \leqslant b} \right\}. $$
Higher-dimensional rectangles with one or both endpoints open are defined analogously, for example,
$$ \left( {a,b]} \right. = \left\{ {x \in \mathbb{R}^d :a \leqslant x \leqslant b} \right\}. $$

Keywords

Probability Measure Random Vector Radon Measure Nondecreasing Function Quantile Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2007

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