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Notation

We spend a tremendous amount of time on notation and conventions, to make them both intuitive for readers from different disciplines and consistent across a wide set of topics. In the short Section 1 here below we point out a few key tenets that appear throughout all the present work. For the full list of notations, keep reading further below.

Key notation tenets

Operators and special functions

Iverson brackets [W]

indicator function of a set [W], notation consistent with Iverson brackets

Dirac delta function (point mass) concentrated at [W]

canonical basis

Fourier transform (Table 16.3)

inverse Fourier transform (Table 16.3)

integration operator applied times (23.41)

inverse-call transformation with parameter (E.1.29)

positive part function

error function [W]

inverse error function [W]

change (difference, increment)

lag operator of order

exponentially weighted moving average at time with half-life and trailing window of size

operator that extracts the main diagonal of an matrix , i.e.

operator that maps an vector into an matrix which has on the main diagonal and null entries elsewhere, i.e.

absolute value of the scalar

determinant of the square matrix

cardinality of the set

floor function [W]

ceiling function [W]

trace of the square matrix , i.e.

operator that maps an arbitrary matrix into a vector, stacking the columns of , i.e.

pseudo-inverse (14.466) of the matrix

conjugate transpose (14.131) of the complex-valued matrix

complex conjugate of the complex number

Euclidean (vector) norm

Frobenius (matrix) norm

element-wise division (if one operand is a scalar and the other is not, the scalar expands into an array of the same size as the other operand)

Hadamard product (14.460)

Kronecker product (14.462)

Kronecker sum [W]

commutation matrix (14.474)

convolution (16.108)

discrete convolution (16.111)

sorting function which allows to obtain from a generic set the set of sorted elements , where and is a permutation of the indexes such that

directly proportional to [W]

Sets

ellipsoid with center , shape and radius (when the subscript is dropped)

-dimensional unit sphere

filled ellipsoid with center , shape and radius (when the subscript is dropped)

-dimensional filled unit ball

unit simplex in

-dimensional unit hypercube

Calculus

multivariate function

gradient

Hessian

Jacobian

See Chapter 15.1.3 for details.

Probability and general distribution theory

real world probability

risk neutral probability

probability density function (pdf) of

cumulative distribution function (cdf) of

characteristic function of

equality in distribution (i.e. )

(or )

conditional pdf of

univariate standard normal pdf

univariate standard normal cdf

multivariate standard normal pdf (correlation matrix: )

multivariate standard normal cdf (correlation matrix: )

univariate standard Student t cdf (degrees of freedom: )

pdf of an elliptical random variable , including normal (), Student () and Cauchy ()

characteristic function of an elliptical random variable , including normal (), Student () and Cauchy ()

information (generator)

prior distribution (pdf)

posterior distribution (pdf)

copula of represented by the pdf of the grades