Notation

General notation  
:= ‘is defined as’
x+, x− max{x, 0}, max{ − x, 0} (or min{x, 0} in Chapter 10)
Sets and spaces  
image, image, image, image positive integers, integers, rational numbers, real numbers
image+ positive real line
imaged d-dimensional Euclidean space
Dc complement of the set D ⊂ imaged
[a,b) half-closed interval
Matrices  
Id d-dimensional Identity matrix
imagep,q(image) the set of p × q real matrices
Processes  
iid independent and Identically distributed
iid (0,1) iid centered with unit variance
(Xt) or (Xt)timageimage discrete-time process
(imaget) GARCH process
image conditional variance or volatility
(χt) strong white noise with unit variance
κχ kurtosis coefficient of ηt
L or B lag operator
image sigma-field generated by the past of Xt
Functions  
double-struck-1A (x) 1 if x image A, 0 otherwise
[x] integer part of x
γx,ρx autocovariance and autocorrelation functions of (Xt)
imagex.imagex sample autocovariance and autocorrelation
Probability  
image(m, Σ) Gaussian law with mean m and covariance matrix Σ
image chi-square distribution with d degrees of freedom
image (α) quantile of order α of the image distribution
image convergence in distribution
a.s. almost surely
υn= op(un) υn/un → 0 in probability
image a equals b up to the stochastic order op(l)
Estimation  
image Fisher information matrix
(κη − 1)J− 1 asymptotic variance of the QML
θ0 true parameter value
Θ parameter set
θ element of the parameter set
image estimators of θ0
image = image (θ) volatility built with the value θ
image as image but with initial values
2113_fmt = 2113_fmtt − 2 log (conditional variance of imaget)
image approximation of 2113_fmtt, built with initial values
VARas Coyas asymptotic variance and covariance
Some abbreviations  
ES expected shortfall
FGLS feasible generalized least squares
OLS ordinary least squares
QML quasi-maximum likelihood
RMSE root mean square error
SACR sample autocorrelation
SACV sample autocovariance
SPAC sample partial autocorrelation
VaR value at risk
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