The following definitions are based on the textbooks (Woodward and Crawley) unless otherwise noted.

Autocorrelation (\(\rho\))
  • TBA
Autocovariance (\(\gamma\))
  • covariance within the same time series
Cauchy-Schwarz inequality
Correlation
  • the degree to which two variables are linearly related; see also Pearson correlation coefficient
Covariance
  • the measure of joint variability of two random variables; positive, negative or near zero
Continuous parameter
  • a numeric parameter that can take any value from a specified range of values, for example \(T = (-\infty, \infty)\); compare against a discrete parameter
Deterministic
  • TBA
Discrete parameter
  • a numeric parameter that can take any value from a specified range of values, limited by a minimum permissible distance between each value, for example \(T = \{0, \pm1, \pm2, ...\}\); compare against a continuous parameter
Ensemble
  • the collection of all possible realizations
Ergodic
  • a process whose ensemble averages can be consistently estimated from a single observation (e.g., u(t) doesn’t depend on t)
Gaussian (normal)
  • TBA
Identically distributed
  • a collection of random variables where each random variable has the same probability distribution as the others; if these random variables are also mutually independent, they may be given the shorthand notation IID
Mean (\(\mu\))
  • the expected value (E) or first raw moment of a real-valued random variable (e.g., \(X(t)\)), oftentimes denoted: \(E[X(t)]\)
Observation
  • a real number from a given random variable
Probabilistic
  • TBA
Random variable
  • a variable whose values depend on outcomes of a random process; it can be thought of as a function that is defined in some sample space whose range is the real numbers
Real number (\({\rm I\!R}\))
  • a value from a continuous quantity that can represent a distance along a line (includes all positive and negative rational and irrational numbers)
Realization
  • a set of real-valued outcomes for a random variable at a fixed location in the random variable’s sample space; compare against an ensemble
Stationary
  • a process that is in a state of statistical equilibrium; examples include strict stationarity, covariance stationarity
Stochastic process
  • a collection of indexed random variables from the same sample space
Time series
  • a special type of stochastic process where the sample space represents time
Univariate
  • having or involving a single variable
Variance (\(\sigma^2\))
  • the expected value of the squared deviation from the mean; see also here and here
White noise
  • TBA