In that case, the Fourier coefficient corresponding to the zero-frequency component (essentially, the average of the ) will also have a non-zero expected value ; and the power spectrum will be flat only over the non-zero frequencies. Discrete-time white noise A discrete-time stochastic process is a generalization of random vectors with a finite number of components to infinitely many components. A discrete-time stochastic process is called white noise if its mean does not depend on the time and is equal to zero, i.e. and if the autocorrelation function has a nonzero value only for , i.e. . Continuous-time white noise In order to define the notion of "white noise" in the theory of continuous-time signals, one must replace the concept of a "random vector" by a continuous-time random signal; that is, a random process that generates a function of a real-valued parameter . Such a process is said to be white noise in the strongest sense if the value for any time is a random variable that is statistically independent of its entire history before . A weaker definition requires independence only between the values and at every pair of distinct times and . An even weaker definition requires only that such pairs and be uncorrelated. As in the discrete case, some authors adopt the weaker definition for "white noise", and use the qualifier independent to refer to either of the stronger definitions. Others use weakly white and strongly white to distinguish between them. However, a precise definition of these concepts is not trivial, because some quantities that are finite sums in the finite discrete case must be replaced by integrals that may not converge. Indeed, the set of all possible instances of a signal is no longer a finite-dimensional space , but an infinite-dimensional function space. Moreover, by any definition a white noise signal would have to be essentially discontinuous at every point; therefore even the simplest operations on , like integration over a finite interval, require advanced mathematical machinery. Some authors require each value to be a real-valued random variable with expectation and some finite variance . Then the covariance between the values at two times and is well-defined: it is zero if the times are distinct, and if they are equal. However, by this definition, the integral over any interval with positive width would be simply the width times the expectation: . This property would render the