concept inadequate as a model of physical "white noise" signals. Therefore, most authors define the signal indirectly by specifying non-zero values for the integrals of and over any interval , as a function of its width . In this approach, however, the value of at an isolated time cannot be defined as a real-valued random variable. Also the covariance becomes infinite when ; and the autocorrelation function must be defined as , where is some real constant and is Dirac's "function". In this approach, one usually specifies that the integral of over an interval is a real random variable with normal distribution, zero mean, and variance ; and also that the covariance of the integrals , is , where is the width of the intersection of the two intervals . This model is called a Gaussian white noise signal (or process). In the mathematical field known as white noise analysis, a Gaussian white noise is defined as a stochastic tempered distribution, i.e. a random variable with values in the space of tempered distributions. Analogous to the case for finite-dimensional random vectors, a probability law on the infinite-dimensional space can be defined via its characteristic function (existence and uniqueness are guaranteed by an extension of the Bochner–Minlos theorem, which goes under the name Bochner–Minlos–Sazanov theorem); analogously to the case of the multivariate normal distribution , which has characteristic function the white noise must satisfy where is the natural pairing of the tempered distribution with the Schwartz function , taken scenariowise for , and . Mathematical applications Time series analysis and regression In statistics and econometrics one often assumes that an observed series of data values is the sum of a series of values generated by a deterministic linear process, depending on certain independent (explanatory) variables, and on a series of random noise values. Then regression analysis is used to infer the parameters of the model process from the observed data, e.g. by ordinary least squares, and to test the null hypothesis that each of the parameters is zero against the alternative hypothesis that it is non-zero. Hypothesis testing typically assumes that the noise values are mutually uncorrelated with zero mean and have the same Gaussian probability distributionin other words, that the noise is Gaussian white (not just white). If there is non-zero correlation between the noise values underlying different observations then the estimated model parameters are still unbiased, but estimates of