Dear Vasana,

The usual test for goodness of fit is a one dimensional chi-squared.

Dear Vasana,

As far as I know this is a problem with no general solution working for any situation. Given multivariate observations, I realize that you are asking for a GOF test for the marginal distribution, do you? There are several results for normality in the time-series framework. In general, to construct a GOF test , it is necessary to specify the marginal distribution (Gamma, Weibull, Poisson, Binomial...) and the correlational model (AR(1), ARMA(p,q), ...) as well. For instance, in the following paper you can find and extension of the classical Fisher dispersion test for the Poisson distribution with an AR(1) structure (INAR(1) model):

Schweer S, Weiß CH (2014) Compound Poisson INAR(1) processes: stochastic properties and testing for overdispersion. Comput Stat Data Anal 77:267–284

Best regards.

The following may work for you, depending both on how formal a test you want, and on how much data you have. First decide on a decorrelation time, so that the values are effectively uncorrelated at this time step. Next create several new time-series by extracting equally-spaced values, at a spacing equal to your decorrelation time. Then apply your standard test (for uncorrelated data) to each of the new time-series separately. This results in a number of pairs of test statistics and the probability points at which they lie under the null distribution. The answer you want may be immediately obvious from this set of results. It will not be clear how to deal will this set of dependent  test statistics more formally but some help might be gained either by doing a simulation study, or by noting that under the null hypothesis the proportion of the (dependent) test statistics exceeding the critical value is expected to be equal to the significance level  of the individual tests.