  For a zero-mean, unit-variance second-order stationary univariate Gaussian
process we derive the probability that a record at the time $n$, say $X_n$,
takes place and derive its distribution function. We study the joint
distribution of the arrival time process of records and the distribution of the
increments between the first and second record, and the third and second record
and we compute the expected number of records. We also consider two consecutive
and non-consecutive records, one at time $j$ and one at time $n$ and we derive
the probability that the joint records $(X_j,X_n)$ occur as well as their
distribution function. The probability that the records $X_n$ and $(X_j,X_n)$
take place and the arrival time of the $n$-th record, are independent of the
marginal distribution function, provided that it is continuous. These results
actually hold for a second-order stationary process with Gaussian copulas. We
extend some of these results to the case of a multivariate Gaussian process.
Finally, for a strictly stationary process satisfying some mild conditions on
the tail behavior of the common marginal distribution function $F$ and the
long-range dependence of the extremes of the process, we derive the asymptotic
probability that the record $X_n$ occurs and derive its distribution function.
