We study the limit distributions of upper and lower record values of a stationary Gaussian sequence (SGS) under an equi-correlated set up, when the random sample size is assumed to converge weakly and independent of the basic variables. Moreover, the class of limit distribution functions (df’s) of joint upper and lower record values of a SGS with random indices are fully characterized. As an application of this result, the sufficient conditions for the weak convergence of record quasi-range, record quasi-mid-range, record extremal quasiquotient and record extremal quasi-product with random indices are obtained. Moreover, the classes of the non degenerate limit df’s of these statistics are derived.