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We study approximation properties of sequences of centered random elements X d , d ∈ N , with values in separable Hilbert spaces. We focus on sequences of tensor product-type and, in particular, degree-type random elements, which have covariance operators of a corresponding tensor form. The average case approximation complexity n X d ( ε ) is defined as the minimal number of continuous linear functionals that is needed to approximate X d with a relative 2-average error not exceeding a given threshold ε ∈ ( 0 , 1 ) . In the paper we investigate n X d ( ε ) for arbitrary fixed ε ∈ ( 0 , 1 ) and d → ∞ . Namely, we find criteria of (un) boundedness for n X d ( ε ) on d and of tending n X d ( ε ) → ∞ , d → ∞ , for any fixed ε ∈ ( 0 , 1 ) . In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics ln n X d ( ε ) = a d + q ( ε ) b d + o ( b d ) , d → ∞ , at continuity points of a non-decreasing function q : ( 0 , 1 ) → R . Here ( a d ) d ∈ N is a sequence and ( b d ) d ∈ N is a positive sequence such that b d → ∞ , d → ∞ . Under rather weak assumptions, we show that for tensor product-type random elements only special quantiles of self-decomposable or, in particular, stable (for tensor degrees) probability distributions appear as functions q in the asymptotics. We apply our results to the tensor products of the Euler integrated processes with a given variation of smoothness parameters and to the tensor degrees of random elements with regularly varying eigenvalues of covariance operator. [ABSTRACT FROM AUTHOR]