Almost sure invariance principles for sums of $B$-valued random variables with applications to random Fourier series and the empirical characteristic process

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We establish an almost sure approximation of the partial sums of independent, identically distributed random variables with values in a separable Banach space $B$ by a suitable $B$-valued Brownian motion under the hypothesis that the partial sums can be ${L^1}$-closely approximated by finite-dimensional random variables. We show that this hypothesis is satisfied if the given random variables are random Fourier series or related stochastic processes. As an application we obtain an almost sure approximation of the empirical characteristic process by a suitable ${\mathbf {C}}(K)$-valued Brownian motion whenever the empirical characteristic process satisfies the central limit theorem.