On vector-valued random Fourier series

Abstract

Consider a (complex) Banach spaceX, such thatX ⊅ C_O, and vectors(X _i ) _i∈ℤ ofX. Consider an independent standard normal sequence(g _i ) _i∈ℤ . Then if anX-valued random Fourier series ∑_|k|⩽ _n e ^ikt g _k x _k satisfies

$$\mathop {\sup }\limits_n E\mathop {\sup }\limits_{\left| t \right| \leqslant 1} \left\| {\sum\limits_{\left| k \right| \leqslant n} {e^{ikt} g_k x_k } } \right\|< \infty $$

it a.s. converges uniformly for each compact subset of ℝ.