= Gaussianity of deterministic Brownian stochastic integrals
{c}
{title2=$\int_0^th(s)dB_s\sim N(0,\int_0^th(s)^2ds)$}
For deterministic square-integrable $h$, approximate by step functions. Their <Itô integrals> are sums of independent normal <Brownian increments>. The <Itô isometry> gives convergence in $L^2$, and the variances converge to $\int h^2$; convergence of <characteristic functions> proves the normal limit. Each linear combination of finitely many such integrals is another deterministic stochastic integral, so they are jointly <Gaussian>.
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