Around the weak limit of iterates of some
random-valued functions

Abstract. Given a probability space \( (\Omega, \mathcal{A}, P)\), a real and separable Banach space \(X\), a linear and continuous \(\Lambda:X \to X\) with \(\|\Lambda \| < 1\), and an \(\mathcal{A}\)-measurable and integrable \(\xi :\Omega \to X\) with the Fourier transform \(\gamma :X^\ast \to \mathbb{C}\) we characterize the weak limit of iterates of the random-valued function \(f: X \times \Omega \to X\) given by \(f(x,\omega )=\Lambda x+\xi (\omega )\) with the aid of the functional equation $$ \varphi (x^\ast )= \gamma (x^\ast )\varphi (x^\ast\circ \Lambda ). $$ Then, making use of this characterization, given a probability Borel measure \(\mu\) on \(X\) we examine continuous at zero solutions \(\varphi : X^\ast \to \mathbb{C}\) of the equation $$ \varphi (x^\ast )= \hat{\mu} (x^\ast )\varphi (x^\ast \circ \Lambda ). $$

Key words and phrases. Random-valued functions, iterates, convergence in law, Fourier transform, iterative functional equations, continuous solutions, Gaussian measures.