Author’s abstract: “Using a semigroup product formula similar to that of P. Chernoff’s, we generalize Marc Berger’s central limit theorem for products of independent random matrices to a central limit theorem for products of dependent random linear and nonlinear operators”.
The basic product formula assumes that \((T(t):\;t\geq 0)\) is a family of linear contractions on a Banach space, as well as other assumptions. The conclusion is that strong \(\lim_{n\to \infty}T(t/n)^{n^ 2}=R_{-1} \exp (t^ 2H)\), where \[ H=R_{-1}T''(0)R_{-1}+R_{-1}T'(0)R_ 0T'(0)R_{-1},\quad (\lambda I-T(0))^{-1}=\sum^{\infty}_{h=- 1}(\lambda -1)^ kR_ k. \] As algebraic representation of a sequence \(\{X_ N\}^{\infty}_ 1\) of r.v. on (\(\Omega\),F,P) is \((\Phi,S(U),p^+,P)\), where \(\Phi\) is a certain closed subspace of M(\(\Omega\),F), the bounded signed measures on (\(\Omega\),F), \[ S(U)\psi (X_ 1\in U_ 1,...,X_ n\in U_ n)=\psi (X_ 1\in U,X_ 2\in U_ 1,...,X_{n+1}\in U_ n), \] and \(p^*(\psi)=\psi (\Omega)\). Thus the joint distribution of \(\psi\in \Phi\) can be calculated by \(\psi (X_ 1\in U_ 1,...,X_ n\in U_ n)=p^*S(U_ n)...S(U_ 1)\psi.\) (The author gives various examples.) The main result assumes S(\({\mathbb{R}})\) is a compact operator and gives necessary and sufficient condition for the existence of \[ \lim_{N\to \infty}P(X_ i\in U_ i\text{ for } 1\leq i\leq m,\quad X_{m+N+j}\in V_ j\text{ for } 1\leq j\leq n)\text{ for all } n,\quad m,\quad U_ i,\quad V_ j. \]