The Central Limit Theorem for random exponents on a Hilbert space in the Weak Operator Topology
S. V. Dzhenzher
TL;DR
The paper proves a central limit theorem for products of random operator exponentials on a Hilbert space under the weak operator topology. By centering at $e^{\mathbb{E}A}$ and normalizing by $\sqrt{n}$, the random operator $\sqrt{n}(e^{A_1/n}\cdots e^{A_n/n} - e^{\mathbb{E}A})$ converges in distribution to a Gaussian operator $N(0,\Sigma)$, where $\Sigma$ is given by a Bochner integral capturing the covariance of $A$ and the semigroup generated by $\mathbb{E}A$. The analysis combines Taylor expansions, martingale difference arrays, and a Hall–Heyde CLT in the WOT, with careful control of remainder terms, to extend CLT-type results to operator-valued random processes on Hilbert spaces. This extends existing LLN/CLT frameworks for random operator products to the weak topology and provides a canonical Gaussian limit for quantum-operator-like dynamics.
Abstract
We consider random linear continuous operators $Ω\to \mathcal{L}(\mathcal{H}, \mathcal{H})$ on a Hilbert space $\mathcal{H}$. For example, such random operators may be random quantum channels. The Central Limit Theorem is known for the sums of i.i.d. random operators. Instead of the sum, there may be considered the composition of random exponents $e^{A_i/n}$. We obtain the Central Limit Theorem in the Weak Operator Topology for centralized and normalized random exponents of i.i.d. linear continuous operators on a Hilbert space.