The Central Limit Theorem for random exponents on a Hilbert space in the Weak Operator Topology

2025/10/05 by S. Dzhenzher, Dzhenzher, S. V.
Decision Sciences · Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #Probability and Risk Models #Random Matrices and Applications #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2510.04077

openalex publication_date 2025/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider random linear continuous operators Ω→ L(H, H) on a Hilbert space 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.

Citations

• The Strong Law of Large Numbers for random semigroups on uniformly smooth Banach spaces
• The law of large numbers for discrete generalized quantum channels
• Towards Optimal Convergence Rates for the Quantum Central Limit Theorem
• Law of large numbers and central limit theorem for ergodic quantum processes
• Upper and lower estimates for rate of convergence in the Chernoff product formula for semigroups of operators
• Random Matrices
• Quantum stochastic equation for a test particle interacting with a dilute Bose gas
• QUANTUM MULTIPOLE NOISE AND GENERALIZED QUANTUM STOCHASTIC EQUATIONS

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• The law of large numbers for discrete generalized quantum channels
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