Almost Surely Invariance Principle for Non-stationary and Random Intermittent Dynamical Systems

Random Young Towers and Quenched Limit Laws

“…Proof Let K n (ω) := i≤n E µω ψ 4 σ i ω • F i ω , then by Lemma 5.1 and ergodic theorem, for a.e. ω ∈ Ω, there is C ω > 0 such that K n (ω) ≤ C ω • n. By similar argument of Lemma 4.5 in [Su19a], there is C ω > 0 s.t.…”

“…The proof in this step is quite similar to Lemma 4.1-4.6 in [Su19a], so we will sketch the same part and focus on the difference: Lemma 5.3 (See also Lemma 4.1 in [Su19a])…”

“…Instead, we will give a simple method, make a clear claim that RYT has QASIP and answer one open question in [BBMD02]. The approach to the limit theorem is the martingale approximation similar to [Liv96] and Skorokhod embedding similar to [Su19a]. Besides, technical lemmas will be set up to overcome the difficulty caused by the unbounded C ω .…”

Sunklodas' approach to normal approximation for time-dependent dynamical systems

“…Statistical properties of time-dependent dynamical systems have been studied in several previous works including [2,11,22,23,36,37,40]. Central limit theorems were obtained by Bakhtin [3,4], Conze and Raugi [8], and more recently by Nándori et al [29] and Nicol et al [30].…”