Some Limit Theorems for Random Functions. I

This paper establishes a number of limit theorems for random functions H(Δ ) in the interval Δ, which satisfy the “strong mixing condition” \[ {lim }τ → ∞ { }_t { }_{A ∈ M_∞ ^t ,B ∈ Mt + τ^∞ } | {{ P}(AB) - { P}(A){ P}(B)} | = 0 \] where Mt₁t₂ is a σ-algebra generated by events \ H(Δ ₁ ) < h₁ , ⋯ ,H(Δ ₙ ) < hₙ \ ,Δ ₁ , ⋯ ,Δ ₙ are arbitrary intervals, Δ ₖ ⊆ (t₁ ,t₂ ),h₁ , ⋯ hₙ are arbitrary real numbers.