Convergence rate in the law of logarithm for negatively dependent random variables under sub-linear expectations

Let ,Xₙ,n≥ 1\ be a sequence of identically distributed, negatively dependent (NA) random variables under sub-linear expectations, and denote Sₙ=∑ᵢ₌₁ⁿXᵢ, n≥ 1. Assume that h(·) is a positive non-decreasing function on (0,∞) fulfulling ∫₁∞(th(t))⁻¹ t=∞. Write Lt=ln max\,t\, ψ(t)=∫₁ᵗ(sh(s))⁻¹ s, t≥ 1. In this sequel, we establish that ∑ₙ₌₁∞(nh(n))⁻¹\|Sₙ|≥ (1+ε)σ√2nLψ(n)\<∞, ∀ ε>0 if (X)=(-X)=0 and (X²)=σ²∈ (0,∞). The result generalizes that of NA random variables in probability space.