Stochastic Regularization and Eigenvalue Concentration Bounds for Singular Ensembles of Random Operators

Figure 1

If 𝐹(𝑠_𝑖)=𝐹(𝑡⁻_𝑖)=𝜗_𝑖, then Γ (the right inverse of 𝐹) has a jump of magnitude 𝛽_𝑖−𝛼_𝑖 at 𝜗_𝑖. Γ has at most a countable number of jumps. We always set Γ(𝜗_𝑖)=𝛼_𝑖, so Γ is lower continuous. Γ is strictly increasing, if 𝐹 is continuous.

(a)

(b)