When is Half‐Line Convergence to an Infinitely Divisible Law v Equivalent to Weak Convergence to v ?

For a wide class of infinitely divisible laws μ numbers R (μ) are evaluated satisfying the property: if R > R (μ) then μ is uniquely determined by its values on the half‐line (−∞, R ); if R < R (μ) then μ is not determined by its values on (−∞, R ). Necessary and sufficient conditions are given for ‘half‐line’ convergence to μ on (−∞, R ) to be equivalent to weak convergence to μ.