Splitting method for strong stable mixtures of normal laws in stable index $\alpha=1$ and $\alpha=2$

Splitting method for strong stable scale mixtures of normal laws is investigated, the mixtures being limit distributions of compound Cox processes. For stable index $\alpha=1$ the precise formula of invariant density distribution relevant to scale transformation was derived. Using classical and asymptotic methods the sample size necessary to determine the difference between normal mixture and the alternative $\alpha=1$ was computed.