Geodesic
On estimates of the stability measure for decompositions of probability distributions into components
Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 3, pp. 527-539

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Let $\mathfrak G_m$ be the class of indecomposable probability laws with bounded spectrum $S(G)$ where \begin{gather*} m=\min(u,v),\ u=G(\{\inf S(G)\}),\ v=G(\{\sup S(G)\}),\\ G(\{x\})=G(x+0)-G(x). \end{gather*} If $G_1\ast G_2\in\mathfrak G_m$, $m>0$, $F_1$ has median 0 and if the uniform metric $\rho(F_1\ast F_2,G_1\ast G_2)\le\varepsilon$ then there exists a constant $\varepsilon_0=\varepsilon_0(G)>0$ such that $$ \min\{\rho(F_1,G_1),\rho(F_1,G_2)\}\le(m-\sqrt{m^2-4\varepsilon})/2 $$ when $0\le\varepsilon\le\varepsilon_0$, and this estimate cannot be improved in the class $\mathfrak G_m$. 
@article{TVP_1978_23_3_a3,
     author = {R. V. Janu\v{s}kevi\v{c}ius},
     title = {On estimates of the stability measure for decompositions of probability distributions into components},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {527--539},
     publisher = {mathdoc},
     volume = {23},
     number = {3},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1978_23_3_a3/}
}
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R. V. Januškevičius. On estimates of the stability measure for decompositions of probability distributions into components. Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 3, pp. 527-539. http://geodesic.mathdoc.fr/item/TVP_1978_23_3_a3/