I.V. Ostrovskii’s Work on Arithmetic of Probability Laws
DOI:
https://doi.org/10.15407/mag20.03.05Анотація
Перша частина статті є коротким оглядом розвитку арифметики ймовірнісних законів від роботи Г. Крамера про компоненти закону Гаусса до досліджень Ю.В. Лінника 1950-х років. У другій частині дано опис внеску Й.В. Островського в арифметику ймовірнісних законів та теорію аналітичних характеристичних функцій.
Mathematical Subject Classification 2020: 60E07, 60E10
Ключові слова:
ймовірнісний закон, характеристична функція, безмежно подільний закон, клас $I_0$, арифметика ймовірнісних законівПосилання
G.P. Chistyakov, Decompositional stability of distribution laws, Teor. Veroyatn. Primen. 31 (1986), No. 3, 433--450 (Russian) https://doi.org/10.1137/1131053 ; Engl. transl.: Theory Probab. Appl. 31 (1987), 375--390. https://doi.org/10.1137/1131053
G.P. Chistyakov, Factorization of probability distributions from Yu.V. Linnik's class ℒ, I, Teor. Funkts. Funkts. Anal. Prilozh. 47 (1987), 3--25 (Russian); Engl. transl.: J. Sov. Math. 48 (1990), No. 6, 619--635. https://doi.org/10.1007/BF01094715
G.P. Chistyakov, Factorization of probability distributions from Yu.V. Linnik's class ℒ, II, Teor. Funkts. Funkts. Anal. Prilozh. 48 (1987), 3--26 (Russian); Engl. transl.: J. Sov. Math. 49 (1990), No. 2, 857--871. https://doi.org/10.1007/BF02205630
G.P. Chistyakov, On the conditions under which probability laws with nonanalytic characteristic functions belong to class $I_0$, Dokl. Acad. Nauk SSSR 201 (1971), 280--283 (Russian); Engl. transl.: Sov. Math., Dokl. 12 (1971), 1654--1658.
G.P. Chistyakov, On a theorem of I.V. Ostrovskii-R. Cuppens, Analytical Methods in Probability Theory and Operator Theory (Ed. V.A. Marchenko), Collection of scientific works. Kiev: Naukova Dumka (1990), 3--14 (Russian).
H. Cramér, Über eine Eigenschaft der normalen Verteilungsfunktion, Math. Z. 41 (1936), 405--414. https://doi.org/10.1007/BF01180430
R. Cuppens, Ensembles indépendants et décomposition des fonctions caractétristiques, C. R. Acad. Sci. 272 (1971), No. 22, A1464--A1466.
R. Davidson, Arithmetic and other properties of certain Delphic semigroups, I, II, Z. Wahrscheinlichkeitstheor. und verw. Geb. 10 (1968), No. 2, 120--145, 146--172. https://doi.org/10.1007/BF00531846
T.-C. Dinh, S. Ghosh, H.-S. Tran, and M.-H. Tran, Quantitative Marcinkiewicz's theorem and central limits theorem: applications to spin systems and point processes, preprint, https://arxiv.org/abs/2107.08469v2
A.E. Eremenko, Valiron deficiencies of entire characteristic functions of finite order, Ukr. Mat. Zh. 29 (1977), No. 6, 807--809 (Russian) https://doi.org/10.1007/BF01085969 ; Engl. transl.: Ukrainian Math. J. 29 (1978), No. 6, 600--601. https://doi.org/10.1007/BF01085969
A.E. Eremenko and A.E. Fryntov, Stability in the Marcinkiewicz theorem, J. Math. Phys. Anal. Geom. 17 (2021), No. 4, 463--467. https://doi.org/10.15407/mag17.04.463
G.M. Feldman, Gaussian distributions on locally compact Abelian groups, Theory Probab. Appl. 23 (1979), No. 3, 529--542. https://doi.org/10.1137/1123062
G.M. Fel'dman, On a decomposition of generalized Poisson distribuions on groups, Teor. Veroyatn. Primen. 27 (1982), No. 4, 725--738 (Russian) https://doi.org/10.1137/1127087 ; Engl. transl.: Theory Probab. Appl. 27 (1983), 780--794. https://doi.org/10.1137/1127087
G.M. Fel'dman, Generalized Poisson distribuions of class $I_0$ on groups, Teor. Veroyatn. Primen. 29 (1984), No. 2, 222--233 (Russian) https://doi.org/10.1137/1129034 ; Engl. transl.: Theory Probab. Appl. 29 (1985), 218--230. https://doi.org/10.1137/1129034
G.M. Feldman, On infinitely divisible distributions of the class $I_0$ on groups, Theory Probab. Appl. 30 (1986), No. 3, 475--484. https://doi.org/10.1137/1130060
G.M. Feldman, Marcinkiewicz and Lukacs theorems on Abelian groups, Theory Probab. Appl. 34 (1990), No. 2, 290--297. https://doi.org/10.1137/1134024
G.M. Fel'dman, Arithmetic of Probability Distributions and Characterization Problems on Abelian Groups, Translations of Mathematical Monographs, 116, Amer. Math. Soc., Providence, RI, 1993. https://doi.org/10.1090/mmono/116
G.M. Feldman and A.E. Fryntov, On the decomposition of the convolution of a Gaussian and Poisson distributions on locally compact Abelian groups, J. Mult. Analysis 13 (1983), No. 1, 148--166. https://doi.org/10.1016/0047-259X(83)90010-6
A.E. Fryntov, On the factorization of infinitely divisible distributions, Teor. Veroyatn. Primen. 20 (1975), No. 3, 661--664 (Russian) https://doi.org/10.1137/1120073 ; Engl. transl.: Theory Probab. Appl. 20 (1975), 648--652. https://doi.org/10.1137/1120073
A.E. Fryntov, On one property of the cone generated by multiplicative shifts of the subharmonic ridge function, Analytical Methods in Probability Theory and Operator Theory. Collection of scientific works (Ed. V.A. Marchenko), Naukova Dumka, Kyiv, 1990, 33--39 (Russian).
A.E. Fryntov and G.P. Chistyakov, On infinitely divisible distributions of class $I_0$, Teor. Funkts. Funkts. Anal. Prilozh. 32 (1979), 91--97 (Russian).
B.N. Ginzburg and N.D. Serykh, On algebraic null surfaces of entire characteristic functions of multidimensional probability distributions, Teor. Funkts. Funkts. Anal. Prilozh. 30 (1978), 30--36 (Russian).
A.A. Goldberg and I.V. Ostrovskii, Value Distribution of Meromorphic Functions, Moscow, Nauka, 1970 (Russian); Engl. transl.: Amer. Math. Soc., Providence, RI, 2008.
A.A. Goldberg and I.V. Ostrovskii, On the growth of entire ridge functions with real zeros, Mat. Fiz. Funkts. Anal. Acad. Nauk Ukr. SSR, Fiz.-Tehn. Inst. Nizkih Temperatur, Kharkov 5 (1974), 3--10 (Russian); Engl. transl.: Sel. Transl. Math. Stat. Probab. 15 (1981), 147-155. https://doi.org/10.1080/09595238100185171
A.A. Goldberg and I.V. Ostrovskii, Application of a theorem of W.K. Heyman to a question in the theory of decompositions of probability laws, Ukr. Math. Zh. 19 (1967), 104--106 (Russian); Engl. transl.: Sel. Transl. Math. Stat. Probab. 9 (1971), 147--151.
A.A. Goldberg and I.V. Ostrovskii, Indicators of entire Hermitian-positive functions of finite order, Sib. Mat.Zh. 23 (1982), No. 6, 55--73 (Russian) https://doi.org/10.1007/BF00968750 ; Engl. transl.: Sib. Math. J. 23 (1983), 804--820. https://doi.org/10.1007/BF00968750
A.A. Goldberg and I.V. Ostrovskii, Paley effect for entire characteristic functions and entire functions represented by Dirichlet series, Teor. Funkts. Funkts. Anal. Prilozh. 44 (1985), 18--23 (Russian); Engl. transl.: J. Sov. Math. 48 (1990), No. 3, 255--259. https://doi.org/10.1007/BF01101241
I.P. Il'inskaya, Arithmetic of semigroups of sequences generated by Jacobi polynomials, Dynamical systems and complex analysis. Collection of scientific works (Ed. V.A. Marchenko), Naukova Dumka, Kyiv, 1992, 119--135 (Russian).
I.P. Il'inskaya, Arithmetic of a semigroup of series in Legendre functions of the second kind, Turk. J. Math. 21 (1997), No. 3, 357--373.
I.P. Il'inskaya, The arithmetic of a semigroup of series of Walsh functions, J. Aust. Math. Soc. A68 (2000), No. 3, 365--378. https://doi.org/10.1017/S1446788700001464
I.P. Il'inskaya, Arithmetic of semigroups of series in multiplicative systems, Ukr. Mat. Zh. 61 (2009), No. 7, 939--947 (Russian) https://doi.org/10.1007/s11253-009-0264-3 ; Engl. transl.: Ukrainian. Math. J. 61 (2009), No. 7, 1113--1122. https://doi.org/10.1007/s11253-009-0264-3
A.I. Il'inskii, On the arithmetic of G. Polya's characteristic functions, Mat. Zametki 21 (1977), No. 5, 717--725 (Russian); Engl. transl.: Math. Notes 21 (1977), 400--405. https://doi.org/10.1007/BF01788238
A.I. Il'inskii, On the arithmetic of B.M. Levitan's generalized characteristic functions, Dynamical systems and complex analysis. Collection of scientific works (Ed. V.A. Marchenko), Naukova Dumka, Kyiv, 1992, 135--150 (Russian).
I.P. Kamynin, A generalization of Marcinkiewicz's theorem on entire characteristic functions of probability distributions, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 85 (1979), 94--103 (Russian).
I.P. Kamynin, Entire ridge functions with real zeros, Stability problems for stochastic models. Proc. Semin., Moscow 1980, Inst. Syst. Stud. (1980), 57--63 (Russian); Engl. transl.: J. Sov. Math. 32 (1986), 47--52. https://doi.org/10.1007/BF01084498
I.P. Kamynin and I.V. Ostrovskii, On zeros of entire ridge functions, Teor. Funkts. Funkts. Anal. Prilozh. 24 (1975), 41--50 (Russian).
I.P. Kamynin and I.V. Ostrovskii, On the zero sets of entire Hermitian-positive functions, Dopov. Akad. Nauk Ukr. RSR, Ser. A (1980), No. 7, 20--23 (Ukrainian).
I.P. Kamynin and I.V. Ostrovskii, Zero sets of entire Hermitian-positive functions, Sib. Mat. Zh. 23 (1982), No. 3, 66--82 (Russian) https://doi.org/10.1007/BF00973491 ; Engl. transl.: Sib. Math. J. 23 (1983), 344--357. https://doi.org/10.1007/BF00973491
D.G. Kendall, Delphic semigroups, infinitely divisible regenerative phenomena, and the arithmetic of p-functions, Z. Wahrscheinlichkeitstheorie und verw. Geb. 9 (1968), 163--195. https://doi.org/10.1007/BF00535637
A.Ya. Khinchin, On the arithmetic of distribution laws, Bull. Moscow State Univ. A1 (1937), 6--17 (Russian).
J.F.C. Kingman, Random walks with spherical symmetry, Acta Math. 109 (1963), No. 1-2, 11--53. https://doi.org/10.1007/BF02391808
L.S. Kudina, On decomposition of radially symmetric distributions, Teor. Veroyatn. Primen. 20 (1975), No. 3, 656--660 (Russian) https://doi.org/10.1137/1120072 ; Engl. transl.: Theory Probab. Appl. 20 (1975), 644--648. https://doi.org/10.1137/1120072
L.S. Kudina, On the components of radially symmetric distributions, Teor. Funkts. Funkts. Anal. Prilozh. 25 (1976), 77--81 (Russian).
L.S. Kudina, Expansions of probability distributions decreasing rapidly at infinity, Sib. Mat. Zh. 19 (1978), No. 2, 336--342 (Russian) https://doi.org/10.1007/BF00970505 ; Engl. transl.: Sib. Math. J. 19 (1978), 235--239. https://doi.org/10.1007/BF00970505
L.S. Kudina, Decompositions of multivariate probability distributions, Teor. Veroyatn. Primen. 25 (1980), No. 2, 430--431 (Russian); Engl. transl.: J. Sov. Math. 32, 60--63. https://doi.org/10.1007/BF01084501
R.G. Laha and V.K. Rohatgi, Decomposition of probability measures on locally compact Abelian groups, Lect. Notes Math. 861 (1981), 83--92. https://doi.org/10.1007/BFb0097317
Yu.V. Linnik, On factorizing the composition of a Gaussian and a Poisson laws, Teor. Veroyatn. Primen. 2 (1957), No. 1, 34--59 (Russian). https://doi.org/10.1137/1102002
Yu.V. Linnik, General theorems on the factorization of infinitely divisible laws, I, II, III, Teor. Veroyatn. Primen. 3 (1958), No. 1, 3--40 https://doi.org/10.1137/1103001 4 (1959), No. 1, 55--85 https://doi.org/10.1016/S0022-3476(59)80177-3 4 (1959), No. 2, 150--171 (Russian). https://doi.org/10.1007/BF00494780
Yu.V. Linnik, Decomposition of Probability Distributions, Leningrad: Izdat. Leningrad. Univ., 1960 (Russian); Engl. transl.: Edinburgh-London: Oliver & Boyd, Ltd, 1964.
Yu.V. Linnik and I.V. Ostrovskii, Decomposition of Random Variables and Vectors, M., Nauka (1972) (Russian); Engl. transl.: Translations of Mathematical Monographs, 48, Amer. Math. Soc., Providence, RI, 1977.
L.Z. Livshits and I.V. Ostrovskii, On multidimensional infinitely divisible laws, all of whose components are infinitely divisible, Dokl. Akad. Nauk SSSR 198 (1971), No. 6, 1273 (Russian); Engl. transl.: Sov. Math., Dokl. 12 (1971), 978--979.
L.Z. Livshits and I.V. Ostrovskii, On multidimensional infinitely divisible laws which have only infinitely divisible components, Mat. Fiz. Funkts. Anal., Acad. Nauk Ukr. SSR, Fiz.-Tehn. Inst. Nizkih Temperatur, Kharkov 2 (1971), 61--75 (Russian); Engl. transl.: Sel. Transl. Math. Stat. Probab. 14 (1978), 13--31.
L.Z. Livshits, On the conditions for an infinitely divisible law to belong to the class $I_0$, Dopov. Akad. Nauk Ukr. RSR, Ser. A (1972), No. 11, 992--993 (Ukrainian).
J. Marcinkiewicz, Sur une propriété de la loi Gauss, Math. Z. 44 (1938), 612--618. https://doi.org/10.1007/BF01210677
I.V. Ostrovskii, On the application of a law established by Wiman and Valiron to the investigation of characteristic functions of probability laws, Dokl. Acad. Nauk SSSR 143(1962), No. 3, 532--535 (Russian); Engl. transl.: Sov. Math., Dokl. 3 (1962), 433--437.
I.V. Ostrovskii, Infinitely divisible laws with unbounded Poisson spectrum, Dokl. Akad. Nauk SSSR, 152 (1963), No. 6, 1301--1304 (Russian); Engl. transl.: Sov. Math., Dokl. 4 (1963), 1552--1555.
I.V. Ostrovskii, On entire functions satisfying some special inequalities connected with the theory of characteristic functions of probability laws, Zap. mekh.-mat. fakulteta KhGU i KhMO 29 (1963), 145--168 (Russian); Engl. transl.: Sel. Transl. Math. Stat. Probab. 7 (1968), 203--234.
I.V. Ostrovskii, On the decompositions of infinitely divisible lattice laws, Vestn. Leningr. Univ., Ser. Mat. Mekh. Astron. 19 (1964), No. 4, 51--60 (Russian); Engl. transl.: Sel. Transl. Math. Stat. Probab. 9 (1971), 127--139.
I.V. Ostrovskii, Decomposition of infinitely divisible laws without Gaussian component, Dokl. Akad. Nauk SSSR, 161 (1965), No. 1, 48--51 (Russian); Engl. transl.: Sov. Math., Dokl. 6 (1965), 372--376.
I.V. Ostrovskii, On the factorization of the composition of Gaussian and Poisson laws, Uspekhi Mat. Nauk 20 (1965), No. 4, 166--171 (Russian).
I.V. Ostrovskii, Some theorems on decompositions of probability laws, Trudy Mat. Inst. V.A. Steklova 79 (1965), 198--235 (Russian); Engl. transl.: Proc. Steklov Inst. Math. 79 (1965), 221--259.
I.V. Ostrovskii, The multidimensional analogue of Yu.V. Linnik's theorem on decompositions of a convolution of Gaussian and Poisson laws, Teor. Veroyatn. Primen. 10 (1965), No. 4, 742--745 (Russian); https://doi.org/10.1137/1110083 Engl. transl.: Theor. Probab. Appl. 10 (1965), 673--677 https://doi.org/10.1137/1110083 Teor. Veroyatn. Primen. 12 (1967), No. 3, 586--587 (Russian); Engl. transl.: Theor. Probab. Appl. 12 (1967), 529--530. https://doi.org/10.1037/007835
I.V. Ostrovskii, On the growth of entire characteristic functions of probability laws, Sovremen. Probl. Teor. Analit. Funktsij, Mezhdunarod. Konf. Teor. Analit. Funktsij, Erevan (1965), 1966, 239 -- 245 (Russian).
I.V. Ostrovskii, Some properties of holomorphic characteristic functions of multidimensional probability laws, Teor. Funkts. Funkts. Anal. Prilozh. 2 (1966), 169--177 (Russian).
I.V. Ostrovskii, Decomposition of multidimensional probability laws, Dokl. Akad. Nauk SSSR 169 (1966), No. 5, 1017--1019 (Russian); Engl. transl.: Sov. Math., Dokl. 7 (1966), 1052--1055.
I.V. Ostrovskii, On decompositions of multidimensional infinitely divisible laws without Gaussian component, Vestn. Kharkov. Univ. (Ser. Mekh.-Mat.) 32 (1966), 51--72 (Russian).
I.V. Ostrovskii, On a problem of S.R. Rao, Teor. Veroyatn. Primen. 14 (1969), No. 2, 317--318 (Russian) https://doi.org/10.1137/1114038 ; Engl. transl.: Theor. Probab. Appl. 14 (1969), 312--313. https://doi.org/10.1137/1114038
I.V. Ostrovskii, On some classes of infinitely divisible laws, Izv. Akad. Nauk SSSR, Ser. Mat. 34 (1970), No. 4, 923--944 (Russian) https://doi.org/10.1070/IM1970v004n04ABEH000938 ; Engl. transl.: Math. USSR, Izv. 4, 931--952 (1971). https://doi.org/10.1070/IM1970v004n04ABEH000938
I.V. Ostrovskii, On infinitely divisible laws having only infinitely divisible components, Dokl. Akad. Nauk SSSR 193 (1970), No. 6, 1238--1240 (Russian); Engl. transl.: Sov. Math., Dokl. 11 (1970), 1110--1113.
I.V. Ostrovskii, On a class of characteristic functions, Trudy Mat. Inst. V.A. Steklova 111 (1970), 195--207 (Russian); Engl. transl.: Proc. Steklov Inst. Math. 111 (1970), 233--247.
I.V. Ostrovskii, On the theory of decompositions of multidimensional infinitely divisible laws, Dopov. Akad. Nauk Ukr. RSR, Ser. A (1972), No. 11, 997--1000 (Ukrainian).
I.V. Ostrovskii, Description of the class $I_0$ in a special semigroup of probability measures, Mat. Fiz. Funkts. Anal., Mater. Sci. Semin., Phys.-Tech. Inst. Low Temp. Akad. Sci. Ukr. SSR, Kiev 4 (1973), 3--12 (Russian); Engl. transl.: Sel. Transl. Math. Stat. Probab. 15 (1981), 1--8.
I.V. Ostrovskii, Description of the $I_0$ class in a special semigroup of probability measures, Dokl. Akad. Nauk SSSR 209 (1973), No. 4, 788--791 (Russian); Engl. transl.: Sov. Math., Dokl. 14 (1973), 525--529. https://doi.org/10.1090/S0002-9939-1973-0318719-3
I.V. Ostrovskii, On an infinitely divisible factorization, Teor. Funkts. Funkts. Anal. Prilozh. 34 (1980), 89--96 (Russian); Engl. transl.: Sel. Transl. Math. Stat. Probab. 16 (1985), 11--17.
I.V. Ostrovskii, On divisors of infinitely divisible distributions representable as Cartesian products, Teor. Veroyatn. Primen. 27 (1982), No. 4, 772--777 (Russian) https://doi.org/10.1137/1127091 ; Engl. transl.: Theory Probab. Appl. 27 (1982), 832--837. https://doi.org/10.1137/1127091
I.V. Ostrovskii, On the growth of ridge functions that are entire or analytic in a half-plane, Mat. Sb., New Ser. 119 (1982), No. 1, 150--159 (Russian); Engl. transl.: Math. USSR, Sb. 47 (1984), 145--154. https://doi.org/10.1070/SM1984v047n01ABEH002635
I.V. Ostrovskii, On the zero sets of entire periodic Hermitian-positive functions, Teor. Funkts. Funkts. Anal. Prilozh. 37 (1982), 102--110 (Russian).
I.V. Ostrovskii, On the zeros of entire characteristic functions of distributions concentrated on the half-line, Teor. Veroyatn. Primen. 33 (1988), No. 1, 180--184 (Russian) https://doi.org/10.1137/1133022 ; Engl. transl.: Theory Probab. Appl. 33 (1988), No. 1, 167--171. https://doi.org/10.1137/1133022
I.V. Ostrovskii and I.P. Trukhina, On the arithmetic of Schoenberg-Kennedy semigroups, Quest. Math. Phys. Funct. Anal., Mater. Sci. Semin., Phys.-Tech. Inst. Low Temp. Akad. Sci. Ukr. SSR, Kiev (1976) 11--19 (Russian).
D.A. Raikov, On decomposition of Poisson law, Dokl. Akad. Nauk SSSR, 14 (1937), 9--12 (Russian).
I.P. Trukhina, A problem related to the arithmetic of probability measures on the sphere, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, 87 (1979), 143--158 (Russian); Engl. transl.: J. Sov. Math. 17 (1981), 2321--2333. https://doi.org/10.1007/BF01085929
I.P. Trukhina, The arithmetic of spherically symmetric measures in Lobachevskij space, Teor. Funkts. Funkts. Anal. Prilozh. 34 (1980), 136--146 (Russian).
A.M. Ulanovskii, $Γ-$sequences and their arithmetic, Problems of stability of stochastic models, Proc. Semin., Moscow, 1981 (1981), 129--138 (Russian); Engl. transl.: J. Sov. Math. 34 (1986), 1556--1564. https://doi.org/10.1007/BF01089796
A.M. Ulanovskii, Description of the class $I_0$ in Kingman's semigroup of standard $p-$functions, Teor. Veroyatn. Primen. 28 (1983), No. 3, 533--543 (Russian) https://doi.org/10.1137/1128051 ; Engl. transl.: Theor. Probab. Appl. 28 (1984), 561--572. https://doi.org/10.1137/1128051
K. Urbanik, Generalized convolutions, Studia Math. 23 (1964), 217--245. https://doi.org/10.4064/sm-23-3-217-245
A.M. Vishnyakova and I.V. Ostrovskii, Analogue of Marcinkiewiz's theorem for entire ridge functions not having zeros in an angular domain, Dokl. Akad. Nauk Ukr. SSR, Ser. A (1987), No. 9, 8--11 (Russian).
A.M. Vishnyakova, On the growth of ridge functions non-vanishing in an angular domain, Analytical Methods in Probability Theory and Operator Theory. Collection of scientific works (Ed. V.A.Marchenko), Naukova Dumka, Kyiv, 1990, 40--48 (Russian); Engl. transl.: Bull. Hong Kong Math. Soc. 1 (1997), No. 2, 351--361.
A.M. Vishnyakova, I.V. Ostrovskii, and A.M. Ulanovskii, On a conjecture of Yu.V. Linnik, Algebra Anal. 2 (1990), No. 4, 82--90 (Russian); Engl. transl.: Leningr. Math. J. 2 (1991), No. 4, 765--773.
V.V. Zimoglyad, On entire functions satisfying special inequalities, Teor. Funkts. Funkts. Anal. Prilozh. 6 (1968), 30--41 (Russian).
