List of publications by A. V. Skorokhod
Monographs:
1.Skorokhod,A.V.:Studiesinthetheoryofrandomprocesses(inRussian).Izdat.Kiev.Univ., Kyiv(1961)
2.Skorokhod,A.V.:Studiesinthetheoryofrandomprocesses(Englishtranslationofthe1961 Russianedition).Addison-WesleyPublishingCo.,Reading,Mass(1965)
3.Skorokhod,A.V.:Studiesinthetheoryofrandomprocesses(Reprintofthe1965English translation).DoverPublishing,NewYork(1982)
4.Skorokhod,A.V.:Randomprocesseswithindependentincrements(inRussian).Nauka, Moscow(1964)
5.Skorokhod,A.V.:Theoryofrandomprocesses(Englishtranslationofthe“Randomprocesseswithindependentincrements”,1964).NationalLendingLibrary,BostonSpa(1971)
6.Gikhman,I.I.,Skorokhod,A.V.:Introductiontothetheoryofrandomprocesses(inRussian). Nauka,Moscow(1965)
7.Gikhman,I.I.,Skorokhod,A.V.:Introductiontothetheoryofrandomprocesses(English translationofthe1965Russianedition).W.B.SaundersCo.,Philadelphia,Pa.-LondonToronto,Ont.(1969)
8.Gikhman,I.I.,Skorokhod,A.V.:Introductiontothetheoryofrandomprocesses(Reprintof the1969Englishtranslation).DoverPublications,Inc.,Mineola,NY(1996)
9.Gikhman,I.I.,Skorokhod,A.V.:Stochasticdifferentialequations(inRussian).Naukova Dumka,Kyiv(1968)
10.Gikhman,I.I.,Skorokhod,A.V.:StochastischeDifferentialgleichungen(Germantranslation ofthe1968Russianedition).Akademie-Verlag,Berlin(1971)
11.Gikhman,I.I.,Skorokhod,A.V.:Stochasticdifferentialequations(Englishtranslationofthe 1968Russianedition).Springer-Verlag,NewYork-Heidelberg(1972)
12.Skorokhod,A.V.,Slobodenyuk,N.P.:Limittheoremsforrandomwalks(inRussian). NaukovaDumka,Kyiv(1970)
13.Gikhman,I.I.,Skorokhod,A.V.:Thetheoryofstochasticprocesses.Vol.I(inRussian). Nauka,Moscow(1971)
14.Gikhman,I.I.,Skorokhod,A.V.:Thetheoryofstochasticprocesses.Vol.I(Englishtranslationofthe1971Russianedition).Springer-Verlag,NewYork-Heidelberg(1974)
15.Gikhman,I.I.,Skorokhod,A.V.:Thetheoryofstochasticprocesses.Vol.I.(Correctedreprint ofthe1974Englishedition).Springer-Verlag,Berlin-NewYork(1980)
16.Gikhman,I.I.,Skorokhod,A.V.:Thetheoryofstochasticprocesses.Vol.I(Reprintofthe 1974Englishedition).Springer-Verlag,Berlin(2004)
17.Gikhman,I.I.,Skorokhod,A.V.:Thetheoryofstochasticprocesses.Vol.II(inRussian). Nauka,Moscow(1973)
18.Gikhman,I.I.,Skorokhod,A.V.:Thetheoryofstochasticprocesses.Vol.II(Englishtranslationofthe1973Russianedition).Springer-Verlag,NewYork-Heidelberg(1975)
19.Gikhman,I.I.,Skorokhod,A.V.:Thetheoryofstochasticprocesses.Vol.II(Reprintofthe 1975Englishedition).Springer-Verlag,Berlin(2004)
20.Skorokhod,A.V.:IntegrationinHilbertspace(inRussian).Nauka,Moscow(1975)
21.Skorokhod,A.V.:IntegrationinHilbertspace(Englishtranslationofthe1975Russianedition).Springer-Verlag,NewYork-Heidelberg(1974)
22.Skorokhod,A.V.:IntegrationinHilbertspace(Reprintofthe1974Englishedition). Springer,London(2011)
23.Skorokhod,A.V.:Foundationsofprobabilitytheoryandrandomprocesses(inUkrainian). VyshchaShkola,Kyiv(1975)
24.Skorokhod,A.V.:Foundationsofprobabilitytheoryandrandomprocesses(Russiantranslationofthe1975Ukrainianedition).VyshchaShkola,Kyiv(1980)
25.Gikhman,I.I.,Skorokhod,A.V.:Thetheoryofstochasticprocesses.Vol.III(inRussian). Nauka,Moscow(1975)
26.Gikhman,I.I.,Skorokhod,A.V.:Thetheoryofstochasticprocesses.Vol.III(Englishtranslationofthe1975Russianedition).Springer-Verlag,Berlin-NewYork(1979)
27.Gikhman,I.I.,Skorokhod,A.V.:Thetheoryofstochasticprocesses.Vol.III(Reprintofthe 1979Englishedition).Springer,Berlin(2007)
28.Gikhman,I.I.,Skorokhod,A.V.:Thetheoryofstochasticprocesses.Vol.III(Reprintofthe 2007Englishedition).Springer,NewYork(2007)
29.Dorogovtsev,A.Ja.,Silvestrov,D.S.,Skorokhod,A.V.,Yadrenko,M.I.:Probabilitytheory: collectionofproblems(inUkrainian).VyshchaShkola,Kyiv(1976)
30.Dorogovtsev,A.Ja.,Silvestrov,D.S.,Skorokhod,A.V.,Yadrenko,M.I.:Probabilitytheory: collectionofproblems(Russiantranslationofthe1976Ukrainianedition).VyshchaShkola, Kyiv(1980)
31.Dorogovtsev,A.Ja.,Silvestrov,D.S.,Skorokhod,A.V.,Yadrenko,M.I.:Probabilitytheory: collectionofproblems(Englishtranslationofthe1976Ukrainianedition).AmericanMathematicalSociety,Providence,RI(1997)
32.Gikhman,I.I.,Skorokhod,A.V.:Controlledstochasticprocesses(inRussian).Naukova Dumka,Kyiv(1977)
33.Gikhman,I.I.,Skorokhod,A.V.:Controlledstochasticprocesses(Englishtranslationofthe 1977Russianedition).Springer-Verlag,NewYork-Heidelberg(1979)
34.Gikhman,I.I.,Skorokhod,A.V.:Introductiontothetheoryofrandomprocesses.Second edition,revised(inRussian).Nauka,Moscow(1977)
35.Korolyuk,V.S.,Portenko,M.I.,Skorokhod,A.V.,Turbin,A.F.:Amanualonprobability theoryandmathematicalstatistics(inRussian).NaukovaDumka,Kyiv(1978)
36.Korolyuk,V.S.,Portenko,M.I.,Skorokhod,A.V.,Turbin,A.F.:Manualdelateoriadeprobabilidadesyestadisticamatematica.(Spanishtranslationofthe1978Russianedition).Mir, Moscow(1981)
37.Korolyuk,V.S.,Portenko,M.I.,Skorokhod,A.V.,Turbin,A.F.:Aide-memoiredetheoriedes probabilitesetdestatistiquemathematique(Frenchtranslationofthe1978Russianedition). Mir,Moscow(1983)
38.Skorokhod,A.V.:Randomlinearoperators(inRussian).NaukovaDumka,Kyiv(1978)
39.Skorokhod,A.V.:Randomlinearoperators(Englishtranslationofthe1978Russianedition). D.ReidelPublishingCo.,Dordrecht(1984)
40.Skorokhod,A.V.:Randomlinearoperators(Reprintofthe1984Englishedition).Springer, Netherlands(2001)
41.Skorokhod,A.V.:Probabilityaroundus(inRussian).NaukovaDumka,Kyiv(1980)
42.Ibramkhalilov,I.Sh.,Skorokhod,A.V.:Consistentestimatorsofparametersofrandomprocesses(inRussian).NaukovaDumka,Kyiv(1980)
43.Gikhman,I.I.,Skorokhod,A.V.:Stochasticdifferentialequationsandtheirapplications(in Russian).NaukovaDumka,Kyiv(1982)
44.Skorokhod,A.V.:Stochasticequationsforcomplexsystems(inRussian).Nauka,Moscow (1983)
45.Skorokhod,A.V.:Stochasticequationsforcomplexsystems(Englishtranslationofthe1983 Russianedition).D.ReidelPublishingCo.,Dordrecht(1988)
46.Skorokhod,A.V.:Stochasticequationsforcomplexsystems(Reprintofthe1988English edition).Springer,Netherlands(2012)
47.Korolyuk,V.S.,Portenko,M.I.,Skorokhod,A.V.,Turbin,A.F.:Amanualonprobability theoryandmathematicalstatistics.Secondedition(inRussian).Nauka,Moscow(1985)
48.Skorokhod,A.V.:Randomprocesseswithindependentincrements.Secondedition(inRussian).Nauka,Moscow(1986)
49.Skorokhod,A.V.:Randomprocesseswithindependentincrements.(Englishtranslationof the1986secondRussianedition).KluwerAcademicPublishersGroup,Dordrecht(1991)
50.Skorokhod,A.V.:Asymptoticmethodsinthetheoryofstochasticdifferentialequations(in Russian).NaukovaDumka,Kyiv(1987)
51.Skorokhod,A.V.:Asymptoticmethodsinthetheoryofstochasticdifferentialequations (Englishtranslationofthe1987Russianedition).AmericanMathematicalSociety,Providence,RI(1989)
52.Skorokhod,A.V.:Probability.Basicconcepts.Structure.Methods;Markovprocessesand probabilisticapplicationsinanalysis;Probability.Appliedaspects(inRussian).In:Gamkrelidze,R.V.(ed.)Currentproblemsinmathematics.Fundamentaldirections.Vol.43.Itogi NaukiiTekhniki,Akad.NaukSSSR,Vsesoyuz.Inst.Nauchn.iTekhn.Inform.,Moscow (1989)
53.Skorokhod,A.V.:Basicprinciplesandapplicationsofprobabilitytheory(Englishtranslation ofthe“Currentproblemsinmathematics.Fundamentaldirections.Vol.43”,1989).SpringerVerlag,Berlin(2005)
54.Portenko,M.I.,Skorokhod,A.V.,Shurenkov,V.M.:Markovprocesses(inRussian).In: Gamkrelidze,R.V.(ed.)Currentproblemsinmathematics.Fundamentaldirections,Vol. 46.ItogiNaukiiTekhniki,Akad.NaukSSSR,Vsesoyuz.Inst.Nauchn.iTekhn.Inform., Moscow(1989)
55.Skorokhod,A.V.:Lecturesonthetheoryofstochasticprocesses(inUkrainian).Lybid,Kyiv (1990)
56.Skorokhod,A.V.:Lecturesonthetheoryofstochasticprocesses(Englishtranslationofthe 1990Ukrainianedition).VSP,Utrecht;TBiMCScientificPublishers,Kiev(1996)
57.Hoppensteadt,F.C.,Salehi,H.,Skorokhod,A.V.:Randomperturbationmethodswithapplicationsinscienceandengineering(inEnglish).Springer-Verlag,NewYork(2002)
58.Hoppensteadt,F.C.,Salehi,H.,Skorokhod,A.V.:Randomperturbationmethodswithapplicationsinscienceandengineering(Reprintofthe2002Englishedition).Springer,London (2012)
Articles:
59.Kostyuchenko,A.G.,Skorokhod,A.V.:OnatheoremofN.K.Bary.UspekhiMatem.Nauk 8,no.5(57),165–166(1953)
60.Skorokhod,A.V.:Asymptoticformulasforstabledistributionlaws.Dokl.Akad.NaukSSSR 98,731-734(1954)
61.Skorokhod,A.V.:Onatheoremconcerningstabledistributions.UspekhiMatem.Nauk 9, no.2(60),189–190(1954)
62.Skorokhod,A.V.:Onlimitingtransitionfromasequenceofsumsofindependentrandom quantitiestoahomogeneousrandomprocesswithindependentincrements.Dokl.Akad. NaukSSSR 104,364–367(1955)
63.Skorokhod,A.V.,Zolotarev,V.M.:Onsomeanalyticpropertiesofstabledistributionfunctions.UspekhiMatem.Nauk 10,no.4,181(1956)
64.Skorokhod,A.V.:OnaclassoflimittheoremsforMarkovchains.Dokl.Akad.NaukSSSR 106,781-784(1956)
65.Skorokhod,A.V.:Limittheoremsforstochasticprocesses.Teor.Veroyatnost.iPrimenen. 1, no.3,289-319(1956)
66.Skorokhod,A.V.:Onthedifferentiabilityofmeasureswhichcorrespondtostochasticprocesses.I.Processeswithindependentincrements.Teor.Veroyatnost.iPrimenen. 2,no.4, 417-443(1957)
67.Skorokhod,A.V.:Limittheoremsforstochasticprocesseswithindependentincrements. Teor.Veroyatnost.iPrimenen. 2,no.2,145-177(1957)
68.Skorokhod,A.V.:Someremarksontherandommeasures.Visn.Kyiv.Univ.Astron.Matem. Mekh. 1,no.1,105–111(1958)
69.Skorokhod,A.V.:LimittheoremsforMarkovprocesses.Teor.Veroyatnost.iPrimenen. 3, no.3,217-264(1958)
70.Skorokhod,A.V.:Certainproblemsofthestatisticsofstochasticprocesses.Bull.KievUniv. 2,no.1,91–109(1959)
71.Skorokhod,A.V.:Alimittheoremforindependentrandomvariables.Dokl.Akad.Nauk SSSR 133,34–35(1960)
72.Skorokhod,A.V.:Onaclassofdiffusionequationswithdiscontinuouscoefficients.Visn. Kyiv.Univ.Matem.Mekh. 1,76–83(1960)
73.Skorokhod,A.V.:OnaproblemofthestatisticsofGaussianprocesses.DopovidiAkad. NaukUkrain.RSR 9,1167-1170(1960)
74.Skorokhod,A.V.:Onthedifferentiabilityofmeasureswhichcorrespondtostochasticprocesses.II.Markovprocesses.Teor.Verojatnost.iPrimenen. 5,no.1,45-53(1960)
75.Skorokhod,A.V.:AlimittheoremforMarkovchains.DopovidiAkad.NaukUkrain.RSR 11,Ser.A,1408–1411(1961)
76.Skorokhod,A.V.:AdditivefunctionalsofaBrownianprocess.Teor.Verojatnost.iPrimenen. 6,no.4,430-439(1961)
77.Skorokhod,A.V.:Densityofthetransitionprobabilitiesofthesolutionofstochasticequations.Visn.Kyiv.Univ.Matem.Mekh. 4,32–36(1961)
78.Skorokhod,A.V.:Somelimittheoremsforadditivefunctionalsofasequenceofsumsof independentrandomvariables.Ukrain.Mat.Zh. 13,no.4,67-78(1961)
79.Skorokhod,A.V.:Stochasticequationsfordiffusionprocesseswithboundaries.I.Teor.Verojatnost.iPrimenen. 6,no.3,287-298(1961)
80.Skorokhod,A.V.:Onintegro-differentialequationsconnectedwiththesolutionsofstochasticequations.DopovidiAkad.NaukUkrain.RSR 7,Ser.A,854858(1961)
81.Skorokhod,A.V.:Ontheexistenceandtheuniquenessofthesolutionsofstochasticdiffusion equations.Sibirsk.Mat.Zh. 2,no.1,129–137(1961)
82.Skorokhod,A.V.:Ontheexistenceofsolutionsofstochasticequations.DopovidiAkad. NaukUkrain.RSR 9,Ser.A,1119-1122(1961)
83.Skorokhod,A.V.:Stochasticequationsfordiffusionprocesseswithboundaries.II.Teor. Verojatnost.iPrimenen. 7,no.1,5-25(1962)
84.Skorokhod,A.V.:AlimittheoremforhomogeneousMarkovchains.Teor.Verojatnost.i Primenen. 8,no.1,67-75(1963)
85.Skorokhod,A.V.:OnhomogeneouscontinuousMarkovprocesseswhicharemartingales. Teor.Verojatnost.iPrimenen. 8,no.4,379-390(1963)
86.Skorokhod,A.V.:BoundaryconditionsforcertainMarkovprocesses.Teor.Verojatnost.i Primenen. 9,no.4,644-654(1964)
87.Skorokhod,A.V.:Branchingdiffusionprocesses.Teor.Verojatnost.iPrimenen. 9,no.3, 492-497(1964)
88.Skorokhod,A.V.:Onmultiplicative,uniformlycontinuousfamiliesofrandomstochastic operators.Dokl.Akad.NaukSSSR 159,no.2,283-285(1964)
89.Skorokhod,A.V.:Absolutecontinuityofafamilyofmeasuresdependingonaparameter. Ukrain.Mat.Zh. 17,no.5,129-135(1965)
90.Skorokhod,A.V.:Constructivemethodsofthedeterminationofstochasticprocesses.UspekhiMat.Nauk. 20,no.3,67–87(1965)
91.Skorokhod,A.V.,Slobodenyuk,N.P.:Limitdistributionsforadditivefunctionalsofasequenceofsumsofindependentidenticallydistributedlatticerandomvariables.Ukrain.Mat. Zh. 17,no.2,97-105(1965)
92.Skorokhod,A.V.,Slobodenyuk,N.P.:Limittheoremsforrandomwalks.I.Teor.Verojatnost. iPrimenen. 10,no.4,660-671(1965)
93.Skorokhod,A.V.:Onabsolutecontinuityofinfintelydivisibledistributionsundertranslations.Teor.Verojatnost.iPrimenen. 10,no.3,510-518(1965)
94.Skorokhod,A.V.:Onthequantityoftheinformationthatistransformedbynon-linearsystemwithinnernoises.Prob.Trans.Inform. 1,no.2,101–107(1965)
95.Skorokhod,A.V.:AsymptoticbehaviorofcertainfunctionalsoftheBrownianmotion. Ukrain.Mat.Zh. 18,no.4,60-71(1966)
96.Gikhman,I.I.,Skorokhod,A.V.:Densitiesofprobabilitymeasuresinfunctionspaces.UspekhiMat.Nauk 21,no.6,83-152(1966)
97.Skorokhod,A.V.,Slobodenyuk,N.P.:Limittheoremsforrandomwalks.II.Teor.Verojatnost.iPrimenen. 11,no.1,56-67(1966)
98.Skorokhod,A.V.:LocalstructureofcontinuousMarkovprocesses.Dokl.Akad.NaukSSSR 169,no.1,38-41(1966)
99.Skorokhod,A.V.:LocalstructureofcontinuousMarkovprocesses.Teor.Verojatnost.i Primenen. 11,no.3,381-423(1966)
100.Skorokhod,A.V.:Nonlineartransformationsofrandomprocesses.Kibernetika(Kyiv) 1, 34–40(1966)
101.Skorokhod,A.V.:Nonlineartransformationsofprobabilisticmeasuresinfunctionspaces. Dokl.Akad.NaukSSSR 168,no.6,1269-1271(1966)
102.Skorokhod,A.V.:HomogeneousMarkovprocesseswithoutdiscontinuitiesofthesecond kind.Teor.Verojatnost.iPrimenen. 12,no.2,258-278(1967)
103.Skorokhod,A.V.,Taratuta,A.S.:Influenceofthediffusionofcarriersinthebasedomainof transistorsonspectraldensityofnoisecurrent.Semi-conductedtechniquesandmicroelectronics 2,50–53(1967)
104.Skorokhod,A.V.:Onthedensitiesofprobabilitymeasuresinfunctionalspaces.In:LeCam, L.M.,Neyman,J.(eds.)Proc.VBerkeleySympos.onMath.Statist.andProb.,Vol.2:ContributionstoProbabilityTheory,Part1,pp.163–182.UniversityofCaliforniaPress,Berkeley,Calif.(1967)
105.Gatun,V.P.,Skorokhod,A.V.:AnapplicationofthetheoryofMarkovchainstodifference equations.DopovidiAkad.NaukUkrain.RSR 10,Ser.A,880-883(1968)
106.Ezhov,I.I.,Skorokhod,A.V.:AcertaininversionformulafortheFouriertransforminHilbert space.Teor.Verojatnost.iPrimenen. 14,no.1,122-127(1969)
107.Gatun,V.P.,Skorokhod,A.V.:DifferenceequationsandMarkovchains.Ukrain.Mat.Zh. 21,no.3,301-311(1969)
108.Ezhov,I.I.,Skorokhod,A.V.:Markovprocesseswhicharehomogeneousinthesecondcomponent.I.Teor.Verojatnost.iPrimenen. 14,no.1,3-14(1969)
109.Skorokhod,A.V.:AnoteonGaussianmeasuresinBanachspace.Teor.Verojatnost.iPrimenen. 15,no.3,519-520(1970)
110.Skorokhod,A.V.:AnoteonunbiasedestimatesoftheparametersofGaussiandistributions inHilbertspace.Teor.Verojatnost.iMat.Statist. 3,195-199(1970)
111.Skorokhod,A.V.:AdmissibleshiftsofmeasuresinHilbertspace.Teor.Verojatnost.iPrimenen. 15,no.4,577-598(1970)
112.Skorokhod,A.V.:AremarkonhomogeneousMarkovprocesseswithdiscretecomponent. Teor.Verojatnost.iMat.Statist. 1,216-221(1970)
113.Ibramkhalilov,I.Sh.,Skorokhod,A.V.:AremarkonthedeterminationofthemeanforGaussiandistributionsinHilbertspace.Teor.Verojatnost.iMat.Statist. 1,86-89(1970)
114.Ryzhov,Yu.M.,Skorokhod,A.V.:Homogeneousbranchingprocesseswithafinitenumber oftypesandwithcontinuouslychangingmass.Teor.Verojatnost.iPrimenen. 15,no.4, 722-726(1970)
115.Skorokhod,A.V.:SurfaceintegralsandtheGreenformulainHilbertspace.Teor.Verojatnost.iMat.Statist. 2,172-175(1970)
116.Skorokhod,A.V.:Positiveadditivefunctionalsofahomogeneousprocesswithindependent increments.Teor.Verojatnost.iMat.Statist. 4,171-178(1971)
117.Dzhafarov,K.M.,Nasirova,T.I.,Skorokhod,A.V.:Thelimitofacertainprocesswithsemiindependentincrements.Teor.Verojatnost.iMat.Statist. 5,51-56(1971)
118.Skorokhod,A.V.:ErgodictheoremforgeneralMarkovprocesseswithdiscreteinterference ofthechance.Teor.Verojatnost.iPrimenen. 17,no.4,795-796(1972)
119.Skorokhod,A.V.,Yadrenko,M.I.:AbsolutecontinuityofmeasuresthatcorrespondtohomogeneousGaussianfields.Teor.Verojatnost.iPrimenen. 18,no.1,30-43(1973)
120.Skorokhod,A.V.:Atheoremonthecontinuityofarandomfunctiononacompactsetin Hilbertspace.Teor.Verojatnost.iPrimenen. 18,no.4,809-811(1973)
121.Nasirova,T.I.,Skorokhod,A.V.:Thedistributionofcertainfunctionalsofaprocesswith semi-independentincrements.Ukrain.Mat.Zh. 25,no.3,400-405(1973)
122.Skorokhod,A.V.:AremarkonconsistentestimatesofaGaussiandistribution.Teor.Verojatnost.iPrimenen. 19,no.1,195-198(1974)
123.Mitropol’skii,Ju.A.,Gusak,D.V.,Skorokhod,A.V.,Turbin,A.F.:Asymptoticmethodsin probabilisticproblems.Ukrain.Mat.Zh. 27,no.4,471-476(1975)
124.Skorokhod,A.V.:Aremarkonsquareintegrablemartingales.Teor.Verojatnost.iPrimenen. 20,no.1,199-202(1975)
125.Skorokhod,A.V.:Onageneralizationofthestochasticintegral.Teor.Verojatnost.iPrimenen. 20,no.2,223-238(1975)
126.Skorokhod,A.V.:Regularfamiliesofdistributions,discriminationofhypotheses,andconsistentestimates.Teor.Verojatnost.iMat.Statist. 12,138-149(1975)
127.Nasirova,T.I.,Skorokhod,A.V.:Functionalsofprocesseswithsemi-independentincrements.Theoryofrandomprocesses,“NaukovaDumka”,Kyiv 3,80-84(1975)
128.Prokopenko,L.N.,Skorokhod,A.V.:Onflowsofeventsinsemi-orderedspaces.Teor.Verojatnost.iPrimenen. 21,no.3,537-547(1976)
129.Prokopenko,L.N.,Skorokhod,A.V.:Ontheexistenceofasolutionforthegeneralmartingaleproblem.Teor.Verojatnost.iMat.Statist. 14,137-147(1976)
130.Shatashvili,A.D.,Skorokhod,A.V.:OntheabsolutecontinuityofaGaussianmeasureunder anonlineartransformation.Teor.Verojatnost.iMat.Statist. 15,139-151(1976)
131.Skorokhod,A.V.:Onarepresentationofrandomvariables.Teor.Verojatnost.iPrimenen. 21,no.3,645-648(1976)
132.Skorokhod,A.V.:Operatormartingalesandstochasticsemigroups.Theoryofrandomprocesses, 4,86-94,Kyiv:Nauk.Dumka(1976)
133.Nasirova,T.I.,Skorokhod,A.V.:Aclassofjumpprocesseswitharetainingbarrier.Teor. Verojatnost.iMat.Statist. 16,75-88(1977)
134.Skorokhod,A.V.:LinearandalmostlinearfunctionalsonameasurableHilbertspace.Teor. Verojatnost.iPrimenen. 23,no.2,397-402(1978)
135.Akhmedova,G.M.,Skorokhod,A.V.:Thedistributionoffunctionalsofsomeprocesseswith independentincrementswithadelayingboundary.Ukrain.Mat.Zh. 31,no.1,54-62(1979)
136.Nasirova,T.I.,Skorokhod,A.V.:Asymptoticbehavioroftheprocessesinaschemeofinventorycontrol.Teor.Verojatnost.iMat.Statist. 23,130-135(1980)
137.Skorokhod,A.V.:Measurabilityofstochasticprocesses.Teor.Verojatnost.iPrimenen. 25, no.1,140-142(1980)
138.Nasirova,T.I.,Skorokhod,A.V.:Anergodictheoremforaclassofprocessesconstructed fromsumsofindependentrandomvariables.Teor.Verojatnost.iMat.Statist. 22,135-145 (1980)
139.Skorokhod,A.V.:Stochasticdifferentialequationsdependingonaparameter.Teor.Verojatnost.iPrimenen. 25,no.4,675-682(1980)
140.Nasirova,T.I.,Skorokhod,A.V.:Limittheoremsforsomeclassesofrandomprocessesthat areconnectedwithsemi-Markovwalks.Teor.Verojatnost.iMat.Statist. 25,125-139(1981)
141.Skorokhod,A.V.:Onrandomlyinteractingparticles.(SummaryofthereportontheKyiv ShevchenkoUniversityseminarinprobabilityandmathematicalstatistics).Teor.Verojatnost.iPrimenen. 26,no.1,210-212(1981)
142.Skorokhod,A.V.:Operatorstochasticdifferentialequationsandstochasticsemigroups.UspekhiMat.Nauk 37,no.6,157-183(1982)
143.Skorokhod,A.V.:Demi-groupesstochastiquesdeHilbert-Schmidtetequationsdifferentielleslineairesstochastiques.C.R.Acad.Sci.ParisSer.IMath. 296,no.22,929-932(1983)
144.Skorokhod,A.V.:Productsofindependentrandomoperators.UspekhiMat.Nauk 38,no.4, 255-280(1983)
145.Skorokhod,A.V.:Theoriegeneraledesdemi-groupesstochastiqueslineaires.C.R.Acad. Sci.ParisSer.IMath. 297,no.3,199-202(1983)
146.Skorokhod,A.V.:Stochasticsemigroups.UspekhiMat.Nauk 41,no.1,218(1986)
147.Skorokhod,A.V.:TopologicallyrecurrentMarkovchains.Ergodicproperties.Teor.Verojatnost.iPrimenen. 31,no.4,641-650(1986)
148.Sarafyan,V.V.,Skorokhod,A.V.:Dynamicalsystemswithfastswitchings.Teor.Verojatnost. iPrimenen. 32,no.4,658-669(1987)
149.Ibramkhalilov,I.Sh.,Skorokhod,A.V.:Distinguishingsimilarhypothesesonacorrelation functionforaGaussiansequence.DopovidiAkad.NaukUkrain.RSR 9,Ser.A,7-11(1988)
150.Skorokhod,A.V.,Stepakhno,V.I.:AgeneralizationofHermitepolynomials.Ukrain.Mat. Zh. 42,no.11,1524-1528(1990)
151.Skorokhod,A.V.,Stepakhno,V.I.:AcentrallimittheoremforHermitepolynomialsinindependentGaussianvariables.Ukrain.Mat.Zh. 42,no.12,1681-1686(1990)
152.Skorokhod,A.V.,Stepakhno,V.I.:Aclassofdistributionsandrelatedtheoremsonlarge deviations.DopovidiAkad.NaukUkrain.RSR 7,36-40(1991)
153.Skorokhod,A.V.:Dynamicalsystemsundertheinfluenceoffastrandomperturbations. Ukrain.Mat.Zh. 43,no.1,3-21(1991)
154.Skorokhod,A.V.,Stepakhno,V.I.:Someempiricalcharacteristicsofthemultidimensional normaldistribution.Teor.Verojatnost.iPrimenen. 36,no.2,386-391(1991)
155.Skorokhod,A.V.: σ algebrasofeventsonprobabilityspaces.Congruenceandfactorization.Teor.Verojatnost.iPrimenen. 36,no.1,127-137(1991)
156.Skorokhod,A.V.:Infinitesystemsofrandomlyinteractingparticles.RandomOper.StochasticEquations 1,no.1,1-13(1993)
157.Dynkin,E.B.,Kuznetsov,A.V.,Skorokhod,A.V.:Branchingmeasure-valuedprocesses. Probab.TheoryRelatedFields 99,no.1,55-96(1994)
158.Salehi,H.,Skorokhod,A.V.:Onasymptoticbehaviorofsolutionsofthewaveequations perturbedbyafastMarkovprocess.UlamQuart. 2,no.4,40–56(1994)
159.Skorokhod,A.V.:OntransformationsofWienerspace.J.Appl.Math.StochasticAnal. 7, no.3,239-246(1994)
160.Skorokhod,A.V.:OntheL ´ evymeasureformeasure-valuedbranchingprocesses.Teor. Imovir.Mat.Stat. 52,154-160(1995)
161.Maidanyuk,R.Ya.,Skorokhod,A.V.:OnaclassofMarkovprocessesongraphs.Teor. Imovir.Mat.Stat. 52,112-116(1995)
162.Hoppensteadt,F.C.,Salehi,H.,Skorokhod,A.V.:RandomlyperturbedVolterraintegral equationsandsomeapplications.StochasticsandStochasticsReports 54,no.1–2,89-125 (1995)
163.Levental,Sh.,Skorokhod,A.V.:Anecessaryandsufficientconditionforabsenceofarbitrage withtameportfolios.Ann.Appl.Probab. 5,no.4,906-925(1995)
164.Khas’minskii,R.Z.,Skorokhod,A.V.:Parameterestimationfromindirectobservations. ProblemyPeredachiInformatsii 32,no.1,70–81(1996)
165.Hoppensteadt,F.C.,Salehi,H.,Skorokhod,A.V.:AnaveragingprinciplefordynamicalsystemsinHilbertspacewithMarkovrandomperturbations.StochasticProcess.Appl. 61,no. 1,85-108(1996)
166.Jacod,J.,Skorokhod,A.V.:JumpingMarkovprocesses.Ann.Inst.H.Poincar ´ eProbab. Statist. 32,no.1,11-67(1996)
167.Hoppensteadt,F.C.,Salehi,H.,Skorokhod,A.V.:MarkovchainwithsmallrandomperturbationswithapplicationstoBacterialGenetics.RandomOper.StochasticEquations 4,no. 3,205-227(1996)
168.Salehi,H.,Skorokhod,A.V.:OnasymptoticbehaviorofoscillatorysolutionsofoperatordifferentialequationsperturbedbyafastMarkovprocess.Probabilisticengineeringmechanics 11,no.4,251–255(1996)
169.Skorokhod,A.V.:Ontheregularityofmany-particledynamicalsystemsperturbedbywhite noise.J.Appl.Math.StochasticAnal. 9,no.4,427-437(1996)
170.Daletskii,Yu.L.,Salehi,H.,Skorokhod,A.V.:Anapplicationoforthogonalandbiorthogonal systemstofilteringproblems.Teor.Imovir.Mat.Stat. 56,169-176(1997)
171.Hoppensteadt,F.C.,Salehi,H.,Skorokhod,A.V.:Discretetimesemigrouptransformations withrandomperturbations.J.Dynam.Different.Equat. 9,no.3,463–505(1997)
172.Skorokhod,A.V.:Measure-valueddiffusion.Ukrain.Mat.Zh. 49,no.3,458-464(1997)
173.Skorokhod,A.V.:Measure-valueddiffusion.TheoryStoch.Processes 3,no.1–2,7–12 (1997)
174.Portenko,M.I.,Salehi,H.,Skorokhod,A.V.:Onoptimalfilteringofmultitargettracking systemsbasedonpointprocessesobservations.RandomOper.StochasticEquations 5,no. 1,1-34(1997)
175.Levental,Sh.,Skorokhod,A.V.:Onthepossibilityofhedgingoptionsinthepresenceof transactioncosts.Ann.Appl.Probab. 7,no.2,410-443(1997)
176.Portenko,M.I.,Salehi,H.,Skorokhod,A.V.:Optimalfilteringintargettrackinginpresence ofuniformlydistributederrorsandfalsetargets.RandomOper.StochasticEquations 6,no. 3,213-240(1998)
177.Portenko,M.I.,Salehi,H.,Skorokhod,A.V.:Filteringinmultitargetsystemswithdiscrete timeobservations.RandomOper.StochasticEquations 7,no.3,207-214(1999)
178.Skorokhod,A.V.:Oninfinitesystemsofstochasticdifferentialequations.MethodsFunct. Anal.Topology 5,no.4,54-61(1999)
179.Skorokhod,A.V.:Onrandomlyperturbedlinearoscillatingmechanicalsystems.Ukrain. Mat.Zh.,no.9,1294–1303(2000)
180.Portenko,M.I.,Salehi,H.,Skorokhod,A.V.:Onlargedeviationsfordynamicalsystems randomlyperturbedbyafastMarkovprocess.TheoryStoch.Processes 6,no.1–2,125–135 (2000)
181.Skorokhod,A.V.:Onrandomlyperturbedmechanicalsystemswithtwodegreesoffreedom. RandomOper.StochasticEquations 9,no.1,1-22(2001)
182.Ibramkhalilov,I.Sh.,Skorokhod,A.V.:Onlargedeviationsformulasforquadraticfunctions ofGaussianrandomvariables.RandomOper.StochasticEquations 9,no.1,23-28(2001)
183.Skorokhod,A.V.:OnStochasticDifferentialEquationsinaConfigurationSpace.Georgian MathematicalJournal 8,no.2,389–400(2001)
184.Skorokhod,A.V.:QuasistablegradientandHamiltoniansystemswithapairwiseinteraction randomlyperturbedbyWienerprocesses.RandomOper.StochasticEquations 8,no.1,1-22 (2001)
185.Skorokhod,A.V.:QuasistablegradientandHamiltoniansystemswithapairwiseinteraction randomlyperturbedbyWienerprocesses.J.Appl.Math.StochasticAnal. 16,no.1,45-67 (2003)
186.Skorokhod,A.V.:Oncontactprocessesincontinuum.Infin.Dimens.Anal.QuantumProbab. Relat.Top. 9,no.2,187-198(2006)
187.Skorokhod,A.V.:HomogeneousMarkovchainsincompactspaces.TheoryStoch.Processes 13,no.3,80-95(2007)
AnatoliiVolodymyrovychSkorokhod
Abriefbiographicaloutline 1
AnatoliiVladimirovichSkorokhodwasbornSeptember10,1930inthetownof Nikopol,Dnipropetrovskregion(inthepast,Ekaterinoslavskayaprovince)inafamilyofteachers.HisfatherVladimirAlexseevichSkorokhodwasofUkrainian peasantfamily.Hismother,NadezhdaAndreevna(maidennameAristova)came fromahighlyeducatedRussianfamily.Herparentsweredoctors.
AnatoliispenthischildhoodinsouthernUkraine.Hisparentstaughtinrural schoolsaroundNikopol.In1935(ayearafterthebirthofhisyoungerbrother Valery)theysettledinasmallminingtownofMarganetz.Familylifewasalwaysverymodest,becausethesalariesofruralteacherswereverysmall.Anatolii’s childhoodtookplaceduringtheverydifficult1930s:ruinaftertheRevolutionand theCivilWarof1919-1922,collectivizationofpeasants,dispossession,exile,and hunger.Skorokhodfamilywasalsoaffectedbythehardtimes:hispaternalgrandfather,aformerroyalnavy northernArkhangelsk
sailor,wasexpropriatedandexiledtothelogginginthe region.
Anatolii’sfather,VladimirAlexseevich,taughtmathematics,physicsandastronomyprimarilyinhighschool.Greatteacher,erudite,hehadasharpanalyticalmind withagreatsenseofhumorthatsometimesgrewintoironyandsarcasm.Hewas endowedwithleadershipabilities.Inadditiontoteachinghealsoheldadministrativepositions:headteacherandschoolprincipal.Despitetheauthoritative,difficult character,hewashighlyrespectedbypeople.VladimirAlexseevich,knowingwell
1 Intheworkonthisbiography,IusedmyownknowledgeaboutA.V.Skorokhod’slife,the autobiographicalbookbyhismother,N.A.Skorokhod,therecollectionssharedwithmebyhis brother,Valerii,thememberoftheUkrainianNationalAcademyofSciences,andotherrelatives, aswellastherecollectionsofAnatolii’sclassmates,friends,students,colleagues,andalsosome articleswrittenbyhim,hisinterviewsandpersonalcorrespondencewithhismotherandme.The photographsgivenbelowbelongtothearchiveofA.V.Skorokhod’sfamily.
SpringerInternationalPublishingSwitzerland20161 A A Dorogovtsev et al (eds ), SelectedWorks
theessenceoftheSovietregime,wasalwayscriticalofit.Inthefamily,father’s authoritywasunquestionable.Anatoliiinheritedfromhisfatheraninquisitive,analyticalmind,andacriticalattitudetowardseverything.Hisfatherplayedamajor roleinthechoiceofhisoldestson’sprofession.Followinghisfather’sadvice,AnatoliidecidedtoapplytotheFacultyofMechanicsandMathematicsofKievState Universityaftercompletionofhighschool.
Anatolii’smother,NadezhdaAndreevna,startedasanelementaryschoolteacher, latertaughtRussianandUkrainianliterature,history,musicandsinging,aswell asmathematics.NadezhdaAndreevnahadmanydifferenttalents.Shewasagood musician:playedthepianoandtheguitar,andsang.Shegavemusicandsinging lessons.Shehadavividdramatictalentfromayoungageparticipatinginamateur theatricals.NadezhdaAndreevnaalsohadgoodwritingskills.Boastinganexcellentstyle,shewrotescripts,stories,wonderfulpoemsfortheoccasions.Nadezhda Andreevnainheritedfromherparentsandbroughtintoherownfamilydeeprespect forpeople,regardlessoftheirsocialstatus,self-sacrificingdevotiontoherwork,and awiseattitudetochildren.Shewasveryresponsive,alwaysreadytohelp.Humble, tolerant,abletoendurehardshipwithoutcomplaintsandtakingallthehardestparts onherself.Sheknewhowtoloveandforgive.Hersonsinheritedloveofliterature,poetry,andclassicalmusicfromtheirmother.Herimpactonthesons’spiritual worldwasprofound.
Anatoliienteredelementaryschoolattheageofseven.HisstudieswereinterruptedbyWorldWarII.ThepartofUkrainewheretheSkorokhodswerelivingwas occupiedbytheenemiesshortlyafterthebeginningofthewarinthesummerof
A. V. Skorokhod with his mother, Nadezhda Andreevna, and his brother, Valery (Kovel, 1947)
1941.Duetoanaggravatedillnessofhisfather,familywasunabletoevacuate.For awhilethechildrenattendedopenedUkrainianschool.Howevertheprogramsof 6thand7thgradesAnatoliihadtostudyathomewithhisparents.
Thepost-waryearsintheSouthernUkrainewereyearsofpoorharvestandin 1946,tryingtoescapefromthehunger,thefamilymovedtoliveinKovelatownat Volyn,intheWesternregionofUkraine,wherethefatherwasofferedapositionof aschoolprincipal.Anatoliiwenttostudyinthe9thgrade,intheschoolwherehis parentstaught.
YearsoflivinginKovel,studyingin9thand10thgrades,weretheyearsofgrowingforAnatolii,andtheformationofhischaracter.Bygraduation,hehadalready awell-establishedpersonalitywithhisownviewoftheworld.
ThefearofpunishmentintheformofdeportationtoSiberiaforstayingduring thewarintheterritoryoccupiedbytheenemy,lifeintheVolynregion,wherethe SovietPowercameonlyin1939,whentheterritorywascededtotheSovietUnion bytheMolotov-RibbentropPact,andwhereSkorokhodwitnessedcollectivization undertakenin1947broughtinformationthatdidnotleaveillusionsaboutthenature oftheSovietregime.Later,whenAnatoliienteredKievStateUniversityin1948,he wasamazedathowmanypeoplestillbelievedtheofficialpropaganda.
A. V. Skorokhod at the time of graduating from the Kyiv University (Kyiv, 1953)
Handsome,athleticallybuiltyoungmanwithdark,wavyhair,thebestmathematicianinhisschool,andthesonoftheschoolprincipal,Anatoliiwasaprominent figureinhissurrounding.Hewasadebater,brave,fearless,easytogetintoafight withhispeers,provinghisrightnessortopunishtheunworthiness.Healsoloved dancingandpartyingwithfriends.ItwasimportantforAnatoliitobeawinnerin gameslikechessandcheckers,aswellassports.Hewasagoodfriend,always helpinghispalswiththeirassignmentsinmathematicsandphysics.Fromthevery beginningofhislifeinKovelAnatoliigothisfirstteachingexperience.Hegave lessonsinmathematicsandothersubjects.Parentsofstudentspaidforthelessons withfood,anditwasagreathelptosupportthefamily.
StudyingwaseasyforAnatolii,withoutanyapparenteffort.Hewasexcellent inallsubjects,andeveryoneofhisteachersthoughtthathis/herparticularsubject shouldbethesubjectofhisfutureprofession.Anatoliihimselfhadfirstthoughtof becomingamilitarypilot,thenanavigator(thislastonemayhadbeeninfluenced bythestoriestoldinhischildhoodbyhisgrandfather,whousedtobeasailor,and alsobyreadingabouttravelers,andloveforastronomy).Anatoliidecidedtoapply totheOdessaNavySchool;howeverthemedicalcommissionfoundmyopiaandhis seadreamnevercametrue.
Aspirationsforthesky,thehorizon,theurgefordistantwanderingswerethereflectionsofaromantic,sensitive,impressionablesoulandatthesametimereflected theyouthfulSkorokhod’sextremism.Hisfatherwaswellawareofthis,andtrying toprotecthisson,didnotallowhimtoapplytotheUniversityofLviv.Vladimir AlexseevichknewthatonceinLviv,Anatoliiwouldbecomeamemberofanationalistorganization(Lvivatthattimewastheideologicalcenterofthenationalist movements),gethimselfintothehandsoftheKGBandbelost.
“YouaregoingtoKiev”–thatwasthedecisionofhisfather.Afterhisgraduation withagoldmedalfromhighschoolin1948,Anatoliisubmittedhisdocumentsto theKievStateUniversitynamedafterTarasShevchenko,andwasenrolledtothe FacultyofMechanicsandMathematics.
Despitethefactthathewasalwaysinterestedinmathematics,duringhisschool yearsAnatoliididnotfeelapredestinationtobecomeamathematician.
Laterinlife,evaluatinghisownchoice,Skorokhodsaidthatunderothercircumstanceshewouldstudyphilosophy,orengageinliterature,whichhewasalways veryfondof.However,studyinghumanitiesmeantdependenceontheSovietsystem andthesubordinationofthoughtstoideology.Mathematicsgavetheopportunityto dosomethingnew,whileremainingindependent.
Inthelate1940samongthestudentsoftheKievStateUniversitywererecenthigh schoolgraduatesaswellaspeoplewhodemobilizedfromthearmyafterthewar. Practicallyallwereverypoorandbadlydressed.Somestudentslivedinthedorm, othersrentedplaces.Howevertheirinterestinmathematicsandthedesiretolearn itwasgenuine.Therewerewonderfullecturersandactivelyworkingresearchers
intheFacultyofMechanicsandMathematics.InhisfreshmanyearSkorokhod wasgreatlyimpressedbythelecturesonmathematicalanalysisgivenbySimon IzrailevichZukhovitskii.AnatoliirepeatedmanytimesthatZukhovitskiiwasthe bestlecturerhehadeverheard.MathematicaltalentquicklydistinguishedAnatolii Skorokhodamongotherstudents.Hedidnotwritelecturenotes,sincetheproofs werecleartohim.Duringtheexams,heoftendemonstratedhisownproofsoftheoremsthatweredifferentfromthosepresentedinthelectures.Atthesametimehe remainedjustaniceguy,wholovedathletics,chess,andhadhiscircleoffriends. Mathematicscertainlywasapriorityforthoseyoungpeople;howevertherewas stillaplacefortheparties,movies,anddances.Later,hewroteaboutthattime“I graduallybegantounderstandsomethinginmathematics.”
DuringhisfirsttwoyearsattheUniversitySkorokhodwasinterestedinthetheoryoffunctionsofcomplexvariables,functionalanalysis.Thiswasnotsurprising becauseatthattimeattheInstituteofMathematicsinKievtherewereoutstanding specialistsonfunctionalanalysis:brothersMarkGrigorievichKreinandSelimGrigorievichKrein,andMarkAlexandrovichKrasnoselskiwhosestudentAnatoliiwas consideredforsometime.Howeverby1950,allofthemwereforcedtoleaveKiev intheongoingbattleagainsttheso-called“rootlesscosmopolitans”(thiswasthe officialnameofanti-Semetizm).ThustherewasnoavailablementorforAnatolii inthisfield.FollowingtheadviceofhisfriendVolodyaMikhalevich,Skorokhod decidedtospecializeinthetheoryofprobability.
Skorokhodbeganindependentscientificworkinhisstudentyears.Togetherwith hisfriendTolyaKostyuchenko(studentofG.E.Shilov),hewrotehisfirstpaper,On atheoremofN.K.Bary”,whichwaspublishedin“UspekhiMatematicheskihNauk” journalin1953.UnderthesupervisionofBorisVladimirovichGnedenko(atthattime Chairmanofthedepartmentofprobabilitytheory)andIosifIllichGikhman(then anassociateprofessorofthisdepartment)Anatoliistartedhisworkinprobability theory.AccordingtoSkorokhod,everythingheknewaboutstochasticprocessesat thattimehelearnedfromI.I.Gikhman.Theresultsofhisresearchofthelimit theoremsforstablelawswerepublishedintheleadingscientificjournalsofthe SovietUnion(“DocladyAcademiiNaukSSSR”and“Usp.Mat.Nauk”).Twoof thesearticleswerepublishedinEnglishtranslationin1961in“SelectedTranslations onMathematicalStatisticsandProbability”ofAmericanMathematicalSociety.At theendofhisstudentyearsbeganSkorokhod’sinvolvementintheresearchrelated tothefamousDonskerinvarianceprinciple.B.V.Gnedenko,whohadanunerring instinctforthepromisingareasofresearch,proposedSkorokhodtoreadthework ofM.Donsker“AnInvariancePrincipleforCertainProbabilityLimitTheorems”, publishedin“MemoirsAmer.Math.Soc.”,6(1951).Hesuggestedthatinthesame wayasitwasdonebyDonsker,theresultcanbegeneralizedtothecasewhere thelimitprocessisnotaWienerprocess,butastableprocessi.e.fromtheprocesses withcontinuoustrajectoriestoproceedtotheconsiderationofprocesseswithjumps. Havingbeguntoponderoverthisproblem,Skorokhodgotconvincedthatitisnot possibletoprovetheconvergenceinthesamewayasitwasdonebyDonsker.Later talkingabouthissearchforapproachestosolvingtheproblemshementionedsome
featureofhiswayofthinking,callingthisfeature“theconservatismofhismind”. Anatoliisaidthatoncehestartedthinkingaboutsomeproblem,hecontinuedtodo soforaslongasthesituationwasnotclearedupforhim.
FinishingUniversitywithcertainscientificachievements(Skorokhodwasbythat timetheauthoroffivearticles),Anatoliiwasanobviouscandidateforadmission intothegraduateschool.HoweverforthisitwasnecessarytogetarecommendationfromtheAcademicCounciloftheFacultyandthePartyCommunistcommittee. AndthenB.V.Gnedenkohadtoendurearealfight.Itwasrememberedthatduring WorldWarIISkorokhodstayedontheterritoryoccupiedbytheenemies.Therefore hewasaccusedofcomplicitywiththeinvaders.Theabsurdityofsuchaccusations wasevident,sinceSkorokhodlivedintheoccupiedterritorywhenhewas11-13 yearsold.B.V.Gnedenkoshowedthecourageandperseveranceinthefightforthe fateofhistalentedstudent.AsaresultofB.V.Gnedenko’seffortstherecommendationtothegraduateschoolhadbeenreceivedandAnatoliibecameagraduate studentoftheFacultyofMechanicsandMathematicsofKievStateUniversity.
B.V.GnedenkoatthattimehadtogotoworkinGermanyandarrangedwith AndreiNikolaevichKolmogorovthatKiev’s“threeheroes”Koroljuk(3rdyeargraduatestudent),Mikhalevich(2ndyeargraduatestudent)andSkorokhod(1styear graduatestudent)wouldstudyinMoscow.Thequestionofadvisorsatthegraduate schoolwasdecidedbyKolmogorov.AsaresultKoroljukandMikhalevichbecame graduatestudentsofKolmogorov,andSkorokhodbecameagraduatestudentof EugeneBorisovichDynkin.
AfterhisgraduationfromtheUniversity,in1953,Anatoliimarriedhisclassmate ValerieRodionova,adaughterofthewell-knownUkrainiangeologist,andacorrespondingmemberofUkrainianAcademyofSciencesS.P.Rodionov.Theyhadtwo children:adaughterTatianaandasonDmitri.BotharegraduatesoftheFaculty ofMechanicsandMathematicsofKievStateUniversity.Thismarriageendedin divorcein1975.
YearsofstudyinthegraduateschoolattheMoscowStateUniversityhadbeen aremarkableperiodinSkorokhod’slifeinmanyways.Atthattime(1950sof20th century)intheFacultyofMechanicsandMathematicsaroundthegreatteachers oftheoldergenerationgatheredabroadaudienceoftalentedyoungpeople,who sawtheirfutureintheserviceoffundamentalscience.A.N.Kolmogorovwasthe recognizedleaderoftheprobabilists,oneofthepillarsoftheworldscience.Itjust sohappenedthatalmostatthesametime,ingraduateschoolandinseniorlevelof undergraduateschooltherewerestudents(Moscovitesandpeoplefromthevarious republicsoftheformerSovietUnion)wholatercreatedhonorandgloryofthe SovietschoolofProbabilityTheory,whichisconsideredtobeoneofthestrongest andmostinfluentialintheworld.
Amongthisgroupofyoungmathematicians,AnatoliiSkorokhodwasdistinguishedbyhisindependenceintheresearchwork,courageandoriginalityofapproachestoproblemsolving.AnatoliiSkorokhod’spersonalityattractedattention:
thesignificanceofhisresultswassurprisingandthenaturalquestionwaswhichone of Kiev’s teachers had an influence on him. In his interview to E. B. Dynkin in 2004 Skorokhodsaidthat“itwasrathertheinfluenceofmathematicsitself,nottheparticularindividuals”.AccordingtoAnatoliithemainthingthathebenefitedfromin Moscow graduate school was the seminar of his advisor E. B. Dynkin, called “Analysis,AlgebraandProbabilityTheory”.Skorokhodattendedtheseminarfortwoyears and this significantly expanded his scientific horizons. E. B. Dynkin said that in fact hedidnotteachSkorokhodanything;insteadthemainobjectiveofhisseminarwas fortheparticipantstolearnfromeachother.
ThePhDthesisofSkorokhodcontaineddescriptionsofnewtopologiesinthe spaceoffunctionswithoutdiscontinuitiesofthesecondkindandtheapplication ofthesetopologiesforprovinglimittheoremsforprocesseswhosetrajectories havenodiscontinuitiesofthesecondkind.Donskerprinciplewasgeneralizedto thecasewhenthelimitprocessisageneralprocesswithindependentincrements. “Thistheorygotlogicalcompletenessandharmonythatdistinguishtheclassics”. Suchcharacteristicwasgiventotheseresultsinsomearticledevotedtotheanalysis of A V Skorokhod’s research In the proofs of the theorems was used the originalmethod,belongingtotheauthor,knownasthe“methodofasingleprobability space”.
AtthemeetingoftheMoscowMathematicalSocietyA.N.Kolmogorovgave atalkabout J -topologyofSkorokhod.In1956inthefirstvolumeofthejournal “TheoryofProbabilityanditsApplications”inissue2waspublishedthearticleby Kolmogorov“Skorokhod’sconvergence”dedicatedtothe J -topology,andinissue 3Skorokhod’sarticle“Limittheoremsforstochasticprocesses”.Thiswasagreat honorforarecentgraduate.Theimportanceoftheideasofaveryyoungmathematicianwasconfirmedbytheentirefuturedevelopmentofthetheoryofstochastic processes.Theterms“Skorokhodtopology”,“Skorokhodspace”and“Skorokhod metric”areincludedinallbasicbooksonthetheoryofstochasticprocesses.Internationalconferences“40YearsoftheSkorokhodtopology”in1996inChile,and
“SkorokhodSpace50yearson”in2007inKiev,Ukraine,weredevotedtothehistory ofthedevelopmentoftheseideas.
AfterfinishinggraduateschoolinJanuaryof1957Skorokhodgotapositionas lecturerintheDepartmentofMathematicalAnalysisofKievStateUniversity.In Mayof1957hedefendedhisdissertationandwasgiventhetitleofCandidateof Science.Infallof1957SkorokhodbecameanAssociateProfessor,andin1961he becameaProfessorofthesamedepartment.In1961inKievStateUniversitywas published the first book of A. V. Skorokhod called “Studies in the theory of random processes”,whichwasthebasisofhisdoctoraldissertation,defendedin1963.At thebeginningof1964attheInstituteofMathematicsoftheAcademyofSciencesofUkrainetheDepartmentofTheoryofRandomProcesseswasopenedand A. V. Skorokhod became a Head of this department. In the same year he was awarded the title of Professor. In 1967 A. V. Skorokhod was elected a corresponding member oftheUkrainianAcademyofSciences.
AfterSkorokhod’sreturnfromMoscowin1957hisfriendshipwithI.I.Gikhman began.Despitethe12-yearagedifference,andthedifferenceintemperamentsthey hadalotincommon.Brightmathematicaltalent,erudition,natural,inexhaustible interesttomathematicalresearch,knowledgeofphilosophyandhistory,loveofliterature,art,classicalmusic,andespeciallypoetrydefinedloyalfriendship,aswellas scientificcooperation,andafurtherlong-termandfruitfulco-authorship.Gikhman andSkorokhodtogetherwrotemanywidelyknownbooks:
A V Skorokhod and B V Gnedenko on the Vilnius conference (1960)
• Introductiontothetheoryofstochasticprocesses.–Moscow:Nauka,1965 (thesecondedition,1977).
• Stochasticdifferentialequations.–Kiev:NaukovaDumka,1968.
• Thetheoryofstochasticprocesses.Vol.1.–Moscow:Nauka,1971.
• Thetheoryofstochasticprocesses.Vol.2.–Moscow:Nauka,1973.
• Thetheoryofstochasticprocesses.Vol.3.–Moscow:Nauka,1975.
• Controlledrandomprocesses.–Kiev:NaukovaDumka,1977.
• Probabilitytheoryandmathematicalstatistics.–Kiev:VyshchaShkola,1979 (co-authorM.Yadrenko).
• Stochasticdifferentialequationsandapplications.–Kiev:NaukovaDumka, 1982.
Allofthesebooksweretranslatedinmanylanguagesandarewidelyknown.In 2004-2007,thefamousthree-volumeeditionwasre-publishedbySpringer-Verlag, Berlinintheseries“ClassicsinMathematics”.
played a prominent role in the development of Ukrainian, par- A. V. Skorokhod
ticularlyKievschoolofprobabilitytheory.Thescaleanddiversityofhisresearch andteachingactivitieswerestriking.Generationsofstudentsgrewuplisteningto hislectures,usinghisownwrittenorco-authoredtextbooksandmonographs.For 35yearsSkorokhodtaughtgeneralandspecialcoursesatKievStateUniversity. Helecturedinmathematicalanalysis,probabilitytheory,mathematicallogic,numbertheory,measurestheory,functionalanalysis,allmaincoursesonthetheoryof stochasticprocesses,suchasgeneraltheoryofstochasticprocesses,Markovprocesses,processeswithindependentincrements,thetheoryofstochasticdifferential equations A V Skorokhod had his own way of teaching special courses He told his studentsaboutthelatestachievementsinthetheoryofstochasticprocesses,formulatedandprovedownresultswhichwerenotevenpublishedyet.Proofsoftentook theformofimpromptu,allowinglistenerstofollowtheflowofthoughts,tobeparticipantsinthecreativeprocess.Ofcourse,suchparticipationwaspossibleonlyto themosttalentedstudents.ForthemlisteningtothelecturesofSkorokhodwasa realschool,introductiontoscientificworkinmathematics.
ThankstoA.V.SkorokhodthenationalseminaronprobabilitytheoryatKiev StateUniversity,whichheledsince1966,hasgainedcredibilityandrelevancenot only in Kiev, but also far beyond A V Skorokhod took the famous seminar of his teacher E B Dynkin at Moscow University as the example The speakers of the seminarweresometimesveryharshlycriticized,whichhadalwaysbeenconstructive andhelpedthemtoseetheflawsandthewaysforcorrectionsandimprovement. Theatmosphereandtheimportanceoftheseminarweredeterminedprimarilyby the personality of A. V. Skorokhod, his high scientific authority, the broadest mathematicalerudition,abilitytoinstantlypenetrateintotheessenceoftheproblem, toseeitsrelationshipwithotherresults,seetheprospectsofthedevelopmentof theideas.Skorokhodabsolutelydidnotallowverbiage,reasoningwithoutproof.At thesametime,hewasaverygenerous,readytoofferanewidea,tosuggestthe directionofresearch,tohelpwithexpertise.
A. V. Skorokhod supervised graduate students at the University, as well as at the InstituteofMathematics.Hewastheadvisorof56PhDstudents.AmonghisgraduatestudentswerenotonlyUkrainianstudents,butyoungscientistsfromIndia, China,Vietnam,EastGermany,Hungary,Nicaraguaandothercountries.Under his consulting were also written 17 doctoral theses. In 2004 E. B. Dynkin, speakingabouthis“genealogytree”notedwithsurprisethatabout400ofhis“children” (abouthalf),including“scientificgrandchildren”,belongtoabranchofSkorokhod. A. V. Skorokhod played a very important role in the development of the school of probabilitytheoryinaRepublicofAzerbaijan.Hisdirectformerstudentsand“scientificgrandchildren”areanactivelyworkingresearchgroup,honoringthememory oftheirteacher.
Anatoliiwasnotausualsupervisor.Hedidnotthinkitwasnecessaryoreven possibleto“press”thestudents,encouragingthemtoresearch,ortotheearlycompletionofthethesis.Howeverhisowncontinuous,veryintenseandcreativeresearch wasitselfaninspiringexampletoothers.
A V Skorokhod did a lot to raise the level of teaching elementary mathematics in Ukraineandforpopularizationofmathematicsamongschoolstudents.Hegave lecturesinelementarymathematicsontelevision,aswellasmathematicaltalks forschoolstudentsandteacherswhiletravelingtoplacesassociatedwiththelife offamousUkrainianmathematiciansM.V.Ostrogradsky,V.Ya.Bunyakovsky,and G.F.Voronoy.Hewrotepopularsciencebooksandtextbooks,andarticlesforpopularsciencejournals.SkorokhodwasaHeadoftheUniversityofYoungMathematicianswhichfunctionedinthe1970-1980syearsatKievInstituteofMathematics.
Everynewschoolyearinthisuniversitywasopenedbyhislecture.However, Skorokhodhadhisownopinionaboutthedevelopmentofsecondaryeducation.He believedthatthereisnoneedsinspecializedmathematicalschoolsbecauseregular secondaryschoolscouldfulfillsocietyrequirementsinmathematicians.However, heinsistedontheneedforthedevelopmentandimprovementofthehumanitarian componentinsecondaryeducation.
ForachievementsinhisresearchSkorokhodwasrewardedin1970Krylov’s PrizeofAcademyofScienceofUkraine,andtwicein1982andin2003hewas awardedtheUkrainianStatePrizeinScienceandTechnology.SinceJanuary1993 SkorokhodworkedattheDepartmentofStatisticsandProbabilityofMichiganState University.Intheearly90s,thePresidentofUkraineLeonidKravchukinaninterviewtothequestion“Howdoyoufeelaboutthefactthatscientistsareleaving Ukraine?”replied:“Positively,becausenowUkrainecannotfeedthem”.ThedecisiontoseekatemporaryjobinAmericawasdictatedsolelybyfinancialnecessity. AsalaryofaheadofthedepartmentatKievInstituteofMathematicsatthattime wasonly$12permonth.
Skorokhod’sresearchandteachingactivitieswhileworkingintheUnitedStates wereintenseanddiverse.Hegavelecturesforgraduatestudents,ledseminarsthat wereattendednotonlybygraduatestudents,butalsobyfacultyofthedepartment.
InAmericaSkorokhodusedtopreparehislecturesinadvanceandwrotethemas smallarticles,thencopiedanddistributedamongthestudentsbeforethelectures.
Hisspecialcourseswereonavarietyoftopicsofstochasticanalysis,suchas absolutecontinuityofmeasures,weakconvergenceofmeasures,theoryofmartingales,dynamicalsystemsundertheinfluenceofrandomperturbationsorclassical sectionsofthetheoryofstochasticprocesses,forwhichtherewerenosuitable, moderntextbooks.HealsowasanadvisoroffourPhDstudents.
AnessentialpartofSkorokhod’swork(andtime)wasdevotedtoworkingwith grants.Hefileddozensof“grantproposals”.Inthecasewhentheproposalwas approvedandthegrantwasreceiveditappearedthepossibilityforcooperationbetweenMichiganStateUniversityandtheDepartmentoftheTheoryofRandomProcessesatKievInstituteofMathematics.InfactmanyscolarsfromUkrainewere invitedtovisittheDepartmentofStatisticsandProbabilityofMichiganStateUniversityforscientificcollaboration.
Skorokhod’sfeelingsaboutlifeintheUnitedStatesweredeterminedbythefact thattheU.S.Constitutionproclaimstherightsandfreedomsofitscitizens,and existingpublicinstitutions,atallcostsactuallyprovideandprotecttheserightsand freedoms.“Americaisafreecountrywithfreepeople”saidandfeltAnatolii.
FromthefirstdaysofworkatMichiganStateUniversitySkorokhodstartedresearchcollaborationwithHabibSalehi(thenChairmanoftheDepartmentofStatisticsandProbability)andFrankHoppensteadt(thenDeanoftheCollegeofNatural Sciences).Thesubjectofresearchwastheinvestigationofdynamicsystemsundertheinfluenceofrandomperturbations.Theymetonaregularbasis,discussing whathadbeendoneandplansforfuturephases.Theresultsofthiscollaboration wereseveralpapersandabook“Randomperturbationsmethodswithapplications inscienceandengineering”publishedin2002bySpringer-Verlag.
SkorokhodalsoworkedwithothermembersoftheDepartmentofStatisticsand Probability.Thearticleonfinancialmathematics“Onthepossibilityofhedging optionsinthepresenceoftransactioncost”waswrittentogetherwithProfessor ShlomoLeventhal.Anotherarticle“AnapproachtothemartingaleproblemfordiffusionstochasticequationsinHilbertspace”SkorokhodwrotetogetherwithProfessorAtmaMandrakar.Thearticle“Parameterestimationfromindirectobservations” publishedin“ProblemsofInformationTransmission”,32(1996),waswrittenwith R.Z.Khasminskii,Skorokhod’sclosefriendfromMoscow,whoworkedatthattime atWayneStateUniversityinDetroit.Hisownresearch,whichAnatoliicalled“the workforthesoul”,wasdevotedtothestudyofsystemswithinfinitenumberofinteractingparticlesundertheinfluenceofrandomperturbations.Thisresearchwasdone incooperationwithY.G.Kondratiev(Bielefeld,Germany).Intensivecollaboration resultedinpublishedarticles,conferencepapers,andplansforthefutureresearch.
DuringhisworkatMichiganStateUniversitySkorokhodattendednumerous scientificconferencesheldintheUnitedStatesaswellasinEurope.Particularly
notableweresmallconferenceswheretheprobabilistsfromformerSovietUnion weregathered.Meetingoldfriends,sharingtheirmemories,talkingabouteverything,sharingcreativeplanswasagreatjoyforallparticipants.ProbabilisticcommunityfromtheformerSovietUnionintheUnitedStatesisveryrespectableand includes A. Wentzell, A. Kagan, N. Krylov, S. Molchanov, B. Rozovsky, M. Freidlin, R. Z. Khasminsky, and of course the patriarch and teacher E. B. Dynkin. In the USSkorokhodalsometwithPaulMalliavinandJeanJacod.Malliavinattendedthe conferenceinEastLansing,Michigan,andJacodatBowlingGreen,Ohio,where Skorokhodworkedduringthespringsemesterof1994.Organizingrepresentative conferencewasapartofhisduties.
IntheDepartmentofStatisticsandProbabilityatMichiganStateUniversityAnatoliihadbeentreatedwithgreatrespect.Hisarrivalatthedepartmentwasconsidered asabigsuccess.Indeed,sowell-knownnamegaveweighttothedepartmentand eventothewholeuniversity.In2000Skorokhodwaselectedamemberofoneof themajorAmericanAcademies–AmericanAcademyofArtsandSciences.
DespitehispermanentresidenceintheUnitedStates,Skorokhod’sconnection andcooperationwithKievInstituteofMathematics,hisdepartmentthere,andthe UkrainianAcademyofScienceshadneverbeeninterrupted.Hetooktohisheartall thatwasgoingoninKiev,tryingtohelpinallthewayshecould.SkorokhodparticipatedinallprobabilityconferencesthattookplaceinUkraineduringsummers. HeparticipatedattheconferencesheldinLviv,Chernivtsi,Sudak,andKiev.His lastappearancetookplacein2007inKievattheconferencededicatedtothe50th anniversaryoftheSkorokhodtopology.
InaninterviewSkorokhodwasaskedhowhefeltaboutsocialactivitiesofthe scientists.TothatAnatoliireplied:“Negative.Ibelievethatascientistshouldbea professional.”
However,underthecircumstancesthatrequireddemonstratingpersonalcourage andprovidingsupportbyhisauthoritytheactionsindefenseofthecivilrightsand freedomsofcitizens,hejoinedtheprotesters.IthappenedinAprilof1968when agroupof139scientists,writersandartists,workersandstudentswrotealetter totheleadersoftheformerUSSRexpressingtheirconcernregardingtherenewed practiceofclosedpoliticaltrialsoftheyoungpeoplefromthemidstoftheartistic andscientificintelligentsia.Anatoliisignedthisletteratthesuggestionofhisfriend, V.A.Vyshenskii,oneoffivepeople,whosignedthepetitionfirst.Participationinthis
event was natural for A.V. Skorokhod - a m an with a sense of dignity, courageous,
independent,whowasneverafraidofanyauthorityandcouldnotbeindifferentto theflagrantfloutingofcivilrightsinthecountry.However,heneversawhimselfasa professionalmemberofthedissidentmovement.Forhimparticipationinthesigning ofthisletterwasadeeplypersonaltestofwillandendurance.Skorokhodalways thoughtthatonedayhisinnerattitudewillleadtoaclashwiththeauthoritiesinthe faceoftheKGB.Itwasimportantforhimtotesthimselfandseehowhewould behaveinsuchcircumstancesasquestioning,arrest,andimprisonment.Takingpart inthesigningoftheletter,heexpectedsucheventsandwasreadyforthem.
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