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Граничні теореми для оцінок параметрів випадкових процесів і полів із довгою пам’яттю та їх уточнення: Автореф. дис… канд. фіз.-мат. наук / М.М. Шар

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EE?ANUeEEE OI?AA?NEOAO ?IAI? OA?ANA OAA*AIEA

OAeE 519.21

Oa?aiia Ieoaeei Ieoaeeiae/

A?AIE*I? OAI?AIE AeEss IOe?IIE IA?AIAO??A

AEIAAeEIAEO I?IOeAN?A ? IIE?A ?C AeIAAITH IAI’ssOOTH

OA ?O OOI*IAIIss

01.01.05 – oai??y eiia??iinoae oa

iaoaiaoe/ia noaoenoeea

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa o?ceei-iaoaiaoe/ieo iaoe

Ee?a – 1999

Aeena?oaoe??th ? ?oeiien.

?iaioo aeeiiaii ia eaoaae?? oai??? eiia??iinoae oa iaoaiaoe/ii?
noaoenoeee iaoai?ei-iaoaiaoe/iiai oaeoeueoaoo Ee?anueeiai oi?aa?neoaoo
?i. Oa?ana Oaa/aiea.

Iaoeiaee ea??aiee: aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?, EAIIAIEI
IEEIEA IEEIEAEIAE*, eaoaae?a oai??? eiia??iinoae oa iaoai. noaoenoeee
iao.-iao. o-oo Ee?anueeiai oi?aa?neoaoo ?i. Oa?ana Oaa/aiea

Io?oe?ei? iiiiaioe:

– aeieoi? o?c.-iao. iaoe, i?ioani? eaoaae?e iaoaiaoe/ii? noaoenoeee
AIEIA IEAENAIAe? AIAe??EIAE*, oaeoeueoao e?aa?iaoeee Ee?anueeiai
oi?aa?neoaoo ?i. Oa?ana Oaa/aiea;

– eaiaeeaeao o?c.-iao. iaoe NOAAEIAA IEAIA AIEIAeEIE??AIA, aeeeaaea/
Ee?anueeiai aeiiii?/iiai ?inoeoooo iaiaaeaeiaioo (AEIIAI).

I?ia?aeia onoaiiaa: ?inoeooo e?aa?iaoeee ?i. aeaaeai?ea A.I. Aeooeiaa
IAI Oe?a?ie.

Caoeno a?aeaoaeaoueny “21”/a?aiy 1999 ?ieo i 14 aiae ia can?aeaii?
niaoe?ae?ciaaii? a/aii? ?aaee Ae 26.00.137 i?e Ee?anueeiio oi?aa?neoao?
?iai? Oa?ana Oaa/aiea ca aae?anith:

252127, i.Ee?a – 127, i?iniaeo aeaae. Aeooeiaa,6, Ee?anueeee
oi?aa?neoao ?iai? Oa?ana Oaa/aiea, iaoai?ei-iaoaiaoe/iee oaeoeueoao.

C aeena?oaoe??th iiaeia iciaeiieoeny a a?ae?ioaoe? Ee?anueeiai
oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea (aoe. Aieiaeeie?nueea, 58, e. 10).

Aaoi?aoa?ao ?ic?neaiee “14” o?aaiy 1999 ?ieo

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee I.I. Iieey/oe

1

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. Caaea/a ioe?ithaaiiy eiao?oe??io?a ?aa?an?? ?
iaei??th c iniiaieo a iaoaiaoe/i?e noaoenoeoe? ?acii ?c caaea/ath
i?iaiicoaaiiy oa o?eueo?aoe??, ui iia’ycaii c iiaeeeainoyie canoinoaaiue
iaoiae?a ?aa?an?eiiai aiae?co ?acii ?c caaea/ath i?iaiicoaaiiy oa
o?eueo?aoe?? aeey aiae?co caeaaeiinoae a aeiiii?oe?, a?ieia??,
aaio?ceoe?, oaoi?oe? oa ?ioeo aaeocyo ciaiue. Inoaii?i /anii ao?oeeai
?icaeaa?oueny iai?yiie, a yeiio “aeiaaeeiaee ooi” ? noaoe?iia?iei
i?ioeanii c aeene?aoiei /e iaia?a?aiei /anii aai iaeii??aeiei
aeiaaeeiaei iieai. O aeiaaeeo, eiee i?ioean aai iiea ia? iaiaaeaiee oa
iaia?a?aiee niaeo? ?noioi? ?acoeueoaoe i?i aeanoeaino? ioe?iie
eiao?oe??io?a ?aa?an?? oa i?iaiicoaaiiy aeiaaeeiaeo i?ioean?a io?eiaee
A?aiaiaea?, ?icaiaeaoo, Oaiiai, Aiaea?nii, A??ee?iaeaea?, ?icaiia,
Oieaai, Aei?iaiaoeaa, Aoeaeea?i, Eica/aiei, Aieia, a aeey iie?a –
ssae?aiei, ?aaiia oa Eaiiaiei. O aeiaaeeo, eiee niaeo?aeueia u?euei?noue
i?ioeano aai iiey ia? ioe?, i?iaeaio ioe?ithaaiiy eiao?oe??io?a ?aa?an??
aeine?aeaeoaaee Aaeainoaaeo, Oieaai, ?anoeia, Naia?ia oa Oaeeo,
ssae?aiei.

Inoaii?i /anii cia/ia oaaaa i?eae?ey?oueny aeine?aeaeaiith noaoenoe/ieo
caaea/ aeey aeiaaeeiaeo ooieoe?e c neeueiith caeaaei?noth (aai aeaeaeith
iai’yooth aai neaaeei niaaeaiiyi ei?aeyoe??). Aiae?oe/ii oea icia/a?, ui
niaeo?aeueia u?euei?noue “aeiaaeeiaiai ooio” ia iaiaaeaia a ioe? aai
ei?aeyoe?eia ooieoe?y ia noiiaia. Oae? aoaeoe aeyaeee I??nii (1902) a
ano?iiii??, Nouethaeaio (1927) i?e aiae?c? o?i?/ieo aeaieo, Ni?n oa O?oe
(1938, 1956) i?e aeine?aeaeaii? n?euenueeiainiiaea?nueeeo a?aeiiinoae,
Eien oa Oaoinaiae (1948) i?e ?icaeyae? aeaieo oaenoeeueii? ?iaeaia???,
Oa?no (1951) i?e aea/aii? a?ae?ieia?/ieo aeaieo, Iaiaeaea?io oa Aaee?n
(1969) ia iniia? e?iaa?noe/ieo aeaieo, Oaniao oa ?aooa?? (1989) na?aae
iaoai?ieia?/ieo aeaieo, Aa?ai, Oa?iai oa Oaeeo (1992), A?ee?iaa?, Oaeeo,
Eaeeaiae, A?enii (1995) o oaeaeiioi?eaoe?eieo aeaieo. Oaeei /eiii,
eaeaoueny i?i no?eeee oaiiiai neeueii? caeaaeiino?, oa?aeoa?iee aeey
aaaaoueio oei?a aeaieo, yeee iio?aao? iaoaiaoe/ieo aeine?aeaeaiue. Oae?
aeine?aeaeaiiy i?iaiaeeeenue Oaeeo, Aeia?ooeiei oa Iaei?ii,
?icaiaeaooii, A??aeo?nii oa No?aa?e?nii, Eienii, Eaiiaieii oa ?ioeie.
Noaoenoe/i? ?acoeueoaoe o oe?e iaeano? iaeaaeaoue Aa?aio, Ethioo,
ssaeaeei?, Naia?iao oa Oaeeo, A??aeo?no oa No?aa?e?no, Eaiiaieo oa
?ioei. I?iaeaie, ?icaeyioo? o aeena?oaoe??, o caaaeaieo ?iaioao ia
aeine?aeaeoaaeenue.

?iaioa i?enay/aia, a iniiaiiio, aeine?aeaeaiith noaoenoe/ieo
oa?aeoa?enoee ioe?iie iaa?aeiieo eiao?oe??io?a ?aa?an?? aeiaaeeiaeo
iie?a ?c neeueiith caeaaei?noth ? i?iaeiaaeo? oa ocaaaeueith? aeaye?
aeine?aeaeaiiy, caaaeai? aeua.

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. Aeena?oaoe?y
aeeiiaia o ?aieao oaie “Aeine?aeaeaiiy ?aa?an?eieo iiaeaeae c neeueiith
caeaaei?noth”, yea i?iaiaeeoueny ia eaoaae?? oai??? eiia??iinoae oa
iaoaiaoe/ii? noaoenoeee Ee?anueeiai oi?aa?neoaoo ?i. O.A. Oaa/aiea.

Iaoa ?iaioe. Aeena?oaoe?y i?enay/aia aeine?aeaeaiith ioe?iie
eiao?oe??io?a ?aa?an?? aaonn?anueeeo oa iaaaonn?anueeeo aeiaaeeiaeo
iie?a c neeueiith caeaaei?noth.

Iaoiaee aeine?aeaeaiue. Iniiaiee iaoiae aeine?aeaeaiiy iieyaa? o
?icaeiaii? ooieoe?iiae?a a?ae iaeii??aeieo iie?a ii i?oiaiiaeueiei
nenoaiai iie?iii?a *aaeoiaa-A?i?oa c iiaeaeueoei aneiioioe/iei aiae?cii
oaeeo ?iceeaae?a, c aeei?enoaiiyi niaeo?aeueii? oai??? aeiaaeeiaeo
iie?a.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a. A aeena?oaoe??:

· Aeine?aeaeaii aaonn?anuee? oa iaaaonn?anuee? a?aie/i? ?iciiae?ee
ia?aii?iiaaieo ioe?iie iaeiaioeo eaaae?ao?a eiao?oe??io?a ?aa?an??
aeiaaeeiaeo i?ioean?a ?c neeueiith caeaaei?noth o aeiaaeeo aeene?aoiiai
/ano;

· Aeine?aeaeaii a?aie/i? ?iciiae?ee eaac??iaanieo ioe?iie eiao?oe??io?a
?aa?an?? aeiaaeeiaeo i?ioean?a ?c neeueiith caeaaei?noth o aeiaaeeo
iaia?a?aiiai /ano, cie?aia aeiaaaeaii, ui eaac??iaani? ioe?iee ca iaaieo
oiia iathoue oaeo ae aneiioioe/io aeenia?n?th, ye ? ioe?iee iaeiaioeo
eaaae?ao?a;

· Noi?ioeueiaaii oa aeiaaaeaii oai?aie ?aaeoeoe??, ye? aeicaieythoue
caiaeeoe aeine?aeaeaiiy iiaeaeae c iaia?a?aiei /anii aei aeine?aeaeaiiy
iiaeaeae c aeene?aoiei /anii ? iaaiaee;

· Ioe?iaii oaeaee?noue ca?aeiino? aei ii?iaeueiiai ?iciiae?eo ioe?iie
iaeiaioeo eaaae?ao?a eiao?oe??io?a ?aa?an?? aeiaaeeiaeo i?ioean?a ?c
neeueiith caeaaei?noth, ui niinoa??aathoueny ia noa??;

· Ciaeaeaii ioe?ieo oaeaeeino? ca?aeiino? aei ii?iaeueiiai ?iciiae?eo
ioe?iie iaeiaioeo eaaae?ao?a eiao?oe??io?a ?aa?an?? aeiaaeeiaeo iie?a ?c
neeueiith caeaaei?noth, ui niinoa??aathoueny ia eoe?.

I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoao?a. ?acoeueoaoe, io?eiai? a
aeena?oaoe??, iathoue oai?aoe/iee oa?aeoa?. Aiie iiaeooue aooe
canoiniaai? o aaeocyo ciaiue, ye? aacothoueny ia noaoenoeoe?
aeiaaeeiaeo iie?a, cie?aia a aeiiii?oe?, iaoai?ieia??, aaio?ceoe?,
noaoenoe/i?e ?aae?io?ceoe? oa aeayeeo ?ioeo iaeanoyo.

Iniaenoee aianie caeiaoaa/a. ?acoeueoaoe /aoaa?oiai, i’yoiai
i?ae?icae?e?a ia?oiai ?icae?eo oa o?aoueiai i?ae?icae?eo ae?oaiai
?icae?eo io?eiai? aeena?oaioii naiino?eii, a ?aooa ?acoeueoao?a
iaea?aeai? aeena?oaioii ca iaoiaeieia?/ii? i?aeo?eiee iaoeiaiai
ea??aieea.

Ai?iaaoe?y ?iaioe. ?acoeueoaoe, io?eiai? a aeena?oaoe??,
aeiiia?aeaeenue ia iaoeiaiio nai?ia?? eaoaae?e oai??? eiia??iinoae
oa iaoaiaoe/ii? noaoenoeee Ee?anueeiai oi?aa?neoaoo ?iai? Oa?ana
Oaa/aiea, ia Nueii?e Iiaeia?iaei?e Iaoeia?e Eiioa?aioei? iiaii
aeaaeaiiea I.I. E?aa/oea (14-16 o?aaiy 1998 ?., i.Ee?a, no.530-532
Iaoa?iaeia Eiioa?aioei?), ia Iiaeia?iaei?e iaoeiai-i?aeoe/i?e
eiioa?aioei? “I?iaeaie aeiiiii/ii? oai?i? i i?aeoeee a no/anieo oiiaao”
(22-24 ea?oiy 1998 ?., Oaoiieia?/iee oi?aa?neoao Iiae?eey o
i.Oiaeueieoeueeiio, no.230-232 cai?ieea iaoeiaeo i?aoeue eiioa?aioei?),
ia Ia?o?e Anaoe?aeinuee?e iaoeiai-i?aeoe/i?e eiioa?aioei? “Aeiiiii/ii
iaoiaee i iiaeaei a aeiiii?oe?: oai??y i i?aeoeea”(2-4 ethoiai 1998 ?.,
Eueaianueeee aea?aeaaiee oiiaa?neoao ii. ?aaia O?aiea, no.37-41
iaoa?iaeia eiioa?aioei?), ia nueii?e A?eueithnuee?e Eiioa?aioe?? c
oai??? eiia??iinoae oa ia 22-io ?a?iiaenueeiio c’?cae? noaoenoee?a
(“Iaoa??aee eiioa?aioe??”, no. 409-410).

Ioae?eaoe??. Iniiai? ?acoeueoaoe aeena?oaoe?? iioae?eiaai? a
?iaioao [1-3].

No?oeoo?a ? ianya aeena?oaoe??. ?iaioa neeaaea?oueny ?c anooio, aeaio
?icae?e?a, aeniiae?a, nieneo e?oa?aoo?e ? aeaio aeiaeaoe?a. Caaaeueiee
ianya ?iaioe 102 noi??iee.

Aaoi? aeneiaeth? ue?o iiaeyeo nai?io iaoeiaiio ea??aieeia?, i?ioani?o
I.I.Eaiiaieo ca iino?eio oaaao aei ?iaioe.

CI?NO ?IAIOE

O anooi? aei ?iaioe iaa?oioiaaii aeooaeuei?noue oaie, iaaaaeaii
nei?i/aiee iaeyae aeine?aeaeaiue, iia’ycaieo c oaiith aeena?oaoe??, a
oaeiae aeeeaaeai? iniiai? ?acoeueoaoe ?iaioe o nei?i/aiiio aa??aio?.

A ?iaio? i?eeiyoi oaeo ioia?aoe?th: ia?oa oeeo?a icia/a? iiia? ?icae?eo,
ae?oaa – iiia? i?ae?icae?eo, o?aoy – iiia? oai?aie aai eaie, iiia?e
oi?ioe acyoi a e?oae? aeoaeee.

O ia?oiio ?icae?e? aeine?aeaeothoueny a?aie/i? ?iciiae?ee eiao?oe??io?a
?aa?an?? aeiaaeeiaeo i?ioean?a ?c neeueiith caeaaei?noth.

Iaoae i?e t0, 1, …, T-1 niinoa??aa?oueny aeiaaeeiaa iine?aeiai?noue
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(1)aea g(t), tZ1, – a?aeiia iaaeiaaeeiaa ooieoe?y ?aa?an??
(o?aiae), – iaa?aeiiee eiao?oe??io ?aa?an??, yeee o?aaa ioe?ieoe ca
?acoeueoaoaie niinoa?aaeaiue, (t), tZ1, – noaoe?iia?iee aeiaaeeiaee
i?ioean (ooi) c E(t)=0 (ia iaia’yceiai aaonn?anueeee). On? aeiaaeeia?
aaee/eie ?icaeyaeathoueny ia iiaiiio eiia??iiniiio i?inoi?? (, F, P).
Ooi ia? aeae (t)=G((t)), aea G(x) L2(R1, (u)du), (u) – u?euei?noue
?iciiae?eo noaiaea?oii? aaonn?anueei? aaee/eie. Oiae? G(x) aeiionea?
?icaeiaiiy o aeaeyae? ?yaeo

picscalex100010009000003930200000300150000000000050000000902000000000400
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00000b0200000000050000000c028004e01f1200000026060f001a00ffffffff00001000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00006b0008000000320ae0028307010000006b0015000000fb0280fe0000000000009001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(1a)
aea Hk(u): k0 – iie?iiie *aaeoiaa-A?i?oa.Oiiaa 1 G(u) ia? a?i?o?a ?aia
m0, oiaoi C1=C2=…=Cm-1=0, Cm0.

Oiiaa 2 c(t), tR1, – ae?eniee aei??iee iaia?a?aiee o n.e. noaoe?iia?iee
aaonn?anueeee i?ioean c Ec(t)=0, Ec2(t)=1 oa ei?aeyoe?eiith ooieoe??th
R(t)=cov(c(0), c(t))=L(|t|)/|t|, 00, – iaa?ae’?iia
ooieoe?y, ui iia?eueii ci?ith?oueny ia iane?i/aiiino? (oiaoi aeey
aeia?eueiiai >0 limtL(ts)/L(t)=1).

Oiiaa 2 ?iaeieo? ?nioaaiiy niaeo?aeueii? u?eueiino? fc()=fc(||),R1,
i?ioeano c(t), tR1, i?e/iio fc() i?e 0, oiaoi ia? i?noea neeueia
caeaaei?noue i?ioeano c(t), tR1.

Oiiaa 3 Iaoae g(t), tR1, – ae?enia aei??ia aeiaeaoia i?e t>0 ooieoe?y,
i?e t>0 (1) g(t) (t), i?e/iio 1) aeey m1 ?c oiiae 1 i?e 00, i?e/iio T>0 WT2=[0,T]g2(t)dt(0,),
00, ? eaac??iaania ioe?iea ? IIE
ia?aiao?a iathoue iaeiaeiaee a?aie/iee ii?iaeueiee ?iciiae?e.

O inoaiiueiio i?ae?icae?e? ia?oiai ?icae?eo ?icaeyaea?oueny ioe?ithaaiiy
eiao?oe??io?a ?aa?an?? noaoe?iia?ieo aeiaaeeiaeo i?ioean?a c aeiaaith
iai’yooth o aeiaaeeo iaia?a?aiiai /ano.

?icaeyiaii noaoe?iia?iee aeiaaeeiaee i?ioean x(t), t[0,T] c Ex(t)=0.

Oiiaa 13 Aeey aeayeiai ae?eniiai /enea H(1/2,1), yea ? o?aaa aoaea
ioe?ieoe, niaeo?aeueia u?euei?noue i?ioeano x(t) i?e 0 ia? aeae
f()L(1/)1-2H.

Aeey ooieoe?? ?iciiae?eo F() i?ioeano x(t) ?icaeyiaii Aeey ooieoe??
?iciiae?eo F() i?ioeano x(t) ?icaeyiaii ioe?ieo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Oiiaa 14 m=2m/T, 1/m+m/T i?e T.

Aeey aeiaaeeiaiai i?ioeano x(t) ?icaeyiaii eiai i?aaenoaaeaiiy Aieueaea
x(t)=[0,+]c(t-s)(ds), aea – i?ioean c i?oiaiiaeueieie i?e?inoaie
(E|(ds)|2=ds), c(t), tR, – aeayea aei??ia ooieoe?y oaea, ui
R|c(t)|2<.oiiaa picscalex100010009000003910100000300150000000000050000000902000000000400 ad0d010000002c0008000000320aa001340801000000290008000000320aa001e7050100 a001f70b01000000640008000000320aa001740601000000640008000000320aa001ad02 picscalex1000100090000039d0200000300150000000000050000000902000000000400 c71101000000290008000000320aa001ec0e01000000280008000000320aa001630d0100 cf1701000000640008000000320aa0019c16010000006c0008000000320aa001ae130100 a001dc1501000000640008000000320aa001790f01000000640008000000320aa001230b a0019e12010000003d000a00000026060f000a00ffffffff01000000000010000000fb02 ca oiia i t picscalex1000100090000038e0100000300150000000000050000000902000000000400 fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d010200 f001010008000000320aed03640001000000460008000000320abd016400010000004600 f00101000300000000000000000000000000 aeey ia h q>0 ?icaeyiaii ioe?ieo

picscalex1470100090000030a0200000400150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02c00620111200000026060f001a00ffffffff00001000
0000c0ffffffbfffffffe01000007f0600000b00000026060f000c004d61746854797065
0000e00009000000fa02000008000000000000002200040000002d010000050000001402
5002ba090500000013025002990e09000000fa0200001000000000000000220004000000
2d010100050000001402a004b006050000001302a0043f1015000000fb0280fe00000000
00009001000000000402001054696d6573204e657720526f6d616e00009c040000002d01
020008000000320a00057a10010000002c0008000000320a2d06d90a020000006c6e0800
0000320a2d06e80901000000320008000000320afd03a30d01000000290008000000320a
fd03530b01000000280008000000320a9d03a00a01000000880008000000320aad01060e
01000000290008000000320aad01f10a01000000280008000000320a4d013e0a01000000
880008000000320ab0024808020000006c6e08000000320a000587040100000031000800
0000320aa004bf0001000000880015000000fb0220ff0000000000009001000000000402
001054696d6573204e657720526f6d616e00009c040000002d01030004000000f0010200
08000000320a60053b02010000002c0015000000fb0280fe000000000000900101000000
0402001054696d6573204e657720526f6d616e00009c040000002d01020004000000f001
030008000000320a2d06470c01000000710008000000320afd03400a0100000046000800
0000320aad01830b01000000710008000000320aad01de0901000000460008000000320a
0005440001000000480015000000fb0220ff000000000000900101000000040200105469
6d6573204e657720526f6d616e00009c040000002d01030004000000f001020008000000
320a5d04be0c010000006d0008000000320a0d02210d010000006d0008000000320a6005
7f0201000000710008000000320a60058c01010000006d0010000000fb0280fe00000000
00009001010000020002001053796d626f6c0002040000002d01020004000000f0010300
08000000320afd03e00b010000006c0008000000320aad01430c010000006c0010000000
fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d010300
04000000f001020008000000320a00057f05010000002d0008000000320a000581030100
00003d000a00000026060f000a00ffffffff01000000000010000000fb02140009000000
0000bc02000000cc0102022253797374656d00cc040000002d01020004000000f0010300
0300000000000000000000000000

i?e/iio ine?eueee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, oi iiaeia
aaaaeaoe, ui q(0,1).

Oai?aia 7 Ca oiia 13-15 i?e q(0,1) oa T
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A ae?oaiio ?icae?e? ?icaeyaea?oueny ca?aei?noue aei ii?iaeueiiai caeiio
?iciiae?e?a IIE eiao?oe??io?a ?aa?an?? aeiaaeeiaeo i?ioean?a ?c
neeueiith caeaaei?noth.Oiiaa 16 Iaoae aeey aeiaaeeiaiai i?ioeano (t)
aeeiio?oueny oiiaa 2, i?e/iio ?nio? niaeo?aeueia u?euei?noue f(),R1, yea
iiiioiiii niaaea? aei 0 ia [0,) aeey aeayeiai 0>0.

Oiiaa 17 Aeey G(u), uR1, EG((0))=0, EG2((0))0 01 – n-aei??iee aaee?ae?a i?ino??, Rn – iaiaaeaia iioeea
iiiaeeia, ui i?noeoue ioeue, a (T) – ia?ac iiiaeeie i?e aiiioaoe/iiio
a?aeia?aaeaii? c oeaio?ii a ioe? oa eiao?oe??ioii aiiioao?? T>0.

?icaeyiaii ?aa?an?eio iiaeaeue aeaeo
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(4)aea g(x) – a?aeiia iaaeiaaeeiaa
ooieoe?y, yea ?ioaa?iaaia c eaaae?aoii ia Rn, – iaa?aeiiee ia?aiao?,
yeee o?aaa ioe?ieoe ia iniia? niinoa?aaeaiue (x), x(T) , a (x), xRn, –
iaeii??aeia aeiaaeeiaa iiea iioeaie c E(x)=0, E2(x)<. oiiaa iaoae xrn ae aei iaia o n.e. iaeii aaonn aeiaaeeiaa iiea ia eiia i c e e2 oa ei ooieoe b aea a n-aei noa sn-1 eg eg2 iie iaa aeae picscalex122010009000003b3020000030015000000000005000000090200000000040 a01000000440008000000320a03044a0801000000440010000000fb0280fe00000000000 a60022702010000003d0015000000fb0220ff00000000000090010000000004020010546 f6d616e00007e040000002d01010004000000f001020008000000320a6002791f0100000 a6002801002000000292908000000320a60021f0f01000000280008000000320a6002be0 d01000000280008000000320a60021a0c01000000290008000000320a6002b90a0100000 a6002751c01000000670008000000320a6002cb1701000000540008000000320a60020b1 . ui g>0 aeey x0 oa ?nio? ne?i/aiia a?aieoey

picscalex122010009000003490300000300150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02a00420211200000026060f001a00ffffffff00001000
0000c0ffffffbeffffffe02000005e0400000b00000026060f000c004d61746854797065
0000000109000000fa02000010000000000000002200040000002d010000050000001402
40028f0a0500000013024002c61105000000140240022314050000001302400230180500
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3d19010000007c0008000000320acd03ea17010000007c0008000000320acd032b140100
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08000000320aad01331101000000290008000000320aad01250f01000000280008000000
320aad01b20d01000000290008000000320aad01910b01000000280009000000320aa002
5d06030000006c696d0008000000320aa002490401000000290008000000320aa002e802
010000002c0008000000320aa002650101000000280015000000fb0220ff000000000000
9001000000000402001054696d6573204e657720526f6d616e0000c2040000002d010200
04000000f001010008000000320a2b031a0d01000000320010000000fb0280fe00000000
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08000000320aa002021f010000003e0008000000320ad703dc1a010000002d0008000000
320a2803611801000000f70008000000320aa302611801000000f70008000000320a2104
611801000000f80008000000320aaa01611801000000f60008000000320a28035d130100
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320aad01ca15010000002d0008000000320aa0022005010000003d0010000000fb02c0fd
0000000000009001000000020002001053796d626f6c0002040000002d01020004000000
f001010008000000320a2a03b70801000000f20008000000320a2a03a30901000000f200
10000000fb0220ff0000000000009001000000020002001053796d626f6c000204000000
2d01010004000000f001020008000000320a4304c20801000000440008000000320a4304
ae0901000000440008000000320a8003cf0701000000a50008000000320a8003e6060100
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fb0280fe0000000000009001010000020002001053796d626f6c0002040000002d010100
04000000f001020008000000320aa002d50101000000610015000000fb0220ff00000000
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010000006b0008000000320a8003430601000000540008000000320a0003b40001000000
6b0015000000fb0280fe0000000000009001010000000402001054696d6573204e657720
526f6d616e0000c2040000002d01010004000000f001020008000000320ad703041c0100
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01000000670008000000320aad01a30f02000000547608000000320aad014c0e01000000
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0f000a00ffffffff01000000000010000000fb021400090000000000bc02000000cc0102
022253797374656d00cc040000002d01020004000000f001010003000000000000000003
0f00,

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Eaaaaa ia noa??, (,F,P) – iiaiee aeiaaeeiaee i?ino??. Iaoae
(,x)=(x):RnR1 aeiaaeeiaa iiea.

Oiiaa 27 Iaoae aeiaaeeiaa iiea

picscalex100010009000003420100000200150000000000050000000902000000000400
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0a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02
000000cc0102022253797374656d00cc040000002d01000004000000f001010003000000
0000(6)aai o noa?e/ieo eii?aeeiaoao

picscalex100010009000003920100000200150000000000050000000902000000000400
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010000002c0008000000320ac001fa0701000000280008000000320a2a011c0701000000
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1c00010000007a0010000000fb0280fe0000000000009001000000020002001053796d62
6f6c0002040000002d01010004000000f001000008000000320ac001a00b010000002b00
08000000320ac0010f05010000003d000a00000026060f000a00ffffffff010000000000
10000000fb021000070000000000bc02000000cc0102022253797374656d00cc04000000
2d01000004000000f0010100030000000000niinoa??aa?oueny ia noa?? s(r), aea
(x)-iaeii??aeia ?cio?iiia iaia?a?aia o n.e. aeiaaeeiaa iiea c
picscalex100010009000003220100000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02000240081200000026060f001a00ffffffff00001000
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2f010100000068000a00000026060f000a00ffffffff01000000000010000000fb021000
070000000000bc02000000cc0102022253797374656d00cc040000002d01000004000000
f0010100030000000000 oa ei?aeyoe?eiith ooieoe??th
picscalex100010009000003ff0100000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02200260191200000026060f001a00ffffffff00001000
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0000500015000000fb0280fe0000000000009001000000000402001054696d6573204e65
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010000002c0008000000320a8001210c01000000280008000000320aea00430b01000000
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7720526f6d616e0000c1040000002d01000004000000f001010008000000320a80018617
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450008000000320a8001b00e01000000750008000000320a800171090100000075000800
0000320a8001060501000000450008000000320a8001460001000000420010000000fb02
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0000f001000008000000320a8001dc1501000000680008000000320a8001301301000000
680008000000320a8001bd0c01000000720008000000320a8001050b0100000068000800
0000320a8001a80701000000720008000000320a8001f00501000000680008000000320a
8001d80101000000720010000000fb0280fe000000000000900100000002000200105379
6d626f6c0002040000002d01000004000000f001010008000000320a8001fa1001000000
3d0008000000320a8001ba03010000003d000a00000026060f000a00ffffffff01000000
000010000000fb021000070000000000bc02000000cc0102022253797374656d00cc0400
00002d01010004000000f0010000030000000000 aea
picscalex100010009000003ee0100000400150000000000050000000902000000000400
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00000b0200000000050000000c02c00280161200000026060f001a00ffffffff00001000
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320008000000320a60027d0a01000000310010000000fb0280fe00000000000090010100
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0002da1401000000670008000000320a0002271101000000720008000000320a0002da0f
01000000720008000000320a0002c80c01000000720008000000320a0002b00901000000
720008000000320a00024c0001000000720010000000fb0280fe00000000000090010000
00020002001053796d626f6c0002040000002d01030004000000f001020008000000320a
0002a80e010000002d0008000000320a0002900b010000002b0008000000320a00025a07
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526f6d616e0000da040000002d01020004000000f001030008000000320a0002ba050100
0000790008000000320a00027a030100000078000a00000026060f000a00ffffffff0100
0000000010000000fb021000070000000000bc02000000cc0102022253797374656d00cc
040000002d01030004000000f0010200030000000000
picscalex1000100090000036a0100000300150000000000050000000902000000000400
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00000b0200000000050000000c024004000c1200000026060f001a00ffffffff00001000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 x=(1,u1), 10, u1s(1), y=(2,u2), 20, u2s(1).

Oiiaa 28 Iaoae g(x)L2 (s(r),(x)) oa aeey aeia?eueiiai r>0
s(r)g2(x)(dx)<. oiiaa iaoae aea x rn n>1, – iaeii??aeia ?cio?iiia
iaia?a?aia o n.e. aaonn?anueea iiea c E(x)=0, E2(x)=1 oa ei?aeyoe?eiith
ooieoe??th

R(x)= R(|x|)=L(|x|)/|x|, 0

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