⛣Ẹ㡯ࡢ࠶ࡿศᯞ㐣⛬ࡢ⤫ィⓗ㏲ḟゎᯒ
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ࡣࡌࡵ࡟
➨୍❶ ணഛ▱㆑࡜ㄽᩥ࡟౑⏝ࡉࢀࡓ୺せ࡞ᐃ⌮࡜ᛶ㉁ࡢࡲ࡜ࡵ
㸯 ᚤศ✚ศᏛࡢ▱㆑
ᩘิ࡜㛵ᩘyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
࣮࣐ࣜࣥ✚ศyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
ᗘㄽࡢᇶ♏
ᗘ✵㛫yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
☜⋡ኚᩘ࡟ࡘ࠸࡚yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
✚ศࡢ▱㆑yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
ᖹᆒyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
⊂❧ᛶ࡜࣐ࣝࢥࣇ㐃㙐yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
኱ᩘࡢἲ๎yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
┤✚ ᗘ࡜ࣇࣅࢽࡢᐃ⌮yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
࣐ࣝࢳࣥࢤ࣮ࣝࡢ⌮ㄽ
᮲௳௜ࡁᮇᚅ್yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
࣐ࣝࢳࣥࢤ࣮ࣝyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
୍ᵝྍ✚ศᛶyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
೵Ṇ᫬yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
㞳ᩓ᫬㛫࣐ࣝࢳࣥࢤ࣮ࣝyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
☜⋡㐣⛬
ࣈࣛ࢘ࣥ㐠ືyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
☜⋡✚ศ࡜ఀ⸨ࡢබᘧyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
%HVVHO 㐣⛬yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
➨஧❶ ༢୍ኚ㔞ศᯞ㐣⛬࡟ࡘ࠸࡚
ศᯞ㐣⛬ࡢẕ㛵ᩘ࡟ࡘ࠸࡚yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
ࣃ࣮࣡ࢩ࣮ࣜࢬศᕸyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
㸱⛣Ẹ㡯ࡢ࠶ࡿศᯞ㐣⛬ࡢ㏲ḟ☜⋡ẚ᳨ᐃࡢ⤖ㄽ
ࣔࢹࣝ࡜≉ᛶ್ࡢ᥎ᐃyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
᭱ᑬ᥎ᐃࡢࢩ࣑࣮ࣗࣞࢩࣙࣥ࡜⤖ᯝyyyyyyyyyyyyyyyyyyyyyyyyyyy
㏲ḟ☜⋡ẚ᳨ᐃyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
㏲ḟ☜⋡ẚ᳨ᐃࡢࢩ࣑࣮ࣗࣞࢩࣙࣥ࡜⤖ᯝศᯒyyyyyyyyyyyyyyyyy
➨୕❶ ⮫⏺್࡟ᑐࡍࡿศᯞ㐣⛬ࡢ㏲ḟ᳨ᐃ
ࣔࢹࣝࡢᇶᮏyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
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ᑬᗘ㛵ᩘ࡟ࡘ࠸࡚yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
)LVKHU ᝟ሗ㔞yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
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೵Ṇ᫬้㸦6WRSSLQJ7LPH㸧࡜㏲ḟ᳨ᐃ㸦6HTXHQWLDO 7HVW㸧
6WRSSLQJ7LPHࡢᐃ⩏yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
㔜せ࡞ᐃ⌮yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
ᐃ⌮ࡢド᫂yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
࣋ࢵࢭࣝ㐣⛬㸦%HVVHO3URFHVV㸧yyyyyyyyyyyyyyyyyyyyyyyyyyyy
ࢩ࣑࣮ࣗࣞࢩࣙࣥ࡜⤖ᯝ
ࢩ࣑࣮ࣗࣞࢩࣙࣥ⌮ㄽⓗ࡞ศᯒyyyyyyyyyyyyyyyyyyyyyyyyyyyy
ࢩ࣑࣮ࣗࣞࢩࣙࣥ⌮ㄽⓗ࡞ศᯒ⤖ᯝyyyyyyyyyyyyyyyyyyyyyyyyy
➨ᅄ❶ 6357 ᳨ᐃ :DOG ࡢ 6357 ᳨ᐃ
:DOG ࡢ 6357 ᳨ᐃࡢព࿡yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
㜈್ࡢỴᐃyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
ศᯞ㐣⛬ࡢ 6357 ᳨ᐃyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
ศᯞ㐣⛬࡟ᑐࡍࡿ 6357 ᳨ᐃࡢࢩ࣑࣮ࣗࣞࢩࣙࣥyyyyyyyyyyyyyyyyyyy
➨஬❶ ⛣Ẹ㡯ࡢ࠶ࡿ ኚ㔞ศᯞ㐣⛬
ࣔࢹࣝ࡟ࡘ࠸࡚ࡢゎㄝ
ࣔࢹࣝ࡜┤ឤࡢព࿡yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
ྛグྕࡢព࿡yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
FULWLFDOLW\SDUDPHWHU࡟ࡘ࠸࡚yyyyyyyyyyyyyyyyyyyyyyyyyyyy
⛣Ẹ㡯ࡢ࠶ࡿ ኚ㔞ศᯞ㐣⛬ࡢ᥎ᐃ
&/6 ࡟ࡘ࠸࡚yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
&/6 ࡼࡾ⛣Ẹ㡯ࡢ࠶ࡿ㸰ኚ㔞ศᯞ㐣⛬ࡢ᥎ᐃyyyyyyyyyyyyyyyyyyy
&/6 ࡼࡾ⛣Ẹ㡯ࡢ࠶ࡿ㸰ኚ㔞ศᯞ㐣⛬ࡢ᥎ᐃyyyyyyyyyyyyyyyyyyy
ࡢࢩ࣑࣮ࣗࣞࢩࣙࣥ
➨භ❶ ཧ⪃ᩥ⊩yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
2
ࡣࡌࡵ࠸࡟
ᮏㄽᩥࡢࢸ࣮࣐ࡣࠕ⛣Ẹ㡯ࡢ࠶ࡿศᯞ㐣⛬ࡢ⤫ィⓗ㏲ḟゎᯒ࡛ࠖ࠶ࡿࠋ
⛣Ẹ㡯ࡢ࠶ࡿศᯞ㐣⛬ࣔࢹࣝ࡟ᑐࡋ࡚ࠊ༢୍ኚ㔞࡜஧ኚ㔞ࡢ஧ࡘࢣ࣮ࢫࢆᐹ
ࡋࡓࠋ⤫ィⓗ࡞ᡭἲ࡛ࠊ⛣Ẹ㡯ࡢ࠶ࡿศᯞ㐣⛬ࣔࢹࣝ࡟ᑐࡋ㏲ḟゎᯒ⌮ㄽࢆཝ
ᐦ࡟ド᫂ࡋࡓ࠺࠼ࠊࢩ࣑࣮ࣗࣞࢩࣙࣥࢆ㏻ࡌࠊ୺ᙇࡍࡿほⅬࢆ┤ほⓗ㦂ドࡋࡓࠋ
➨㸯❶࡛ࡣࠊᮏㄽᩥ࡟౑ࢃࢀࡿ⌮ㄽ᰿ฎ࡜ࡋ࡚ᩘᏛⓗ࡞ᐃ⩏࡜ᐃ⌮ࢆࡲ࡜ࡵ
࡚࠾ࡿࠋᚤศ✚ศᏛࡢ⌮ㄽࠊ ᗘ✵㛫⌮ㄽࠊ࣐ࣝࢳࣥࢤ࣮ࣝࡢ⌮ㄽ࡜☜⋡㐣⛬
୺࡞ᇶ♏࡜࡞ࡿ▱㆑࡛࠶ࡿࠋ
➨㸰❶࡛ࡣࠊ༢୍ኚ㔞ࣔࢹࣝ࡟ࡘ࠸࡚┤ほⓗ࡞ゎㄝࡍࡿ࠺࠼ࠊ⛣Ẹ㡯ࡢ࠶ࡿ
ศᯞ㐣⛬ࡢ㏲ḟ☜⋡ẚ᳨ᐃࢆ⾜ࡗࡓࠋࡇࡢ㒊ศࡣ➹⪅ࡢಟኈ᫬௦ࡢ◊✲㢟┠࡛
࠶ࡿࠋศᯞ㐣⛬ࣔࢹࣝ z0
1, z n
zn1
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i 1
ࢆȣ࡜Ȣࢆ᥎ᐃࡋ࡚ࠊࢩ࣑࣮ࣗࣞࢩࣙࣥࡼࡾࣄࢫࢺࢢ࣒࡛ࣛ⾲♧ࡋࡓࠋࡑࡋ࡚
ᖐ↓௬ㄝȣ㸻㸯࡜ᑐ❧௬ㄝȣ㸼㸯ࢆタᐃࡋࠊ㏲ḟ☜⋡ẚ᳨ᐃࢆ࠾ࡇ࡞ࡗࡓࠋ⤫
P1 z1 , y1 㻸 zn , yn zn1 ィ㔞ࡣ㛵ᩘᑬᗘẚ L
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P0 z1 \1 㻸 zn , yn zn 1 °
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min ®t t 1: log 1 1 1 2
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࡟‶ࡓࡍ᫬้㹲ࢆྲྀࡾฟࡋࠊࢧࣥࣉࣜࣥࢢࢆࢫࢺࢵࣉࡍࡿࠋࡇࡢࡼ࠺࡞೵Ṇ᫬
้ࢆࢩ࣑࣮ࣗࣞࢩࣙࣥࡼࡾほ ࡉࢀࡓࠋࡑࡋ࡚ࡇࡢ᳨ᐃࡢ஧✀㢮ㄗࡾࢆࢩ࣑ࣗ
࣮ࣞࢩࣙࣥࡼࡾ⟬ฟࡋࡓࠋ⌮ㄽ್࡜኱య୍⮴ࡢ⤖ㄽࢆྲྀᚓࡋࡓ࠺࠼ࠊྛ≉ᛶ್
ࡢ≉ᚩࢆ᫂☜࡟ࡋࡓࠋ
➨㸱❶࡛☜⋡ẕ㛵ᩘ࡞࡝ᛶ㉁ࢆ฼⏝ࡋࠊ⛣Ẹ㡯ࡢ࠸࠶ࡿศᯞ㐣⛬ࣔࢹࣝ࡟ᑐ
ࡋ࡚㏲ḟゎᯒࢆ࠾ࡇ࡞ࡗࡓࠋࡑࡋ࡚⮫⏺್㹫㸻㸯࡟࠾࠸࡚㏲ḟ᳨ᐃࢩ࣑࣮ࣗࣞ
ࢩ ࣙ ࣥ ࡢ ࢆ ⾜ ࡗ ࡓ ࠋ ࡇ ࡢ 㒊 ศ ࡛ ࡢ ⌮ ㄽ ࡣ ,9%DVDZD 6HTXHWQWLDO
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㸦㸧 6HTXHWQWLDO (VWLPDWLRQ 7HVW )RU 7KH &ULWLFDOLW\ 2I
%UDQFKLQJ3URFHVVHV ࢆཧ⪃ࡋࡓࠋ6WRSSLQJ 7LPH ࢆᐃ⩏ࡍࡿࡓࡵࣇ࢕ࢩ࣮ࣕ
᝟ሗ㔞ࡢព࿡࡜ࣃ࣮࣡ࢩ࣮ࣜࢬࢆヲࡋࡃ⤂௓ࡋࡓࠋ.HLML1DJDL ࡢㄽᩥ࡛ᐃ⩏ࡉ
ࢀࡓᐃ⌮ࢆᩘᏛⓗ࡞ཝᐦ࡟ド᫂ࢆࡋࡓ࠺࠼ࠊ%HVVHO 㐣⛬ࡼࡾࢩ࣑࣮ࣗࣞࢩࣙࣥ
n

½
ࡢ⪃࠼᪉ࢆᥦ᱌ࡋࡓࠋ Tc inf ®n t 0; ¦ Z i 1 / Vˆ n2 t c 2 ¾ , c o f ࡛ᐃ⩏ࡉࢀࡿ࡜ࡁࠊ
i 1
¯
¿
ࡇࡢ᪉ἲ࡛ࡣࢧࣥࣉࣜࣥࢢࡢつᶍࢆ᝿ᐃࡋࡓࡽ %HVVHO 㐣⛬ࡢ▱㆑ࡼࡾࢧࣥࣉࣜ
ࣥࢢࡢ࡜ࡲࡿ᮲௳㹡ࢆᩘ㔞ⓗ࡟⪃ᐹ࡛ࡁࡿࠋࢩ࣑࣮ࣗࣞࢩ࡛ࣙࣥほ ࡉࢀࡓ
6WRSSLQJ7LPH ࡀ᝿ᐃࡋࡓࢧࣥࣉࣜࣥࢢࡢつᶍ኱య୍⮴࡛࠶ࡿࡢ࡛┠ⓗࢆᐇ⌧ࡋ
T
ࡓࠋࡉࡽ࡟⤫ィ್ c mˆ Tc 1 ࡢศᕸࢆࣄࢫࢺࢢ࣒࡛ࣛほ ࡉࢀࡓࠋ⌮ㄽⓗ࡛ࡣࡇ
ࡢ⤫ィ್ࡀᶆ‽ࣈࣛ࢘ࣥ㐠ື࡟ᚑ࠺ࡀࠊࢩ࣑࣮ࣗࣞࢩ࡛ࣙࣥ኱యṇつศᕸࡢࣄ
ࢫࢺࢢ࣒ࣛࡀฟ࡚࠸ࡓࡓࡵࠊࣔࢹࣝࡢ㏲ḟゎᯒ⌮ㄽࢆ㦂ドࡋࡓࠋ
➨㸲❶࡛ࡣ➨㸱❶ࡢゎᯒ⌮ㄽࡢ࠺࠼ 6357 ᳨ᐃࢆ⾜ࡗࡓࠋ༢୍ኚ㔞⛣Ẹ㡯ࡢ࠶
3
ࡿศᯞ㐣⛬࡟࠾࠸࡚ࠊ$EUDKD :DOG ᖺ࡛ࡀ㛤Ⓨࡋࡓ≉ᐃࡢ㏲ḟ௬ㄝ᳨ᐃࡢ
ᡭẁ࡛ 6357 ᳨ᐃࢆ⾜࠸ࠊࢩ࣑࣮ࣗࣞࢩ᳨࡛ࣙࣥᐃຊ࡟ࡘ࠸࡚⪃ᐹࡍࡿࠋ஧✀㢮
ㄗࡾࡢ⟬ฟ᪉ἲࢆ᫂☜ࡋࡓ࠺࠼ࠊࢩ࣑࣮ࣗࣞࢩ್ࣙࣥࢆィ⟬ࡋࡓࠋࡑࡋ࡚ࠊ
ᮏㄽᩥ࡛ࡣ 6357 ᳨ᐃࡢୖࠊ6WRSSLQJ 7LPH ࡢ೵Ṇ᮲௳࡟ྍኚ㔞ࡢ௜ࡃሙྜࡢࡶ
⪃࠼࡚ࠊࢩ࣑࣮ࣗࣞࢩࣙࣥࢆ⾜ࡗࡓࠋ⡆༢࡛ㄝ᫂ࡍࡿ࡜೵Ṇ᫬้ࡣ
Tac,b inf ^n t 1: log / cn a, b `
࡛ᐃ⩏ࡉࢀࡿࠋࡇࡢሙྜࠊᕼᮃ࡞ࢧࣥࣉࣜࣥࢢつᶍ
ࢆᚓࡿࡓࡵࠊࣔࢹࣝ≉ᛶ್࡛࠶ࡿ㹫ࡢࡳࡢຊ࡞ࡽ୙༑ศ࡛࠶ࡿ࡜ࡁࠊࡉࡽ࡟㹴
ࢆ⏝࠸࡚ࠊ⢭ᐦ࡞೵Ṇ᮲௳ࢆぢࡘࡅࡿࡇ࡜ࡀ࡛ࡁࡿࠋ≀⌮⣔ࡢ◊✲࡞࡝࡟ࡼࡃ
౑ࢃࢀࡿศᯞ㐣⛬ࣔࢹࣝ࡟ᑐࡋ࡚ࡣࠊ㹴ࡢᏑᅾࡀ᳨ᐃࡢ⢭ᗘࢆ኱ᖜ࡟ᨵၿ࡛ࡁ
ࡿ࡛ࡣ࡞࠸࠿࡜⪃࠼࡚࠾ࡿࠋ౛࠼ࡣᆅ㟈ࡢணሗࡸᶵᲔࡢࢥࣥࢺ࣮ࣟࣝ࡞࡝ࠋࡇ
ࡢ㒊ศ࡟㛵ࡍࡿ⌮ㄽࡣᮍᡂ⇍࡛࠶ࡿࡀ Qࠊ௒ᚋࡢ◊✲᪉ྥ࡜ࡋ࡚ດຊࡍࡿḧࡀ࠶
ࡿࠋ
ª X k ,1 º X k 1,1 ª [ k , j ,1,1 º X k 1,2 ª[ k , j ,2,1 º ª H k ,1 º
➨㸳❶࡛ࡣ஧ኚ㔞ࣔࢹࣝ «
» ¦ «
» « » ࡟ᑐࡋ࡚ࠊල
» ¦«
¬ X k ,2 ¼ j 1 ¬[ k , j ,1,2 ¼ j 1 ¬[ k , j ,2,2 ¼ ¬H k ,2 ¼
యⓗ࡞౛࡛ゎㄝࡋࡓୖ㏲ḟ᳨ᐃࢆࡋࡓࠋ
஧ኚ㔞ࣔࢹࣝࡢ೵Ṇ᫬้ၥ㢟ࢆࢩ࣑࣮ࣗࣞࢩࣙࣥࡼࡾ᥈⣴ࡋ࡚ࡳࡓࠋኚ㔞ࣔࢹ
ࣝࡢ೵Ṇ᫬้ࡢᐃ⩏ࠊ᮲௳タᐃ࡞࡝ࡢࡇ࡜࡟ࡘ࠸࡚ᥦ᱌ࡋࡓࠋ
ᮏㄽᩥ࡛ࣃ࣓࣮ࣛࢱࡢ᥎ᐃࠊ೵Ṇ᫬้ၥ㢟ࡢ᥈⣴ࠊ㏲ḟ᳨ᐃ࡞࡝ၥ㢟ࢆ⪃ᐹࡍ
ࡿࡇ࡜ࡼࡾࠊ⛣Ẹ㡯ࡢ࠶ࡿ༢୍ኚ㔞࡜ ኚ㔞ศᯞ㐣⛬࡟㛵ࡍࡿ≉ᛶࢆ᫂☜࡟ࡋ
ࡓࠋ㏲ḟゎᯒࡣ⤫ィⓗ᥎ ࡟ᚲせ࡞ࢹ࣮ࢱࢆᚲせ᭱ᑠ㝈ᚓࡿࡇ࡜࡛ࠊ᪩࠸᫬Ⅼ
࡛᮲௳ࢆ‶ࡓࡍࢧࣥࣉࣝࢆ⋓ᚓࡋࠊ᪩ᮇ㆙ሗࢩࢫࢸ࣒ࢆᥦ౪࡛ࡁࡿⅬ࡟࡞ࡿࠋ
㏲ḟゎᯒࡢ᪉ἲࡣ୍ኚ㔞ࣔࢹࣝ࠿ࡽ஧ኚ㔞ࣔࢹࣝ࡬ᣑᙇࡋ࡚ࠊ᪂ࡓ࡞⪃࠼࡛࠶
ࡿࠋ
⮬↛⏺࡛ࡣ⏕≀ࡢฟ⏕࡜Ṛஸࡀ⮬↛࡞ࡶࡢࡔࠋᏊ࡜ࡶࢆ⏕ࡳࡔࡋ⥆ࡅࡿࡢ࡛
≀✀ࡀ⤯⁛ࡏࡎࠊྛୡ௦ࡀ୍ࡘࡢ㞟ᅋ࡟࡞ࡿࠋ⛉Ꮫ࡛ࡣ㟁Ꮚಙྕࡢ㌿㏦ࡍࡿࡼ
࠺࡟ࠊ୍ᮏಙྕ※࠿ࡽฟ࡚࠸ࡃ㟁Ꮚࡀ㟁Ꮚᯈࢆ㏻ࡿ᫬ఱᮏࡢ᪂ࡋ࠸㟁Ꮚࢆ⏕ࡳ
ࡔࡍࠋᡃࠎࡣᏊࡢࡼ࠺࡞ᩘࡢቑ࠼᪉࡟Ꮚ࡜ࡶ࠿ࡽ┤ほⓗ༳㇟ࡀ࠶ࡿࠋᮏㄽᩥ࡛ࠊ
ᩘ⌮⤫ィࡢ᪉ἲࢆ㏻ࡌࠊࡇࡢቑ࠼᪉ࢆᐃ㔞࡟◊✲ࢆ⾜࠺ᚋࠊศᯞ㐣⛬࡟ᑐࡍࡿ
ㄆ㆑ࡀ῝ࡃ࡞ࡗࡓࠋ≉ᛶ್ࡢィ⟬ࡼࡾࠊศᯞ㐣⛬࡟ᚑ࠺࠶ࡿ㞟ᅋࡢつᶍࠊࡇࡢ
ඛ࡝࠺ࡢࡼ࠺࡟ኚ໬ࡍࡿࡢࡀ᫂ࡽ࠿࡟▱ࡗࡓࠋࡉࡽ࡟ࢩ࣑࣮ࣗࣞࢩࣙࣥࡀศᯞ
㐣⛬ࢆ㔞ⓗ࡟෌⌧ࡋࠊ⌮ㄽ᥎ᐃࡢ⤖ᯝࢆ᳨㦂ࡍࡿࡇ࡜ࡀฟ᮶ࡿࠋࢥࣥࣆ࣮ࣗࢱ
ᢏ⾡ࡢඛ㐍ᛶࢆᐇឤ࡛ࡁࡿࠋ
㞟ᅋつᶍࡢ⦰ᑠࡋ࡚⾜ࡗ࡚࡯ࡋ࠸࠿ࠊᣑ኱࠸࡚⾜ࡗ࡚࡯ࡋ࠸࠿ࠊྛศ㔝ࡢ◊✲
࡟ᑐࡋ࡚ᕼᮃࡶ␗࡞ࡿࠋ㏲ḟゎᯒࢆ⏝࠸࡚ࠊᕼᮃ᮲௳࡟‶ࡓࡍࢧࣥࣉࣜࣥࢢࢆ
᥈ࡍࡇ࡜ࡀ࡛ࡁࡿࠋࡇࡢ᥈ᰝࡀᐇ⌧࡛ࡁࡓࡽࠊᡃࠎࡢ♫఍⏕ά࡛ࡣ῝้ࡢᙳ㡪
ࢆ୚࠼ࡿࠋࡲࡓᮏㄽᩥࡢ౛ࢆゝ࠺࡜ࠊ㏲ḟゎᯒࡼࡾྛศ㔝ࡢ◊✲ཬࡪ♫఍⏕ά
࡟ᑐࡍࡿ✚ᴟ࡞ព⩏ࡀ࠶ࡿࠋࢥࣥࣆ࣮ࣗࢱࡣ஧ࡘࡢ㡿ᇦࢆࡘ࡞ࡂྜࢃࡏࠊ࠸ࢁ
࠸ࢁ࡞◊✲ᡂᯝࢆඹ㏻࡟฼⏝ࡍࡿࡼ࠺࡟࡞ࡗࡓࠋ
4
➨୍❶ ணഛ▱㆑࡜ㄽᩥ࡟౑⏝ࡉࢀࡓ୺せ࡞ᐃ⌮࡜ᛶ㉁ࡢࡲ࡜ࡵ
➨୍❶࡛ࡣࠊᮏㄽᩥ࡟㛵㐃࡛࠶ࡿᩘᏛࠊ⤫ィᏛࡢᐃ⩏࣭ᐃ⌮࣭࿨㢟࣭⿵㢟ࢆ
ࡲ࡜ࡵ࡚࠾ࡿࠋࡇࢀࡽࡢᇶ♏▱㆑ࡣຮᙉࡢ㔜せෆᐜࡢ୍ࡘ࡜ࡋ࡚ࠊ◊✲ࡢᇶ┙
࡜ ࡞ ࡿ ࠋ ḟ ࡢ ᮏ ࢆ ཧ ⪃ ࡋ ࡓ 㸸 %LOOLQJVOH\3> ཧ ⪃ ᩥ ⊩ @'DYLG
:LOOLDPV>ཧ⪃ᩥ⊩ @➟ཎᬊྖ㸦㸧>ཧ⪃ᩥ⊩ @ࠋ
㸯 ᚤศ✚ศᏛࡢ▱㆑
ᚤศ✚ศᏛࡣ ୡ⣖㡭࠿ࡽḟ➨࡟ࡑࡢຊᙉ࠸ጼࢆ⌧ࡋጞࡵࢽ࣮ࣗࢺ࣭ࣥࣛ࢖
ࣉࢽࢵࢶ࡟ࡼࡿయ⣔໬ࢆ⤒࡚௨ᚋᩘୡ⣖࡟ࢃࡓࡾ⮬↛⛉Ꮫࡢ᭱ࡶᇶᮏⓗ࡞㐨ල
࡜ࡋ࡚⏝࠸ࡽࢀࠊྠ᫬࡟ࡲࡓࡑࢀ⮬㌟ࡶ኱ࡁࡃⓎᒎࡋ࡚ࡁࡓே㢮ඹ㏻ࡢ㈈⏘࡛
࠶ࡿࠋᚤศ✚ศᏛࡢᑟ㛵ᩘ㛵ಀࠊ₞㏆ᒎ㛤ࠊᩘิࡢ཰᮰ࠊ࣮࣐ࣜࣥ✚ศ࡞࡝ࡢ
ၥ㢟ࡣ◊✲࡟ࡼࡃ⏝࠸ࡽࢀࡓࡢ࡛ࠊᩚ⌮ࡋ࡚࠾ࡃࠋ
ᩘิ࡜㛵ᩘ ᐃ⩏ ᩘิ ^an ` n 1, 2,... ࡀ㹬Ѝ҄ࡢ᫬ a R ࡟཰᮰ࡍࡿ࡜ࡣ
aࡢ࡝ࢇ࡞H ㏆ഐUH a ࢆ࡜ࡗ࡚ࡶ᭷㝈ಶࡢ␒ྕ Q ࢆ㝖࠸࡚ࡍ࡭࡚ࡢ an UH a ࡜࡞ࡿࡇ࡜࡛࠶ࡿࠋࡇࡢ᫬ lim an
n of
aࡲࡔan o aࠉ
Q o f ࡜᭩ࡃࠋ
ᐃ⩏ ᩘิ ^an ` n 1, 2,... ࡀ㹬Ѝ҄ࡀࢥ࣮ࢩ࣮ิ࡛࠶ࡿ࡜ࡣ௵ពȜ࡟ᑐࡋ
࡚᭷㝈ಶࡢ␒ྕ Q ࢆ㝖࠸࡚ࡍ࡭࡚ࡢ an ࡢ┦஫㊥㞳ࡀ Ȝ ௨ୗ࡟࡞ࡿࡇ࡜࡛࠶ࡿࠋ
ࡍ࡞ࢃࡕ௵ព
Ȝ㸼 ࡟ᑐࡋ࠶ࡿ␒ྕ Q ࠿ࡽඛࡢࡍ࡭࡚ࡢ SࠊT ࡟ᑐࡋ a p aq H ࡜࡞ࡿࡼ࠺
࡟ Q ࡀỴࡵࡽࢀࡿࡇ࡜࡛࠶ࡿࠋ
ᐃ⌮ ཰᮰ิࡀࢥ࣮ࢩ࣮ิ࡛࠶ࡿࠋࡲࡓࠊ㏫࡟ࢥ࣮ࢩ࣮ิࡣ཰᮰ิ࡛࠶ࡿࠋ
ᐃ⌮ lim f x ࡀᏑᅾࡍࡿࡓࡵࡢᚲせ༑ศ᮲௳ࡣ௵ពࡢȜ㸼 ࡟ᑐࡋ *!
x of
ࢆ㐺ᙜ࡟࠼ࡽࢇ࡛࡝ࢇ࡞ x1 ! G , x2 ! G ࡟ࡘ࠸࡚ࡶ f x1 f x2 H ࡜࡞ࡿࡼ࠺
࡟࡛ࡁࡿࡇ࡜࡛࠶ࡿ㸦ࡇࡢ᮲௳ࢆࢥ࣮ࢩ࣮ࡢ᮲௳࡜࿧ࡪ㸧ࠋ
ᐃ⌮ 㸦୰㛫್ᐃ⌮㸧 f x ࢆ᭷⏺㛢✵㛫 I > a, b@ ࡟࠾࠸࡚㐃⥆࡞㛵ᩘ࡜ࡍ
ࡿ࡜ f a ࡜ f b ࡢ୰㛫ࡢ௵ពࡢ್ l ࡟ᑐࡋ l
f [ ࡜࡞ࡿ್ [ I ࡀᏑᅾࡍࡿࠋ
ᐃ⩏ 㞟ྜ $ ୖ࡛ᐃ⩏ࡉࢀࡓ㐃⥆㛵ᩘࡀ $ ୍࡛ᵝ㐃⥆࡜ࡣ௵ពࡢȜ㸼 ࡟
ᑐࡋț㸼 ࢆ㸦Ȝ࡟ᛂࡌ࡚㸧㐺ᙜ࡟㑅ࡪ࡜┦஫㊥㞳ࡀ ț ௨ୗࡢ௵ពࡢ㸰Ⅼ
x A, y A, x y G ࡟ࡘ࠸࡚ࡘࡡ࡟ f x f y H ࡀᡂ❧ࡍࡿࡇ࡜ࢆ࠸࠺ࠋᩜ
5
࡛᭩ࡃ࡜ lim f x f y 0 ࡛࠶ࡿࠋᬑ㏻ࡢ㐃⥆ᛶࡀ x0 ࢆᅛᐃࡍࡿࡈ࡜࡟Ȝ
x y o0
࡟ᑐᛂࡍࡿț㸼 ࢆࡁࡵ࡚ f UG ( x0 ) UH (y0 ) ࡜࡞ࡿࡢ࡟ᑐࡋ୍ᵝ㐃⥆ᛶࡣࡇ
ࡢț࡟ x0 ࡟↓㛵ಀ࡟ $ ඲యࢆ㏻ࡌ࡚ྠࡌ್࡟ྲྀࢀࡿࡇ࡜ࢆ♧ࡋ࡚࠸ࡿࠋ
ᐃ⌮ ᭷⏺㞟ྜ $ ୖ࡛ᐃ⩏ࡉࢀࡓ୍ᵝ㐃⥆㛵ᩘࡣ $ ࡲ࡛㐃⥆㛵ᩘ࡜ࡋ࡚ࡣᣑ
ᙇ࡛ࡁࡿࠋࡑࡋ࡚ࡑࡢᣑᙇࡣ୍ពⓗ࡛࠶ࡿࠋ㏫࡟᭷⏺㛢㞟ྜ . ୖ࡛㐃⥆࡞㛵ᩘ
ࡣ . ୍࡛ᵝ㐃⥆࡛࠶ࡿࠋ
ᐃ⌮ 㸦ࣛ࢖ࣈࢽࢵࢶ㸧 n
§n· k
n
n k
f x , g x ࡀx0࡛nᅇᚤศྍ⬟ࡢ࡜ࡁ fg x0 ¦ ¨ ¸ f x0 g x0 . k
k 0© ¹
ᐃ⩏ ࢸ࢖࣮ࣛࡢබᘧ
f ( x ) ࡀ x0 ࡛㐃⥆࡞ࡽ f x f ( x0 ) ࡣ x o x0 ࡢ᫬↓㝈ᑠ࡛࠶ࡿࠋ f x f ( x0 ) ࢆ඾
ᆺⓗ↓㝈ᑠࡢ㸯ḟ⤖ྜ࡛⾲ࡍࡇ࡜ࡀ࡛ࡁࢀࡤࠊ f ( x ) ࡀ x0 ࡟㏆࡙ࡃ᫬ࡢᵝᏊࡀࡼ
ࡃࢃ࠿ࡿࡇ࡜࡟࡞ࡿࠋ඾ᆺⓗ࡞↓㝈ᑠ࡜ࡋ࡚ࡣ࠸ࢁ࠸ࢁ࡞㛵ᩘࡀ⪃࠼ࡽࢀࡿࡀ
᭱ࡶᇶᮏⓗ࡞ࡶࡢ࡜ࡋ࡚ࠊ ( x x0 )D (D ! 0) ࡀ࠶ࡆࡽࢀࡿ࡛࠶ࢁ࠺ࠋࡑࡇ࡛
f ( x)
f ( x0 ) a1 x x0 1 .... an x x0 n o x x0 a
a
an
0 a1 a2 ... an ࡜࡛ࡁࡿ࠿࡜࠸࠺ၥ㢟ࢆ⪃࠼ࡼ࠺ࠋࡶࡋࠊ f ( x) ࡀ x0 ࡛ᚤศ
ྍ⬟࡞ࡽ f ( x) f ( x0 ) f c( x0 ) x x0 o x x0 ࠉࠉ
x o x0 ࡔ࠿ࡽࡑࡢࡼ࠺࡞
඾ᆺⓗ࡞↓㝈ᑠࡢ୍ࡘ x x0 ࡀ☜࠿࡟ྲྀࢀࡿࠋࡶࡗ࡜ᚤศྍ⬟ᛶࡢᅇᩘࡀቑࡸ
ࡏࡤࡇࡢࡼ࠺࡞㡯ࡣቑ࠼ࡿࡔࢁ࠺ࠋࡍ࡞ࢃࡕḟࡢᐃ⌮ࡀฟࡿ㸸
㸦ࢸ࢖࣮ࣛබᘧ㸧 > x0 , x @ ࡛ f n 1 ࡀ㐃⥆࡛࠶ࡗ࡚ x0 , x ࡛f n ࡀᏑᅾࡍࡿ࡞ࡽࡤ
f x
f x0 ࡇࡇ࡛ࡣRn x f c x0 1!
x x0 f cc x0 2!
x x0 2
... f ( n 1) x0 n 1!
x x0 ( n1)
Rn x , (n)
f ( n ) [ x x0 ࡜࡞ࡿࡼ࠺࡞ [ x0 , x ࡀᏑᅾࡍࡿࠋ
n!
ᐃ⌮ ࢹ࢕ࢽࡢᐃ⌮ ᭷⏺㛢㞟ྜ . ୖ࡛ ^ f n x `n
㸦ࡲࡓࡣῶᑡ㸧ࡋ࡚ f x ࡟ྛⅬ཰᮰ࡋࠊ^ f n x `n
1,2,...
1,2,...
ࡀ㹬࡜ඹ࡟༢ㄪ࡟ቑຍ
ࡶ f x ࡶ㐃⥆㛵ᩘ࡞ࡽࡤ
ࡇࡢ཰᮰ࡣ . ୖ୍࡛ᵝ཰᮰࡛࡞ࡅࢀࡤ࡞ࡽ࡞࠸ࠋ
ᐃ⌮ ᭷⏺࡞✵㛫 I > a, b@ ୖ࡛ ^ fn x ` ࡀ㐃⥆㛵ᩘิ࡛࠶ࡗ࡚ࠊ f x ࡟୍ᵝ
b
཰᮰ࡍࢀࡤ lim ³ f n t dt
n of a
b
³ f t dt a
5LHPDQQ ✚ศ 6
ᐃ⌮ ᚤศ✚ศᏛࡢᇶᮏᐃ⌮
༊㛫ୖࡢ㐃⥆㛵ᩘ f t ࡟ᑐࡋࠊᐃ✚ศࡀཎጞ㛵ᩘࡢ୍ࡘࢆ୚࠼ࡿ㸸
d t
f s ds f t ࠋゝⴥࢆ᥮࠼࡚࠸ࡿ࡜ࠊ F t dt ³a
ศྍ⬟࡛࠶ࡾࠊ F c t f t ࡜࡞ࡿࠋ
t
³ f s ds ࡜࠾ࡃ࡜ࠊ)Wࡣᚤ
a
ᐃ⌮ f x ࠊg x ࡀ࠶ࡿ✵㛫࡛ᚤศྍ⬟࡛ࠊ f c x ࠊg c x ࡀ㐃⥆࡞ࡽࡑࡢ
༊㛫ෆࡢ௵ពࡢ DE ࡟ࡘ࠸࡚
b
b
b
࠸ࡾ
f c t g ( t ) d t ª¬ f t g ( t ) º¼ a ³ f t g c( t ) d t .
³
a
a
ᐃ⌮ f x ࡣ✵㛫 , ୖ࡛㐃⥆࡜ࡋࠊM s ࡣ✵㛫 - ୖ࡛ᚤศྍ⬟࡛ࠊM c s ࡣ
㐃⥆࡜ࡍࡿࠋࡲࡓࠊ M s Iࠊs J ࡜ࡍࡿࠋࡇࡢ࡜ࡁ x M s ࡜ࡍࡿ࡜௵ពࡢ
ȘࠊșѰ- ࡟ᑐࡋࠊ ³
ME M D f ( x)dx
E
³D
f (M ( s))M , ( s)ds. ᗘ✵㛫⌮ㄽࡢᇶ♏
ᗘ✵㛫
ᐃ⩏ 㸦6Ȋ㸧ࢆྍ ✵㛫ࡍ࡞ࢃࡕȊࡣ 6 ୖࡢȪຍἲ᪘࡛࠶ࡿ࡜ࡍࡿࠋ
෗ീȣ㸸ȊЍ>҄@ࡣȣࡀྍ⟬ຍἲⓗ࡛࠶ࡿ࡜ࡁ㸦6Ȋ㸧ୖࡢ ᗘ࡜࿧ࡤࢀࡿࠋ
୕ࡘ⤌㸦6Ȋȣ㸧ࡣ ᗘ✵㛫࡜࿧ࡪࠋ
ȣV㸺࡛҄࠶ࡿ࡜ࡁࠊ ᗘ㸦ȣ ࠶ࡿ࠸ࡣ ᗘ✵㛫㸦6Ȋȣ㸧㸧ࡣ᭷㝈࡛࠶ࡿ࡜
࠸࠺ࠋ
Ȋ ࡢせ⣲ࡢิ Sn : n N ࡛ P S n f n N , ࠿ࡘ㼁Sn S ࡜࡞ࡿࡶࡢࡀᏑ
㹬
ᅾࡍࡿ࡜ࡁࠊ ᗘ ȣ ࢆ Ȫ᭷㝈࡜ࡼࡪࠋ
ᗘ ȣ ࡣ ȣ6㸻㸯࡛࠶ࡿ࡜ࡁ☜⋡ ᗘ࡜࿧ࡤࢀࠊ
㸦6Ȋȣ㸧ࡣ☜⋡ ᗘ࡜࿧
ࡪࠋ
ᐃ⌮ ࢝ࣛࢸ࢜ࢻࣜࡢᣑᙇᐃ⌮
6 ࢆ㞟ྜȊ ࢆ 6 ୖࡢ୍ࡘࡢຍἲ᪘࡜ࡋȊ㸻ȪȊ࡜ࡋࡼ࠺ࠋࡶࡋȣ ࡀྍ⟬ຍ
ἲⓗ࡞෗ീȣ㸸ȊЍ>҄@࡛࠶ࡿ࡜ࡍࡿ࡜6Ȋୖࡢ ᗘȣ࡛࠶ࡗ࡚Ȋ ࡢ
ୖ࡛ ȣ㸻ȣ ࡜࡞ࡿࡶࡢࡀᏑᅾࡍࡿࠋ
⿵㢟 ᇶᮏⓗ࡞୙➼ᘧ
7
S , ¦, P ࢆ ᗘ✵㛫࡜ࡋࡼ࠺ࠋࡇࡢ࡜ࡁࠊ
(1) P A㼁 B d P A P B ࠉࠉ
A, B ¦ . §
·
(2) P ¨ 㼁 Fi ¸ d ¦ P Fi ࠉࠉࠉ
F1 , F2,.... Fn ¦ .
© idn ¹ idn
(3)ࡉࡽ࡟P 㹱㸺
fࢆ௬ᐃࡍࡿ࡜
P A㼁 B P A P B P A㻵 B ࠉࠉࠉ
A, B ¦ .
§
·
(4) F1 , F2,.... Fn ¦࡟ᑐࡋ࡚P ¨ 㼁 Fi ¸㸻¦ P Fi ¦ ¦ P Fi 㻵 Fj i j dn
© idn ¹ idn
ࠉࠉࠉࠉࠉࠉࠉࠉ
+¦¦¦ P Fi 㻵 F j 㻵 Fk ... 1
n 1
P F1 㻵 F2 㻵 ...㻵 Fn ࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ
⿵㢟 ᗘࡢ༢ㄪ཰᮰ᛶ
㸦1㸧Fn ¦(n N)࡛࠶ࡾࠊ࠿ࡘFn n F࡜ࡍࡿ࡜, P Fn n P F ࡀᡂ❧ࠋ
ྍ⟬ಶࡢP 㞽㞟ྜࡢ࿴ࡣࡲࡓP 㞽㞟ྜ࡛࠶ࡿࠋ
☜⋡ኚᩘ
ᐃ⩏ 㸦6Ȋ㸧ࢆྍ ✵㛫࡜ࡍࡿࠋȊྍ 㛵ᩘK6Ѝ55 ࡜ࡋࡼ࠺ࠋ$Ӭ5 ࡟
ᑐࡋ࡚ h1 A ^s S : h(s) A` ࡜ᐃ⩏ࡍࡿࠋ㹦ࡀ Ȋྍ ࡛࠶ࡿ࡜ࡣ㸸
h 1 : % o ¦ . ࡍ࡞ࢃࡕ A % ࡟ᑐࡋ h 1 (A) ¦ ࡛࠶ࡿࡇ࡜ࠋࡺ࠼࡟Ȋྍ 㛵ᩘ
K ࡟ࡘ࠸࡚ḟࡢᘧࡀᡂ❧㸸
h
S o R,
h1
¦m %
⿵㢟 ศᕸ㛵ᩘࡢᛶ㉁
࠶ࡿ☜⋡ኚᩘ ; ࡢศᕸ㛵ᩘ ) )[ ࡜ࡋࡼ࠺ࠋࡇࡢ᫬
(1) F : R o > 0,1@ , F n (ࡍ࡞ࢃࡕx d \ F x F y ). (2) lim F x 1, lim F x 0 x of
x of
3 Fࡣྑ㐃⥆࡛࠶ࡿࠋ
ᐃ⌮ ༢ㄪ᪘ᐃ⌮
+ ࢆ 6 ࠿ࡽ 5 ࡬ࡢ᭷⏺㛵ᩘࡢࡘࡃࡿ୍ࡘࡢ㢮㸦ࢡࣛࢫ㸧࡛ࠊ࠸࠿ࡢᛶ㉁ࢆ‶ࡓࡍ
ࡶࡢ࡜ࡍࡿࠋ
㸦㸧+ ࡣ 5 ୖࡢ࣋ࢡࢺࣝ✵㛫࡛࠶ࡿࠋ
㸦㸧ᐃᩘ㛵ᩘ㸯ࡣ + ࡢせ⣲࡛࠶ࡿࠋ
㸦㸧 f n ࡀ + ࡢ㠀㈇㛵ᩘิ࡛ࠊ f n n f ࠋࡇࡇ࡟ f ࡣ 6 ୖࡢ᭷⏺㛵ᩘ࡜ࡍࡿ࡜
8
+ ࡛࠶ࡿࠋ
ᐃ⩏ 㹼ᐃ⩏ ࡣ☜⋡ኚᩘࡢ཰᮰࡟ࡘ࠸࡚࡛࠶ࡿࠋ
ᩘᏛࡢ☜⋡ㄽࡢศ㔝࡟࠾࠸࡚ࠊ☜⋡ኚᩘࡢ཰᮰㸦FRQYHUJHQFHRIUDQGRP
YDULDEOHV㸧࡟㛵ࡋ࡚ࡣࠊ࠸ࡃࡘ࠿ࡢ␗࡞ࡿᴫᛕࡀ࠶ࡿࠋ☜⋡ኚᩘิࡢ࠶ࡿᴟ㝈
࡬ࡢ཰᮰ࡣࠊ☜⋡ㄽࡸࠊࡑࡢᛂ⏝࡜ࡋ࡚ࡢ⤫ィᏛࡸ☜⋡㐣⛬ࡢ◊✲࡟࠾ࡅࡿ㔜
せ࡞ᴫᛕࡢ୍ࡘ࡛࠶ࡿࠋࡼࡾ୍⯡ⓗ࡞ᩘᏛ࡟࠾࠸࡚ྠᵝࡢᴫᛕࡣ☜⋡཰᮰
㸦VWRFKDVWLFFRQYHUJHQFH㸧࡜ࡋ࡚▱ࡽࢀࠊࡑࡢᴫᛕࡣࠊᮏ㉁ⓗ࡟ࣛࣥࢲ࣒࠶
ࡿ࠸ࡣண ୙ྍ⬟࡞஦㇟ࡢิࡣࠊࡑࡢิ࠿ࡽ༑ศ㞳ࢀ࡚࠸ࡿ࢔࢖ࢸ࣒ࢆ◊✲ࡍ
ࡿሙྜ࡟࠾࠸࡚ࠊࡋࡤࡋࡤࠊᮏ㉁ⓗ࡟୙ኚ࡞ᣲື࡬࡜ⴠࡕ╔ࡃࡇ࡜ࡀண᝿ࡉࢀ
ࡿࡇ࡜ࡀ࠶ࡿࠊ࡜࠸࠺⪃࠼ࢆᐃᘧ໬ࡍࡿࡶࡢ࡛࠶ࡿࠋ␗࡞ࡿ཰᮰ࡢᴫᛕ࡜ࡣࠊ
ࡑࡢࡼ࠺࡞ᣲືࡢ≉ᚩ࡙ࡅ࡟㛵㐃ࡍࡿࡶࡢ࡛࠶ࡿ㸸ࡍࡄ࡟ศ࠿ࡿ஧ࡘࡢᣲື࡜
ࡣࠊࡑࡢิࡀ᭱⤊ⓗ࡟ᐃᩘ࡜࡞ࡿ࠿ࠊ࠶ࡿ࠸ࡣࡑࡢิ࡟ྵࡲࢀࡿ್ࡣኚືࢆ⥆
ࡅࡿࡀ࠶ࡿ୙ኚ࡞☜⋡ศᕸ࡟ࡼࡗ࡚ࡑࡢኚືࡀ⾲⌧ࡉࢀࡿࠊ࡜࠸࠺ࡼ࠺࡞ࡶࡢ
࡛࠶ࡿࠋ
ᐃ⩏ ᙅ཰᮰
Fn࡜F ࡀศᕸ㛵ᩘ࡛࠶ࡾࠊ) ࡀ㐃⥆㛵ᩘ࡛࠶ࡿࠋ௵ព x ࡟ࡓ࠸ࡋ࡚ࡣ
limn Fn x F x ࡛࠶ࡿ࡞ࡽࡤࠊ FnࡀF ࡟ᙅ཰᮰ࡍࡿ࡜ゝ࠸ࠊ)㹬䋻㻲㻌䛸᭩䛟䚹㻌
P n࡜P 䛿☜⋡ ᗘ䛷䛒䜚䚸䛭䜜䛮䜜ศᕸ㛵ᩘ Fn࡜F ࢆᣢࡘ࡜ࡁࠊ)㹬䋻㻲㻌䛻䛺䜛᮲௳
䛿 limn Pn A
P A 䛜ᡂ❧䛩䜛䛣䛸䚹䛣䛣䛷䛿 A (f, x], P x 0. Fn un vn F x ࡜ Fn an bn G x ࡜௬ᐃࡍࡿ un ! 0, an ! 0. ࡑ࠺ࡍࡿ࡜
ḟࡢᘧ࡟‶ࡓࡍ a, b ࡀᏑᅾࡍࡿࠋ
b v an
o a, n n o b, F ax b G x un
un
ࡇࡢᘧ࡟ᚑࡗ࡚ḟࡢ஬ࡘ⤖ㄽࡀ࠶ࡿ㸸
i Fn F , an o a, bn o b࡞ࡽࡤࠉFn an xn bn F ax b .
1 x ! 0
® ࠉ
¯0 x 0
iii Fn F , bn↓཰᮰࡞ࡽࡤFn x bn ࡀᙅ཰᮰࡟࡞ࡽ࡞࠸
Fn an x ii Fn F , an o f࡞ࡽࡤ lim
n
iv Fn F , Fn an xn bn G x ࡞ࡽࡤ0㸺LQIn an d sup n an f,sup n bn
v ࡍ࡭࡚ࡢx࡟࠾࠸࡚ࡣF x F ax b ࡀᡂ❧࡞ࡽࡤa 1, b 0.
f.
ᐃ⩏ ศᕸ཰᮰
ࡇࡢࢱ࢖ࣉࡢ཰᮰࡟ࡼࡾࠊ࠶ࡿ୚࠼ࡽࢀࡓ☜⋡ศᕸ࡟ࡼࡗ࡚ࡼࡾⰋࡃࣔࢹࣝ໬
ࡉࢀࡿࡼ࠺࡞ࣛࣥࢲ࣒ᐇ㦂ࡢิ࡟࠾ࡅࡿ⤖ᯝࢆᮇᚅࡍࡿࡇ࡜ࡀฟ᮶ࡿࠋศᕸ཰
᮰ࡣࠊࡇࡢグ஦ෆ࡛㏙࡭ࡽࢀࡿ඲࡚ࡢ௚ࡢࢱ࢖ࣉࡢ཰᮰ࡶព࿡ࡍࡿ࡜࠸࠺Ⅼ࡟
࠾࠸࡚ࠊ᭱ࡶᙅ࠸཰᮰࡛࠶ࡿࠋࡋ࠿ࡋ࡞ࡀࡽࠊᐇ㝿ࡢ⌧ሙ࡟࠾࠸࡚ࠊศᕸ཰᮰
9
ࡣ㠀ᖖ࡟ࡼࡃ฼⏝ࡉࢀࡿ᭱ࡶࡼࡃ⌧ࢀࡿࡢࡣࠊ୰ᚰᴟ㝈ᐃ⌮ࡢᛂ⏝࡟࠾࠸࡚
࡛࠶ࡿࠋ Fn F ࡞ࡽࡤ ;㹬ࡣศᕸ཰᮰ࡲࡓἲ๎཰᮰࡜ゝ࠸ࠊ X n X ࡜᭩ࡃࠋ
X n࡜X ࡣ☜⋡ኚᩘ࡛ࠊࡑࢀࡒࢀศᕸ㛵ᩘ Fn࡜F ࢆࡶࡘࠋ
ᐃ⩏ ☜⋡཰᮰
☜⋡✵㛫㸦ȐF3㸧࡛ࡢ☜⋡ኚᩘ X , X 1 , X 2 ... ࡟ᑐࡋ࡚ࠊ;㹬ࡣ☜⋡㸯࡛ ; ࡟཰᮰
ࡍࡿ࡜ P ª¬ X n X t H ¼º 0 ࠉࠉ
(H ! 0) ࡘࡲࡾ lim n P ª¬ X n X t H ¼º 0 ࠋ
ࡇࡢ᫬ ;㹬ࡀ ; ☜⋡཰᮰࡜࿧ࡪࠋ X n o p X ࡜᭩ࡃࠋ☜⋡཰᮰ࡀḟࡢᛶ㉁ࢆᣢࡘ
㸦1㸧☜⋡཰᮰ࡣࠊศᕸ཰᮰ࢆព࿡ࡍࡿ.
㸦2㸧 X n X ࡑࡋ࡚X n Yn 0࡞ࡽࡤYn X
㸦3㸧 ௵ពX㸦X o f㸧࡟ᑐࡋࠊX n
u
X (n o f), X n X ࡑࡋ࡚
u
u
OLP uOLPVXS n P ª X n Yn t H º
¬
¼
u
0 ࠉ㸦H㸼0㸧࡞ࡽࡤYn X ࠋ
ᐃ⩏ ᴫ཰᮰
☜⋡ኚᩘิ ;Q ࡀ ; ࡬࡜ᴫ཰᮰࠶ࡿ࠸ࡣ࡯࡜ࢇ࡝☜ᐇ࡟཰᮰ࠊ࡯࡜ࢇ࡝⮳ࡿᡤ࡛
཰᮰ࠊ☜⋡ ࡛཰᮰࠶ࡿ࠸ࡣᙉ཰᮰ࡍࡿ࡜ࡣḟࡢᘧࢆᡂ❧ࡍࡿࡇ࡜ࠋ
P lim X n
n of
X
1
ࡀୖᘧࡣࠊ;Qࡀ;࡬࡜཰᮰ࡋ࡞࠸஦㇟ࡀ㉳ࡁࡿ☜⋡ࡀ࡛࠶ࡿ࡜࠸࠺ព࿡࡛ࠊ
;Qࡢ್ࡀ;ࡢ್࡬࡜㏆௜ࡃࡇ࡜ࢆព࿡ࡍࡿࠋ☜⋡✵㛫㸦Ȑ))33㸧ࢆᐃࡵࠊȐ࠿
ࡽ5࡬ࡢ㛵ᩘ࡜ࡋ࡚ࡢ☜⋡ኚᩘࡢᴫᛕࢆ฼⏝ࡍࡿࡇ࡜࡛ࠊୖᘧࡣḟࡢᘧ࡜ྠ್
࡟࡞ࡿ㸸
P Z : : lim X n Z n of
X Z ࡲࡓᴫ཰᮰ࡢྠ್࡞ᐃ⩏࡟ࡣࠊ௨ୗࡶ࠶ࡿ㸸
1 ࠋ
1ࠉࠉ
(H ! 0) ࠋ
P lim inf ^Z : :| X n Z X Z | H `
ᴫ཰᮰ࡣࠊࡋࡤࡋࡤࠊ཰᮰ࢆ⾲ࡍ▮༳ࡢୖ࡟グྕDV㸦DOPRVWVXUHO\ ࡢ␎㸧
ࢆ௜ࡅຍ࠼ࡿࡇ࡜࡟ࡼࡗ࡚⾲⌧ࡉࢀࡿ㸸
a.s.
X n o X ࠋ
㊥㞳✵㛫6Gୖࡢ୍⯡ⓗ࡞☜⋡せ⣲^;Q`࡟ᑐࡋ࡚ࡶࠊྠᵝ࡟ᴫ཰᮰ࡀᐃ⩏
ࡉࢀࡿ㸸
P(Z : : d X n Z , X Z o 0) ࠋ
n of
ᴫ཰᮰ࡣ☜⋡཰᮰ࢆព࿡ࡋࠊࡋࡓࡀࡗ࡚ศᕸ཰᮰ࢆព࿡ࡍࡿࠋ኱ᩘࡢᙉἲ๎࡛
⏝࠸ࡽࢀࡿᴫᛕࡣࠊᴫ཰᮰࡛࠶ࡿࠋ
ᐃ⩏ ☜ᐇ཰᮰
࠶ࡿ☜⋡✵㛫ୖᐃ⩏ࡉࢀࡿิ࠶ࡿ࠸ࡣ☜⋡ኚᩘ;Q㸦ࡍ࡞ࢃࡕࠊ☜⋡㐣⛬㸧ࡀ;
10
࡬☜ᐇ཰᮰㸦VXUHFRQYHUJHQFH㸧࠶ࡿ࠸ࡣྛⅬ཰᮰ࡍࡿ࡜ࡣࠊ
lim X n Z X Z , Z : ࠋ
n of
ࡀᡂ❧ࡍࡿࡇ࡜ࢆゝ࠺ࠋࡇࡇ࡛Ȑࡣࠊ☜⋡ኚᩘࡀᐃ⩏ࡉࢀࡿ☜⋡✵㛫࡟ྵࡲ
ࢀࡿᶆᮏ✵㛫࡛࠶ࡿࠋࡇࢀࡣࠊ㛵ᩘิࡢྛⅬ཰᮰ࡢᴫᛕࢆ☜⋡ኚᩘࡢิ࡬࡜ᣑ
ᙇࡋࡓࡶࡢ࡛࠶ࡿ㸦☜⋡ኚᩘࡣࡑࢀ⮬㌟ࡀ㛵ᩘ࡛࠶ࡿࡇ࡜࡟ὀពࡉࢀࡓ࠸㸧ࠋ
^Z : lim X Z n of
n
`
X Z :
☜⋡ኚᩘࡢ☜ᐇ཰᮰ࡣࠊୖ㏙ࡢ௚ࡢ඲࡚ࡢ཰᮰ࢆព࿡ࡍࡿࠋࡋ࠿ࡋࠊᴫ཰᮰ࡢ
௦ࢃࡾ࡟☜ᐇ཰᮰ࢆ⏝࠸ࡿࡇ࡜ࡢ࣓ࣜࢵࢺࡣ☜⋡ㄽ࡟࠾࠸࡚ࡣ࠶ࡲࡾ↓࠸ࠋࡑ
ࢀࡽ஧ࡘࡢ཰᮰ࡢ㐪࠸ࡣࠊ☜⋡ࢮࣟࡢ㞟ྜ࡟㛵ࡍࡿⅬࡢࡳ࡟Ꮡᅾࡍࡿࠋࡇࡢࡇ
࡜ࡀࠊ☜ᐇ཰᮰ࡢᴫᛕࡀ⁛ከ࡟⏝࠸ࡽࢀࡿࡇ࡜ࡢ↓࠸⌮⏤࡛࠶ࡿࠋ
✚ศࡢ▱㆑
ᐃ⩏ ୍⯡ࡢྍ 㛵ᩘ f : : o > f, f@ ࡟ࡘ࠸࡚ࡣࠊࡑࡢṇ㒊ศ f ࡜㈇㒊ศ
f ࢆࡑࢀࡒࢀ f f 0, f f 0 ࡜ᐃ⩏ࡍࡿࠋࡇࢀࡽࡀ㠀㈇್ྍ 㛵ᩘ࡜ࡇ࡜
ࡣࡼ࠸ࠋࡑࡋ࡚ࠊ ³ f d P fࡲࡓࡣ³ f d P f ࡛࠶ࡿ࡜ࡁ f ࡢ✚ศࢆ
³ fd P㸻³ f
d P -³ f - d P f ࡜ᐃ⩏ࡍࡿࠋ
㠀㈇ྍ 㛵ᩘ࡟ᑐࡋࡢࡇࡢ✚ศࡢᐃ⩏ࡣ௵ពࡢ༢㛵ᩘȭ࡟ࡘ࠸࡚ࡶࡇࡢ✚ศࡢ
ᐃ⩏ࡣ ,ȭ࡜୍⮴ࡍࡿࡇ࡜ࡣᐜ᫆࡟☜ㄆ࡛ࡁࡿࠋ᭦࡟ ³ f d P f ࠿ࡘ
³f
d P f ࡛࠶ࡿ࡜ࡁ f ࡣ✚ศྍ⬟㸦ྍ✚࣮ࣝ࣋ࢢ✚ศ࣮ࣝ࣋ࢢྍ✚㸧࡛࠶
ࡿ࡜࠸࠺ࠋࡲࡓࠊ௵ពࡢྍ 㞟ྜ $ ࡟ࡘ࠸࡚ f ࡢ $ ୖࡢ✚ศࢆ ³ A fd P
³1
A
fd P ࡜
ᐃ⩏ࡍࡿࠋྍ 㞟ྜ $ ࡟ᑐࡋࠊ1A f ࡀ✚ศྍ⬟࡛࠶ࡿ࡜ࡁ f ࡣ $ ୖ࡛✚ศྍ⬟࡛
࠶ࡿ࡜࠸࠺ࠋ
f : : o > f, f @ࡣ✚ศྍ⬟࡞ࡽࡤࠉࠉ³ fd P ¦ n
f
¦ n 1 An
⿵㢟 f , g ࢆ௵ពࡢྍ 㛵ᩘF ࢆ௵ពࡢᐇᩘ࡜ࡍࡿࠋ
f
f t 0 ࡞ࡽࡤࠊ ³ f d P t 0 ࡛࠶ࡿࠋ
f t 0 ࠿ࡘ ³ f d P
1
³
fd P ࠉࡀᡂ❧ࡍࡿࠋ
An
0 ࡞ࡽࡤࠊ P ^Z : : f Z ! 0` ࡛࠶ࡿࠋ
f ࡀ✚ศྍ⬟࡞ࡽࡤࠊ cf ࡶ✚ศྍ⬟࡛࠶ࡾࠊ ³ cfd P
c ³ fd P ࠋ
f , g ࡀ✚ศྍ⬟࡞ࡽࡤ f g ࡶ✚ศྍ⬟࡛࠶ࡾࠊ ³ f g d P
³ fd P ³ gd P
11
ࡀᡂ❧ࡍࡿࠋ
ᐃ⌮ ࣮ࣝ࣋ࢢࡢ཰᮰ᐃ⌮
㛵ᩘิ f nࡀf ࡟ᴫ཰᮰ࡋࠊ✚ศྍ⬟㛵ᩘ㹥ࡀᏑᅾࡋࠊ࡯࡜ࢇ࡝⮳ࡿ࡜ࡇࢁ࡛
f n Z d g Z ࡀᡂ❧ࡍࡿ࡞ࡽࡤ f ࡣ✚ศྍ⬟࡛ lim ³ f n d P
n of
³ fd P ࡀᡂ❧ࡍࡿࠋ
ᐃ⌮ ༢ㄪ཰᮰ᐃ⌮0RQRWRQH&RQYHUJHQFH7KHRUHP021
㸦㸧 f n ࢆ m ¦ ࡢせ⣲ࡢิ࡛ f n n f ࢆ‶ࡓࡍ࡜ࡋࡼ࠺ࠋࡇࡢ࡜ࡁ
P f n n P f d f, ࡲࡓࡣ௚ࡢグྕ࡛ࡣ ³ f n s P s n ³ f n s P ds ࡀᡂ❧ࡍࡿࠋ
s
s
ࡇࡢᐃ⌮ࡣ✚ศㄽࡢࡲࡗࡓࡃࡍ࡭࡚ࡀࡇࡇ࠶ࡿ࡜࠸࠼ࡿᐃ⌮࡛࠶ࡿࠋ
㸦㸧୚࠼ࡽࢀࡓ f m ¦ ࡟ᑐࡋ࡚༢⣧㛵ᩘิ f r ࡛f r n f ࡜࡞࠺࠺ࡿࡶࡢࢆ
␒┠ࡢ㝵ẁ㛵ᩘ D 㹰㸸
>0, f @ o >0, f @ ࢆḟࡢࡼ࠺࡟ᐃ⩏ࡍࡿࠋ
D
r
x
x 0
0
°
r
r
(i 1)2 x d i 2r d r i N ®(i 1)2 ࠉࠉࠉ
° r
x!r
¯
ࡇࡢࡼ࠺࡟ᐃࡵࡿ࡜ f r ࡗ࡚
D r 㼓 f ࡣf r SF+ ࢆ‶ࡓࡋࠊ f r n f ࡛ࠊ㸦021㸧࡟ࡼ
P f n lim P f r n lim P0 f r ࡇࡇ࡛ࠊ D r ࡣᕥ㐃⥆࡛࠶ࡿࡢ࡛ࠊ f n n f ࡞ࡽࡤ D r f n n D r f ࡛࠶ࡿࠋ
㸦㸧 f , g m6 ࡛f
g (a .e.)࡞ࡽࡤP f P g ࡛࠶ࡿࠋ
㸦㸧 f m6 ࡛ࠊ
f n ࡣ m6 ࡢิ࡛ ȣ㞽㞟ྜ 1 ୖࢆ㝖࠸࡚ f n n f ࡜ࡋࡼ࠺ࠋ
ࡇࡢ᫬ P f n n P f ࡀᡂ❧ࡍࡿࠋ
ᐃ⌮ ඃ཰᮰ᐃ⌮'RPLQDWHG&RQYHUJHQFH7KHRUHP'20
f n , f m6 ࡣ 6 ࡢྛⅬ㹱࡛ f n (s) o f s ࡜ࡋ㛵ᩘิ f n ࡣ L1 S , 6, P ࡢ୍ࡘせ
⣲㹥࡟ࡼࡗ࡚ᢲࡉ࠼ࡽࢀ࡚࠸ࡿࠋࡍ࡞ࢃࡕ
f n s d g s ࠉࠉࠉ
s S , n N ࡜ࡍࡿࠋࡓࡔࡋࡇࡇ࡛ ȣJ࡛҄࠶ࡿࠋࡇࡢ࡜ࡁ f n o f ࡀL1 (S, 6, P ) ࡛ᡂ❧ࡍ
ࡿࠋࡍ࡞ࢃࡕ P f n f o 0 ࡛࠶ࡿࠋ࡞ࡢ࡛ P f n o P f ࡜࡞ࡿࠋ
⿵㢟 )$728 ࡢ⿵㢟
K ࢆྍ✚ศ࡞㛵ᩘ ^ f n ` ࢆྍ 㛵ᩘิ࡜ࡍࡿࠋ
12
f n t h ࠉࠉࠉࠉ
P a.e.ࡢ࡜ࡁ³ lim inf fd P d lim inf
n of
㹶
n of
f n d hࠉࠉࠉࠉ
( P a.e.)࡜ࡁࠉ
³ lim inf fd P t lim inf
㹶
n of
n of
³ f d P n
X
³ f dP n
X
ᖹᆒ
ᐃ⩏ ࣐ࣝࢥࣇࡢ୙➼ᘧ
࣐ࣝࢥࣇࡢ୙➼ᘧࡣࠊ ᗘㄽⓗ࡟ࡣࠊ;Ȋȣࢆ ᗘ✵㛫࡜ࡋࠊIࢆᣑᙇᐇ
ᩘ್㸦↓㝈኱ࡶ࡜ࡾ࠺ࡿ㸧ྍ 㛵ᩘ࡜ࡋࠊW!࡜ࡍࢀࡤࠊ
^
` 1t ³ fd P ࡛࠶ࡿࡇ࡜ࢆ㏙࡭ࡿࠋ
P x X f x t t d
x
✵㛫ࡢ ᗘࡀ࡛࠶ࡿ≉ู࡞ሙྜ㸦ࡘࡲࡾ☜⋡✵㛫࡛࠶ࡿ㸧࡟ࡣࠊḟࡢࡼ࠺࡟
ゝ࠸᥮࠼ࡽࢀࡿ㸸;ࢆ௵ពࡢ☜⋡ኚᩘ࡜ࡋࠊD!࡜ࡍࡿ࡜
P X t a d
E x a
ᐃ⩏ /㹮ӌS㸺҄✵㛫ࡢᐃ⩏
p
㸦㸧SӋ ࢆᅛᐃࡍࡿࠋ; ୖ DH࡛ᐃ⩏ࡉࢀࡓྍ 㛵ᩘ I;Ѝ& ࡛ࠊ³ f d P㸺 f ࢆ
‶ࡓࡍࡶࡢࡢ㞟ྜࢆ /㹮/㹮[࡜᭩ࡃࠋ
㸦㸧IѰ/㹮࡟ᑐࡋࠊ f
p
^³
㸦㸧IѰ/㹮QѰ1ࡀ lim f n f
n of
p
f dP
p
`
1/ p
ࢆ㹤ࡢ /㹮࣮ࣀ࣒ࣝ࡜࠸࠺ࠋ
0 ࢆ‶ࡓࡍ࡜ࡁࠊI㹬ࡣ I ࡟㹮ḟᖹᆒ཰᮰ࡍࡿ࡜
࠸࠺ࠋࡲࡓࠊ lim f n 㹶㸻f 㹶 [DHࢆ‶ࡓࡍ࡜ࡁࠊI㹬ࡣ I ࡟ᴫ཰᮰ࡍࡿ࡜࠸
n of
࠺ࠋ
ᐃ⩏ 6FKZDU] ࡢ୙➼ᘧ
f , g L2࡞ࡽࡤ ³ fg d P d
³
2
2
f dP³ g dP ᐃ⩏ +̉OGHU ࡢ୙➼ᘧ
1 1
㹮㸼㸯, 1࡜ࡍࡿࠋf Lp , g Lq࡞ࡽࡤ
p q
³
㹶
fg P dx d
³ f dP ³ g dP
p
1/ p
q
1/ q
㸦㸧ࢫ࣮࢝ࣛಸ࡜࿴࡟ࡘ࠸࡚㛢ࡌ࡚࠸ࡿ✵㛫㸦㞟ྜ㸧ࢆ⥺ᆺ✵㛫㸦࣋ࢡࢺࣝ✵
㛫࡜࠸࠺㸬
㸦㸧⥺ᆺ✵㛫;࡟ᐇᩘ್㛵ᩘ࣭̀̀;Ѝ5ࡀᐃ⩏ࡉࢀ࡚࠸࡚㸪
㠀㈇̀ÌӋ㸪
୍ព㸸̀Ì ࡞ࡽࡤI 㸪
13
୍ḟ㸸̀DÌ ̀D̀̀Ì㸪
୕ゅ୙➼ᘧ㸸̀IJ̀ӊ̀Ì̀J̀㸪
ࡀᡂࡾ❧ࡘ࡜ࡁ㸪ࡇࡢ✵㛫;࣭̀̀ࢆ⥺ᆺࣀ࣒ࣝ✵㛫㸪࣭̀̀ࢆࣀ࣒ࣝ࡜
࠸࠺㸬
㸦㸧㠀㈇㸸ȨIJӋ㸪
୍ព㸸ȨIJ ࡞ࡽࡤI J㸪
ᑐ⛠㸸ȨIJ ȨJI㸪
୕ゅ୙➼ᘧ㸸ȨIKӊȨIJȨJK㸪
ࡀᡂࡾ❧ࡘ㸰ኚᩘ㛵ᩘࢆ㊥㞳࡜࠸࠺㸬࣭̀̀ࡀࣀ࣒ࣝࡢ࡜ࡁ㸪ȨIJ ̀IJ
̀ࡣ㊥㞳࡛࠶ࡿ㸬ࡇࢀࢆࣀ࣒ࣝࡀᐃ⩏ࡍࡿ㊥㞳࡜࠸࠺㸬
⥺ᆺࣀ࣒ࣝ✵㛫;࣭̀̀ࡀ㸪ࣀ࣒ࣝࡀᐃ⩏ࡍࡿ㊥㞳࡟㛵ࡋ࡚᏶ഛ࡞
࡜ࡁ㸪; ;࣭̀̀ࢆ%DQDFK✵㛫࡜࠸࠺㸬
ᐃ⌮/SB࣭̀̀Sࡣ%DQDFK✵㛫࡛࠶ࡿ㸬
⊂❧ᛶ࡜࣐ࣝࢥࣇ㐃㙐
ᐃ⩏ ⊂❧ᛶ࡟㛵ࡍࡿㅖᐃ⩏
㸦㸧⊂❧࡞Ȫຍἲ᪘
) ࡢ㒊ศ Ȫຍἲ᪘ **ࡀ⊂❧࡛࠶ࡿ࡜ࡣࠊ*LѰ*MLѰ1࡛ L͐LQ ࡀ┦
␗࡞ࡿ࡜ࡁࠊࡘࡡ࡟
P Gi1 㻵 ...㻵 Gin n
P G ik
k 1
ࡀᡂࡾ❧ࡘࡇ࡜࡛࠶ࡿࠋ
⊂❧࡞☜⋡ኚᩘ
☜⋡ኚᩘ ;;͐ࡀ⊂❧࡛࠶ࡿ࡜ࡣȪຍἲ᪘Ȫ;Ȫ;͐ࡀ⊂❧࡛࠶ࡿࡇ
࡜࡛࠶ࡿࠋ☜⋡ㄽࡢ㔜せ࡞ㅖၥ㢟ࡢከࡃࡣࠊ☜⋡ኚᩘࡢิ࡛⊂❧ྠศᕸ
㸦LQGHSHQGHQWDQGLGHQWLFDOO\GLVWULEXWHG,,'㸧࡛࠶ࡿࡶࡢ࡟㛵ࢃࡗ࡚࠸ࡿࠋ
⊂❧࡞஦㇟
஦㇟ (㸯(͐ࡀ⊂❧࡛࠶ࡿ࡜ࡣࠊȪຍἲ᪘ H1 , H 2 ... ࡀ⊂❧࡛࠶ࡿࡇ࡜࡛࠶ࡿࠋࡓ
ࡔࡋࡇࡇ࡛ H n ࡣȪຍἲ᪘ ^I , En , : \ En , :` ࡛࠶ࡿࠋ
ᐃ⩏ ࣐ࣝࢥࣇ㐃㙐
࣐ࣝࢥࣇ㐃㙐࡜ࡣࠊ☜⋡㐣⛬ࡢ୍✀࡛࠶ࡿ࣐ࣝࢥࣇ㐣⛬ࡢ࠺ࡕࠊ࡜ࡾ࠺ࡿ≧ែ
࡞ᩓⓗ㸦᭷㝈ࡲࡓࡣྍ⟬㸧࡞ࡶࡢ㸦㞳ᩓ≧ែ࣐ࣝࢥࣇ㐣⛬㸧ࢆ࠸࠺ࠋࡲࡓ≉࡟ࠊ
᫬㛫ࡀ㞳ᩓⓗ࡞ࡶࡢ㸦᫬้ࡣῧ࠼Ꮠ࡛⾲ࡉࢀࡿ㸧ࢆᣦࡍࡇ࡜ࡀከ࠸㸦௚࡟㐃⥆
᫬㛫࣐ࣝࢥࣇ㐣⛬࡜࠸࠺ࡶࡢࡶ࠶ࡾࠊࡇࢀࡣ᫬้ࡀ㐃⥆࡛࠶ࡿ㸧
ࠋ࣐ࣝࢥࣇ㐃㙐
ࡣࠊᮍ᮶ࡢᣲືࡀ⌧ᅾࡢ್ࡔࡅ࡛Ỵᐃࡉࢀࠊ㐣ཤࡢᣲື࡜↓㛵ಀ࡛࠶ࡿ㸦࣐ࣝ
ࢥࣇᛶ㸧
ࠋྛ᫬้࡟࠾࠸࡚㉳ࡇࡿ≧ែኚ໬㸦㑄⛣ࡲࡓࡣ᥎⛣㸧࡟㛵ࡋ࡚ࠊ࣐ࣝࢥ
ࣇ㐃㙐ࡣ㑄⛣☜⋡ࡀ㐣ཤࡢ≧ែ࡟ࡼࡽࡎࠊ⌧ᅾࡢ≧ែࡢࡳ࡟ࡼࡿ⣔ิ࡛࠶ࡿࠋ
≉࡟㔜せ࡞☜⋡㐣⛬࡜ࡋ࡚ࠊᵝࠎ࡞ศ㔝࡟ᛂ⏝ࡉࢀࡿࠋ
࣐ࣝࢥࣇ࡜࠸࠺ྡ๓ࡣࣟࢩ࢔ࡢᩘᏛ⪅࢔ࣥࢻࣞ࢖࣭࣐ࣝࢥࣇ࡟ࡼࡿ㸦⥛ࡾࡣ
14
0DUNRY㸧
ࠋ≀⌮Ꮫࡸ⤫ィᏛࡢࣔࢹࣝ࡟౑ࢃࢀࡿ௚ࠊ᝟ሗᏛ࡟࠾࠸࡚ࡣእࡍࡇ࡜ࡢ
࡛ࡁ࡞࠸㔜せ࡞⪃࠼᪉࡛࠶ࡿࠋ
࣐ࣝࢥࣇ㐃㙐ࡣࠊ୍㐃ࡢ☜⋡ኚᩘ;;;࡛͐ࠊ⌧ᅾࡢ≧ែࡀỴࡲࡗ࡚࠸ࢀࡤࠊ
㐣ཤ࠾ࡼࡧᮍ᮶ࡢ≧ែࡣ⊂❧࡛࠶ࡿࡶࡢ࡛࠶ࡿࠋᙧᘧⓗ࡟ࡣࠊ
P( X n 1 x | X n xn ,...., X 1 x1 , X 0 x0 ) P( X n 1 x | X n xn ) ;Lࡢ࡜ࡾ࠺ࡿ್ࡣࠊ㐃㙐ࡢ≧ែ✵㛫࡜࿧ࡤࢀࠊྍ⟬㞟ྜ 6ࢆ࡞ࡍࠋ࣐ࣝࢥࣇ㐃
㙐ࡣ᭷ྥࢢࣛࣇ࡛⾲⌧ࡉࢀࠊ࢚ࢵࢪ࡟ࡣ࠶ࡿ≧ែ࠿ࡽ௚ࡢ≧ែ࡬㑄⛣ࡍࡿ☜⋡
ࢆ⾲♧ࡍࡿࠋ࣐ࣝࢥࣇ㐃㙐ࡢ୍౛࡟᭷㝈≧ែᶵᲔࡀ࠶ࡿࠋࡇࢀࡣࠊ᫬้ Q࡟࠾
࠸࡚≧ែ\࡟࠶ࡿ࡜ࡍࡿ࡜ࠊࡑࢀࡀ᫬้ Q࡟࠾࠸࡚≧ែ [࡟ືࡃ☜⋡ࡣࠊ
⌧ᅾࡢ≧ែ࡟ࡔࡅ౫Ꮡࡋࠊ᫬้ Q࡟ࡣ౫Ꮡࡋ࡞࠸ࠋ
᫬㛫ⓗ࡟ᆒ୍࡞㸦ᩧ᫬ⓗ㸧࣐ࣝࢥࣇ㐃㙐࡜ࡣࠊࡍ࡭࡚ࡢ Q࡟ᑐࡋ
P( X n 1 x | X n y ) P ( X n x | X n 1 y ) ࡛࠶ࡿࡼ࠺࡞㐣⛬ࢆ࠸࠺ࠋ୍⯡ࡢࠊ᫬㛫ⓗ࡟ᆒ୍࡛࡞࠸࣐ࣝࢥࣇ㐃㙐ࡣࠊࡇࡢ
➼ᘧࢆ‶ࡓࡉ࡞࠸ࠋึᮇ≧ែ L࠿ࡽ᫬้ Q࡛≧ែ M࡟⛣ࡿ☜⋡ࡣࠊ
Pij n P( X n j | X 0 i ) ࡛ᐃ⩏ࡉࢀࠊ༢୍ẁ㝵ࡢ㑄⛣ࡣ
Pij P ( X 1
j | X0
i) ࡛ᐃ⩏ࡉࢀࡿࠋQẁ㝵㑄⛣ࡣࠊ௵ពࡢNQ࡟ᑐࡋ࡚ḟࡢࢳ
ࢳࣕࢵࣉ࣐࣭ࣥࢥࣝ
ࣔࢦࣟࣇࡢ➼ᘧࢆ‶ࡓࡍ㸸
n
k nk p ¦p p ij
rS
ir
rj
᫬้ Q࡛ࡢ≧ែ࡟㛵ࡍࡿ☜⋡㸦࿘㎶☜⋡㸧ࡣḟࡢࡼ࠺࡟᭩ࡅࡿ㸸
P( X n
j)
¦ p P X
rS
rj
n 1
r
¦p
rS
n
rj
P X0
r ࡇࡇ࡛ྑୖ௜ࡁῧ࠼ᏐQࡣᩚᩘ್࡛࠶ࡿࠋࡶࡋ࣐ࣝࢥࣇ㐃㙐ࡀ᫬㛫࡟ᑐࡋ࡚ᐃ
ᖖⓗ࡞ࡽࡤࠊࡇࡢῧ࠼ᏐࡣQ ஌࡜࠸࠺ព࿡࡟࡜ࡗ࡚ࡶࡼ࠸ࠋ
኱ᩘࡢἲ๎
኱ᩘࡢἲ๎㸦ODZRIODUJHQXPEHUV㸧ࡣࠊ☜⋡ㄽ࣭⤫ィᏛ࡟࠾ࡅࡿᴟ㝈ᐃ⌮ࡢ
ࡦ࡜ࡘ࡛ࠊࠕ⤒㦂ⓗ☜⋡࡜⌮ㄽⓗ☜⋡ࡀ୍⮴ࡍࡿࠖ࡜࠸࠺ࠊ⣲ᮔ࡞ព࿡࡛ࡢ☜
⋡ࢆព࿡௜ࡅࠊᐃ⩏௜ࡅࡿἲ๎࡛࠶ࡿࠋ
ཝᐦ࡟ࡣࠊࣖࢥࣈ࣭࣋ࣝࢾ࣮࢖࡟ࡼࡿ኱ᩘࡢᙅἲ๎://1:HDN/DZRI/DUJH
1XPEHUV࡜ࠊ࢚࣑࣮࣭ࣝ࣎ࣞࣝࡸ࢔ࣥࢻࣞ࢖࣭ࢥࣝࣔࢦࣟࣇ࡟ࡼࡿ኱ᩘࡢᙉἲ
๎6//16WURQJ/DZRI/DUJH1XPEHUV࡜ࡀ࠶ࡿࠋ༢࡟ࠕ኱ᩘࡢἲ๎ࠖ࡜ゝ
ࡗࡓሙྜࠊ࡝ࡕࡽࢆᣦࡋ࡚࠸ࡿࡢ࠿ࡣᩥ⬦࡟ࡼࡾุ᩿ࡍࡿᚲせࡀ࠶ࡿ
ᐃ⌮ ᮇᚅ್ȣ࡛࠶ࡿࡼ࠺࡞ྍ✚ศ⊂❧ྠ᫬ศᕸ☜⋡ኚᩘิ;;ࡢ⟬⾡ᖹ
ᆒ
15
X 1 X 2 ... X n
n
> Xn @
ࡢ࡜ࡿ್ࡣࠊ༑ศ኱ࡁ࡞ Q ࡲ࡛⪃࠼ࢀࡤࠊ࡯࡜ࢇ࡝ࡢ Q ࡛࠾࠾ࡼࡑȣ࡛࠶ࡿ㸦>;Q@
ࡀȣ࠿ࡽ኱ࡁࡃእࢀࡿࡼ࠺࡞ Q ࡢ⌧ࢀࡿ☜⋡ࡣQࢆ↓㝈࡟኱ࡁࡃࡍࡿ࡜ ࡟㏆
࡙ࡃ㸧㸸 lim > X n @ P ࠉ
LQSUREDELOLW\
n of
ࡇࢀࢆ኱ᩘࡢᙅἲ๎࡜࠸࠺ࠋࡲࡓྠࡌ᮲௳ୗ࡛ࠊQЍ҄࡜ࡍࡿ࡜ࡁࠊ>;Q@ࡣȣ࡟
࡯࡜ࢇ࡝☜ᐇ࡟㸦DOPRVWVXUHO\☜⋡ ࡛㸧཰᮰ࡍࡿ㸸 lim > X n @ P ࠉࠉa.s. n of
ࡇࢀࢆ኱ᩘࡢᙉἲ๎࡜࠸࠺
ᐃ⌮ ࢳ࢙ࣅࢩ࢙ࣇࡢ୙➼ᘧ
ࢳ࢙ࣅࢩ࢙ࣇࡢ୙➼ᘧࡣࠊ୙➼ᘧ࡛⾲ࡉࢀࡿࠊ☜⋡ㄽࡢᇶᮏⓗ࡞ᐃ⌮࡛࠶ࡿࠋ
ࣃࣇࢾࢸ࢕࣭ࢳ࢙ࣅࢩ࢙ࣇ࡟ࡼࡾึࡵ࡚ド᫂ࡉࢀࡓࠋ
ᶆᮏ࠶ࡿ࠸ࡣ☜⋡ศᕸࡣࠊᖹᆒࡢࡲࢃࡾ࡟ࠊ࠶ࡿᶆ‽೫ᕪࢆࡶࡗ࡚ศᕸࡍࡿࠋ
ࡇࡢศᕸ࡜ᶆ‽೫ᕪࡢ㛫࡟ࠊ࡝ࡢࡼ࠺࡞ᶆᮏ࣭☜⋡ศᕸ࡛ࡶᡂࡾ❧ࡘ㛵ಀࢆ♧
ࡋࡓࡢࡀࠊࢳ࢙ࣅࢩ࢙ࣇࡢ୙➼ᘧ࡛࠶ࡿࠋ౛࠼ࡤࠊᖹᆒ࠿ࡽ ᶆ‽೫ᕪ௨ୖ㞳
ࢀࡓ್ࡣ඲యࡢࢆ㉸࠼ࡿࡇ࡜ࡣ࡞ࡃࠊ୍⯡࡟ Q ᶆ‽೫ᕪ௨ୖ㞳ࢀࡓ್ࡣ඲
యࡢQࢆ㉸࠼ࡿࡇ࡜ࡣ࡞࠸ ; ࢆࠊᮇᚅ್ࡀȣࠊ᭷㝈ࡢศᩓࡀȪ ࡛࠶ࡿ☜⋡
1
ኚᩘ࡜ࡍࡿ࡜ࠊ௵ពࡢᐇᩘ N! ࡟ᑐࡋ࡚ P X P t kV d 2 ࡓࡔࡋ N!ࡢሙ
k
ྜ࡟ࡔࡅព࿡ࡀ࠶ࡿࠋ
┤✚ ᗘ࡜ࣇࣅࢽࡢᐃ⌮
㸦;㸯)ȣ㸯㸧࠾ࡼࡧ㸦;㸰)㸰ȣ㸰㸧ࡣྛྛȪ̿᭷㝈࡜௬ᐃࡍࡿࠋ
(i ) f : X 1 u X 2 o [0, f]ࡣF1 F2 % R ࡛ྍ ࡜ࡍࡿࠋࡇࡢ࡜ࡁ
x o f x, y P㸯 dy ࡣF1 % R ྍ ³
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(ii ) f : X 1 u X 2 o RࡣP1 P2ྍ✚ศ࡜ࡍࡿࠋࡇࡢ᫬ ࡜ࡀᡂ❧ࠋ
࣐ࣝࢳࣥࢤ࣮ࣝࡢ⌮ㄽ
᮲௳௜ࡁᮇᚅ್
16
ᐃ⩏ ☜⋡✵㛫㸦Ȑ)3㸧࡛ ; ࡣ☜⋡ኚᩘ࡛ E X f ࡜ࡋࠊ* ࡣ ) ࡢ㒊ศ Ȫ
ຍἲ᪘࡜ࡍࡿࠋࡇࡢ࡜ࡁ୍ࡘࡢ☜⋡ኚᩘ <࡛ḟࡢㅖᛶ㉁ࢆ‶ࡓࡍࡶࡢࡀᏑᅾࡍ
ࡿࠋ
ࡍ࡞ࢃࡕ
* ྍ ࠋ
E Y f YdP ³ XdP * ࡢ࡝ࡢ㞟ྜ $ ࡟ᑐࡋ࡚ࡶ ³A
A
ࡀᡂࡾ❧ࡘࠋ
ࡉࡽ࡟ࡶࡋ Y㻑ࡀ௚ࡢ☜⋡ኚᩘ࡛ࡇࢀࡽࡢㅖᛶ㉁ࢆ‶ࡓࡍ࡞ࡽࡤࠊY㻑 Y a.s. ࠋࡍ
࡞ࢃࡕ P ª¬Y㻑 Y º¼ ࡛࠶ࡿࠋ
㸦㸧㹼㸦㸧ࡢㅖᛶ㉁ࢆ‶ࡓࡍ୍ࡘࡢ☜⋡ኚᩘ < ࢆࠊ
* ࢆ୚࠼ࡓ࡜ࡁࡢ ; ࡢ᮲௳௜ࡁᖹᆒ E X G ࡢ୍ࡘࡢኚᙧ࡜ࡼࡧࠊ Y E X G 㸦a.s.㸧࡜⾲ࡍࠋ
᮲௳௜ࡁᮇᚅ್ࡢᛶ㉁㸸
㸦 X , Y , X n ࡣྍ✚ศ࡛࠶ࡿࠋ a, b ࡣᐇᩘ࡛࠶ࡿࠋ
㸧
㸦㸧 E aX bY G aE X G bE Y G ࠉࠉࠉa.s. 㸦㸧 X t 0 ࡛࠶ࢀࡤ E X G t 0 ࠉࠉa.s. 㸦㸧 E 1 G 1ࠉࠉࠉa.s. 㸦㸧 XࡣG ࡛ྍ ࠊ X , Y ྍ✚࡞ࡽࡤࠊ E X G 㸦㸧 H G ࡢ࡜ࡁࠊ E E X G H
X ࠉࠉࠉa.s. E X H ࠉࠉa.s. 㸦㸧/ ࡛ X n o X ࡞ࡽࡤ E X n G o E X G ࠋ
㸦㸧-HQVHQ ୙➼ᘧ\ : R1 o R1 ࡣฝ࡛࠶ࡾࠊ\ X ྍ✚ศ࡞ࡽࡤࠊ
\ E X G d E \ X G ࠉࠉࠉa.s. ୍⯡ⓗ࡛ࡣࠊ E X G d E X G ࡑࡋ࡚ E X G d E X
2
2
G ࠋ
; ࡀ * ࡟౫Ꮡࡋ࡞࠸ࠋ࣎ࣞࣝྍ 㛵ᩘ I[ࡀྍ✚ศࡢ᫬ࡢࡳࠊ
E f x G E f x ࠋ
=
= ࡣ * ྍ ࡞ࡽࡤࠊ E zX G zE X G ࠋ
⊂❧ᛶ X rv L 㸪 G, H ࡀȪDOJ࡛ V V x , G ⊂❧࡛࠶ࡿࠋ
ࡑࡢ࡜ࡁ E X E ¬ª X V G , H ¼º
E ¬ª X G ¼º ࠋ
ᐃ⩏ ṇ๎᮲௳௜☜⋡࡜☜⋡ᐦᗘ㛵ᩘSGI
㹹SȰ$㹻ȰѰȐ$Ѱ) ࡀḟࡢ୕ࡘ᮲௳ࢆ‶ࡓࡍ࡞ࡽࡤࠊṇ๎᮲௳ḟ௜ࡁ☜⋡࡜࿧ࡪࠋ
Ȱࢆᅛᐃࡋ࡚ࠊ$ЍSȰ$ࡣ㸦Ȑ)㸧
17
㸦㸧$ ࢆᅛᐃࡋ࡚ࠊ$Ѱ)ȰЍSȰ$ࡀྍ 㸹
㸦㸧 ௵ព$ Fࠊ% G,P($ 㻵 %)= ³㹮Z , A P (d Z )ࠋ
B
ࡘࡲࡾ௵ព㠀㈇ ; ࡜ %Ѱ* X : B
½
³ ®¯ I Z Z³ X ZcP Z, dZc¾¿P(d Z ) B
:
³Z X Z cP Z , dZ c ࡀ E ( X G )Z ࡜୍⮴ࡍࡿ DVࠋ
࣐ࣝࢳࣥࢤ࣮ࣝ
ᐃ⩏ 㞳ᩓ᫬㛫ࡢ࣐ࣝࢳࣥࢤ࣮ࣝ
᫬้ࡢ㞟ྜࡣ 7 ^͐`࡜ࡋࠊ᝟ሗቑ኱⣔^)Q`QѮ7 ࡀ୚࠼ࡽࢀࡓ࡜ࡁࠊᐇ
ᩘ್㞳ᩓ᫬㛫☜⋡㐣⛬;QQѮ7ࡀ࣐ࣝࢳࣥࢤ࣮࡛ࣝ࠶ࡿ࡜ࡣ
㸦㸧;ࡣ㐺ྜ
㸦㸧௵ពࡢ᫬้Q࡟ࡘ࠸࡚(_;Q_㸺҄
㸦㸧௵ពࡢ᫬้Q࡟ࡘ࠸࡚(>;Q_)Q@ ;Q DV Q㸼
ࡀᡂ❧ࡍࡿࡇ࡜࡛࠶ࡿࠋ
࣐ࣝࢳࣥࢤ࣮ࣝ⌮ㄽࡀ኱ኚ࡟㔜せ࡛࠶ࡿࡢࡣ࣐ࣝࢳࣥࢤ࣮ࣝࡀ㠀ᖖ࡟ከࡃࡢ
ሙ㠃࡟ጼࢆ⌧ࡍ࡜࠸࠺ࡇ࡜࡟᰿ᣐࡀ࠶ࡿࠋ⡆༢ࡢ౛ࢆぢࡿࠋ
㸦㸧⊂❧࡛ᖹᆒ ࡢ☜⋡ኚᩘࡢ࿴ࡀ࣐ࣝࢳࣥࢤ࣮࡛ࣝ࠶ࡿࠋ ;;͐ࢆ⊂❧
࡞☜⋡ኚᩘࡢิ࡛ࠊ௵ព㹬࡟ᑐࡋ࡚ (_;㹩_㸺࡛҄࠶ࡾࠊ࠿ࡘ (;㹩㸻 ࡜ࡍࡿࠋ
Sn X1 X 2 ... X n ࠉ
S0 0 , Fn V X1 X 2 ... X n ࡜ࡍࡿࠋࡇࡢ࡜ࡁ QӍ㸯࡟
ᑐࡋ DVࠋ
㸦㸧ᖹᆒ㸯ࡢ㠀㈇⊂❧☜⋡ኚᩘࡢ✚ࡀ࣐ࣝࢳࣥࢤ࣮࡛ࣝ࠶ࡿࠋ ;;͐ࢆ⊂
❧࡞㠀㈇☜⋡ኚᩘ࡜ࡍࡿࠋ M 0 1, F0 ^0, :` ࡜ࡋ࡚
Mn
X1 X 2 ... X n , Fn V X1 , X 2 ,..., X n ࡜ᐃ⩏ࡍࡿࠋࡇࡢ᫬ࠊQӋ ࡟ᑐࡋ
E (M n | Fn1 ) E M n1 X n | Fn1 M n1 E X n | Fn1 M n1 E X n M n1 ࢆ㸦DV㸧ࡢព࿡࡛ᚓࡿ࠿ࡽ M ࡣ࣐ࣝࢳࣥࢤ࣮࡛ࣝ࠶ࡿࠋ
㸦㸧୍ࡘࡢ☜⋡ኚᩘ࡟㛵ࡍࡿࢹ࣮ࢱࡢ⣼✚ࡀ࣐ࣝࢳࣥࢤ࣮࡛ࣝ࠶ࡿ Fn ࢆࣇ
࢕ࣝࢺ࣮ࣞࢩࣙࣥ࡜ࡋ [ L1 :, F , P ࡜ࡍࡿࠋ㐺ᙜ࡞ኚᙧࢆ࡜ࡗ࡚
Mn
E [ Fn ࡜ᐃ⩏ࡋࡼ࠺ࠋሪࡢᛶ㉁࡟ࡼࡗ࡚
E (M n | Fn1 ) E [ | Fn | Fn1 E [ | Fn1 M n1ࠉࠉa.s. ࢆᚓࡿࠋࡋࡓࡀࡗ࡚ࠊ M ࡣ࣐ࣝࢳࣥࢤ࣮࡛ࣝ࠶ࡿࠋ
ᐃ⩏ ඃ࣐ࣝࢳࣥࢤ࣮ࣝ࡜ຎ࣐ࣝࢳࣥࢤ࣮ࣝ
^Fn ` , P ࡟ࡘ࠸࡚ඃ࣐ࣝࢳࣥࢤ࣮ࣝࡣࠊᐃ⩏ ࡢ㸦㸧㸦㸧ࡀᡂࡾ❧ࡘୖ
ࢆ (>;Q_)Q@ӌ;Q DV Q㸼࡟⨨ࡁ᥮࠼࡚ᐃ⩏ࡉࢀࡿࠋࡑࡋ࡚ຎ࣐ࣝࢳࣥࢤ
18
࣮ࣝࡣࢆ (>;Q_)Q@Ӎ;Q DV Q㸼࡟⨨ࡁ᥮࠼࡚ᐃ⩏ࡉࢀࡿࠋ
ᐃ⩏ ࣐ࣝࢳࣥࢤ࣮ࣝኚ᥮
f n , n N ࢆ㠀㈇ྍண 㐣⛬࡜ࡍࡿࠋຎ㸦ඃ㸧࣐ࣝࢳࣥࢤ࣮ࣝ࡟ᑐࡋࠊ
Y0
0, Yn
n
¦ f X
k
k
X k 1 , n t 1 k 1
࡜࠾ࡃࠋྑ㎶ࡢྛ㡯ࡀྍ✚ศ࡞ࡽࡤ Yn ࡣຎ㸦ඃ㸧࣐ࣝࢳࣥࢤ࣮࡛ࣝ࠶ࡿࠋ
ᐃ⌮ X n , n N ࢆ㸦ຎ㸧࣐ࣝࢳࣥࢤ࣮ࣝ࡜ࡍࡿࠋF x : R o R ࢆ㸦ቑ኱ࡍࡿ㸧
ฝ㛵ᩘ࡜ࡍࡿࠋ Yn
F X n ࡀࡍ࡭࡚ࡢ㹬࡛ྍ✚ศ࡞ࡽࡤ Yn , n N ࡣຎ࣐ࣝࢳࣥ
ࢤ࣮࡛ࣝ࠶ࡿࠋ
ࡇࡢᐃ⌮ࡣ -HQVHQ ࡢ୙➼ᘧࡼࡾゝ࠼ࡿࠋ
f X n
s
f E ¬ª X n 1 Fn ¼º d E ¬ª f X n 1 Fn ¼º . ୍ᵝྍ✚ศᛶ
ᮏ⠇࡛ࡣࠊ࣐ࣝࢳࣥࢤ࣮ࣝࡢㅖᛶ㉁࡟ࡘ࠸࡚⪃ᐹࡍࡿࠋ௨ୗࠊ㏻ᖖ᮲௳ࢆ‶
ࡓࡍ᝟ሗቑ኱⣔௜ࡁ☜⋡✵㛫 ሺȐǡ ‫ܨ‬ǡ ‫ܩ‬ǡ ܲሻǡ ‫ ܩ‬ൌ ሼ‫ܨ‬௧ Ǣ ‫ ݐ‬൒ Ͳሽ ࡜ ሺȐǡ ‫ܨ‬ǡ ‫ܩ‬ǡ ܲሻ ୖ࡛ᐃ⩏
ࡉࢀࡓ☜⋡㐣⛬ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ ݐ‬൒ Ͳሽ ࢆᡤ୚࡜ࡍࡿࠋ
☜⋡㐣⛬ࡢ୍ᵝྍ✚ศᛶࢆᐃ⩏ࡍࡿ๓࡟ࠊࡑࡢ⌮ゎࢆᐜ᫆࡟ࡍࡿࡓࡵ࡟☜⋡
ኚᩘࡢྍ✚ศᛶࡢᚲせ༑ศ᮲௳࡟ࡘ࠸࡚♧ࡍࠋ
⿵㢟 ☜⋡ኚᩘࡀྍ✚ศࠊࡍ࡞ࢃࡕࠊ‫ܧ‬ሾȁܺȁሿ ൏ ࡛҄࠶ࡿ࡞ࡽࡤࠊ
‫׊‬Ԗ ൐ Ͳǡ ‫ ߜ׌‬൐ ͲǢ ‫ܧ‬ሾȁܺȁͳீ ሿ ൏ ߳Ǣ ‫ܨ א ܩ׊‬Ǣ ܲሺ‫ܩ‬ሻ ൏ ߜ ࡇࡢ⿵㢟ࢆド᫂ࡍࡿ࡟ࡣࠊ⫼⌮ἲࢆ⏝࠸ࠊ%RUHO&DQWHOOL ࡢ⿵㢟࡜ )DWRX ࡢ⿵
㢟㸦⿵㢟㸧ࢆ฼⏝ࡍࡿࠋ
ᐃ⌮ ☜⋡ኚᩘࡀྍ✚ศࠊࡍ࡞ࢃࡕࠊ‫ܧ‬ሾȁܺȁሿ ൏ ࡛҄࠶ࡿࡓࡵࡢᚲせ༑ศ᮲௳
ࡣࠊ
‫ܧ‬ሾȁܺȁͳሼȁ௑ȁவ௡ሽ ሿ ൌ Ͳ
௡՜҄
࡜࡞ࡿࡇ࡜࡛࠶ࡿࠋ
ド ᫂ 㸸 ࡀ ྍ ✚ ศ ࡞ ࡽ ࡤ ࠊ ȁܺȁͳሼȁ௑ȁவ௡ሽ ൑ ȁܺȁ ൏ ҄ DV ࠊ ࡋ ࡓ ࡀ ࡗ ࡚
௡՜҄ ȁܺȁͳሼȁ௑ȁவ௡ሽ ൌ ͲDV࡛࠶ࡿ࠿ࡽࠊ/HEHVJXH ࡢ᭷⏺཰᮰ᐃ⌮ࡼࡾࠊᐃ⌮ ࡀᡂ❧ࡍࡿࠋ
ᐃ⩏ ☜⋡㐣⛬ሼܺሺ‫ݐ‬ሻǢ ‫ ܴ א ݐ‬ା ሽࡀࠊ‫݌ݑݏ‬௧‫א‬ோశ ‫ܧ‬ሾܺሺ‫ݐ‬ሻ௣ ሿ ൏ ҄ǡ ‫ ݌‬൒ ͳࢆ‶ࡓࡍ᫬ࠊ‫݌‬஌
ྍ✚ศ㸦VTXDUHLQWHJUDEOH㸧࡛࠶ࡿ࡜࠸࠺ࠋ≉࡟‫ ݌‬ൌ ͳࡢ࡜ࡁࡣࠊྍ✚ศ࡛࠶ࡿ
࡜࠸࠺ࠋ
ᐃ⩏ ☜⋡㐣⛬ሼܺሺ‫ݐ‬ሻǢ ‫ܴ א ݐ‬ା ሽࡀࠊ
19
‫݌ݑݏ‬௧‫א‬ோశ ‫ܧ‬ሾȁܺሺ‫ݐ‬ሻȁͳሼȁ௑ሺ௧ሻȁவ௡ሽ ሿ ൌ Ͳ
௡՜҄
࡜࡞ࡿ࡜ࡁࠊܺࡣࠊ୍ᵝྍ✚ศ㸦XQLIRUPLO\LQWHJUDEOH㸧࡛࠶ࡿ࡜࠸࠺ࠋ
ᐃ⌮ ܺ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ܴ א ݐ‬ା ሽࡀࠊ୍ᵝྍ✚ศ࡛࠶ࢀࡤࠊ‫݌ݑݏ‬௧‫א‬ோశ ‫ܧ‬ሾȁܺሺ‫ݐ‬ሻȁሿ ൏ ҄ ࠊ
ࡍ࡞ࢃࡕࠊܺࡣྍ✚ศ࡛࠶ࡿࠋ
ᐃ⌮ ☜⋡㐣⛬ܺ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ ܴ א ݐ‬ା ሽ࡟ᑐࡋ࡚ࠊ
ȁܺሺ‫ݐ‬ሻȁ ൏ ܻǡ ‫ ܴ א ݐ‬ା ǡ ‫ܧ‬ሾȁܻȁሿ ൏ ҄
࡜࡞ࡿ☜⋡ኚᩘܻࡀᏑᅾ࡞ࡽࡤࠊܺࡣ୍ᵝྍ✚ศ࡛࠶ࡿࠋ
ド᫂㸸ȁܺሺ‫ݐ‬ሻȁ ൏ ܻǡ ‫ ܴ א ݐ‬ା ࡼࡾ
‫݌ݑݏ‬௧ஹ଴ ‫ܧ‬ሾȁܺሺ‫ݐ‬ሻȁͳሼȁ௑ሺ௧ሻȁவ௡ሽ ሿ ൑ ‫ܧ‬ሾȁܻȁͳሼȁ௒ȁவ௡ሽ ሿ
࠿ࡘࠊܻ࡟㛵ࡋ࡚ᐃ⌮ ࢆ⏝࠸ࢀࡤࠊ
‫݌ݑݏ‬௧ஹ଴ ‫ܧ‬ሾȁܺሺ‫ݐ‬ሻȁͳሼȁ௑ሺ௧ሻȁவ௡ሽ ሿ ൑ ‫ܧ‬ሾȁܻȁͳሼȁ௒ȁவ௡ሽ ሿ ൌ Ͳ
௡՜҄
௡՜҄
ᐃ⌮ ࡼࡾࠊ┤ࡕ࡟ḟࡢ⣔ࢆᚓࡿ
‫ܧ‬ሾ‫݌ݑݏ‬௧‫א‬ோశ ȁܺሺ‫ݐ‬ሻȁሿ ൏ ࡛҄࠶ࡿ࡞ࡽࡤࠊܺ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ܴ א ݐ‬ା ሽࡣ୍ᵝྍ✚ศ࡜࡞ࡿࠋ
ᐃ⌮ ‫ ݌‬൐ ͳ࡜ࡋ࡚ܺ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ܴ א ݐ‬ା ሽࡀ‫݌‬஌ྍ✚ศ࡞ࡽࡤࠊ୍ᵝྍ✚ศ࡜࡞ࡿࠋ
ା
ᐃ⌮ ࡼࡾࠊܺ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ ܴ א ݐ‬ሽࡀ ஌ྍ✚ศ࡞ࡽࡤࠊܺࡣ୍ᵝྍ✚ศ࡜࡞ࡿࠋ
ᐃ⌮ 'RRE ࡢ࣐ࣝࢳࣥࢤ࣮ࣝ ܻࢆྍ✚ศ࡞☜⋡ኚᩘ࡜ࡋ࡚ࠊ
ܺሺ‫ݐ‬ሻ ൌ ‫ܧ‬ሾܻȁ‫ܨ‬௧ ሿǡ ‫ܴ א ݐ‬ା ࡜࠾ࡃ࡜ࠊܺ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ܴ א ݐ‬ା ሽࡣ୍ᵝྍ✚ศ࡞࣐ࣝࢳࣥࢤ࣮ࣝ࡜࡞ࡿࠋ
ド᫂㸸ܺࡀ࣐ࣝࢳࣥࢤ࣮ࣝ࡜࡞ࡿࡇ࡜ࡣ᫂ࡽ࠿ࠋࡋࡓࡀࡗ࡚ࠊ୍ᵝྍ✚ศᛶ࡟ࡘ
࠸࡚ド᫂ࡍࡿࠋ
ܻ ൒ Ͳ ࡢ ሙ ྜ ࡟ ࡘ ࠸ ࡚ ド ᫂ ࡍ ࡿ ࠋ ܺሺ‫ݐ‬௡ ሻ ՛ ҄ሺ‫ݐ‬௡ ՛ ҄ሻ ࡜ ࡞ ࡿ ༢ ㄪ ቑ ຍ ิ
㹹‫ݐ‬௡ Ǣ ݊ ‫ ܰ א‬㹻ࡀᏑᅾࡋࡓ࡜ࡍࡿ࡜ࠊ༢ㄪ཰᮰ᐃ⌮ࡼࡾࠊሾܺሺ‫ݐ‬௡ ሻሿ ՛ ҄࡜࡞ࡿࡀࠊ
ࡇࢀࡣࠊ‫ܧ‬ሾܺሺ‫ݐ‬௡ ሻሿ ൌ ‫ܧ‬ሾܻሿ ൏ ҄࡟▩┪ࡍࡿࠋࡋࡓࡀࡗ࡚ࠊܺ ‫ כ‬ൌ ‫݌ݑݏ‬௧‫א‬ோశ ܺሺ‫ݐ‬ሻ ൏ ҄ࠋ
ሼܺሺ‫ݐ‬ሻ ൐ ݊ሽ ‫ ؿ‬ሼܺ ‫ כ‬൐ ݊ሽ࡜᮲௳௜ࡁᮇᚅ್ࡢᐃ⩏ࡼࡾࠊ
‫ܧ‬ሾȁܺሺ‫ݐ‬ሻȁͳሼȁ௑ሺ௧ሻȁவ௡ሽ ሿ ൌ ‫ܧ‬ሾܻͳሼȁ௑ሺ௧ሻȁவ௡ሽ ሿ ൑ ‫ܧ‬ሾܻͳሼȁ௑ ‫ כ‬ȁவ௡ሽ ሿ
‫כ‬
ࡇࡇ࡛ࠊ ܺ ࡀ᭷㝈࡛࠶ࡿࡇ࡜࡜ࠊܻࡀྍ✚ศ࡛࠶ࡿࡇ࡜࡟ὀពࡍࡿ࡜ࠊ/HEHVJXH
ࡢ᭷⏺཰᮰ᐃ⌮ࡼࡾ
‫ܧ‬ሾܻͳሼȁ௑ ‫כ‬ȁவ௡ሽ ሿ ൌ Ͳ
௡՜҄
ࡋࡓࡀࡗ࡚ࠊܺࡣ୍ᵝྍ✚ศ࡜࡞ࡿࠋ
ᐃ ⌮ 㸦 ᭷ ⏺ ཰ ᮰ ᐃ ⌮ 㸧 ܺǡ ܺ௡ ǡ ݊ ‫ ⋡ ☜ ࢆ ܰ א‬ኚ ᩘ ࡜ ࡍ ࡿ ࠊ ࡇ ࡢ ࡜ ࡁ ࠊ
‫ܴ א ܭ׌‬Ǣ ȁܺ௡ ȁ ൑ ‫ܭ‬ǡ ‫ ܰ א ݊׊‬
࠿ࡘ௡՜҄ ܺ௡ ൌ ܺ࡞ࡽࡤࠊ
‫ܧ‬ሾȁܺ௡ െ ܺȁሿ ൌ Ͳ
௡՜҄
20
ド᫂௡՜҄ ܺ௡ ൌ ܺ࠿ࡽࠊ௵ពࡢԖ ൐ Ͳ ࡟ᑐࡋ࡚ࠊ
߳
߳
‫݊׌‬଴ ‫ܰ א‬Ǣ ܲሺȁܺ௡ െ ܺȁ ൐ ሻ ൏
‫ ݊׊‬൒ ݊଴ ͵
͵‫ܭ‬
ࡲࡓࠊ‫ ܭ‬൒ ȁܺ௡ ȁࡼࡾࠊ
ଵ
ଵ
ଵ
ሼȁܺȁ െ ൐ ሽ ‫ ؿ‬ሼȁܺȁ െ ȁܺ௡ ȁ ൐ ሽ ‫ ؿ‬ሼȁܺ െ ܺ௡ ȁ ൐ ሽ‫݇׊‬ǡ ݊ ‫ܰ א‬
௞
ଵ
௞
ଵ
௞
࡞ࡢ࡛ ܲሼȁܺȁ െ ‫ ܭ‬൐ ሽ ൑ ܲሼȁܺ െ ܺ௡ ȁ ൐ ሽ
௞
ଵ
௞
ࡼࡗ࡚௡՜҄ ܺ௡ ൌ ܺ ֜ ܲሼȁܺȁ ൐ ‫ ܭ‬൅ ሽ ൌ Ͳ
ଵ
௞
ࡼࡗ࡚ܲሺ‫ڂ‬௞ୀଵሼȁܺȁ ൐ ‫ ܭ‬൅ ሽሻ ൌ ܲሺȁܺȁ ൐ ‫ܭ‬ሻ ൌ Ͳ
҄
௞
ࡍ࡞ࢃࡕࠊ‫ ܭ‬൒ ȁܺȁDV௨ୖ࡟ࡼࡾ
‫ܧ‬ሾȁܺ௡ െ ܺȁሿ ൌ ‫ܧ‬ሾȁܺ௡ െ ܺȁͳȁ௑೙ ି௑ȁவച ሿ ൅ ‫ܧ‬ሾȁܺ௡ െ ܺȁͳȁ௑೙ି௑ȁஸച ሿ ൑ ʹ‫ܲܭ‬ሺȁܺ௡ െ ܺȁ
య
య
߳
߳
൐ ሻ ൅ ൑ ߳
͵
͵
ᐃ⌮ ᖹᆒ཰᮰ࡢᚲせ༑ศ᮲௳ܺǡ ܺ௡ ‫ܮ א‬ଵ ሺȐǡ ‫ܨ‬ǡ ܲሻሻǡ ݊ ‫࡜ࡢࡇࠋࡿࡍ࡜ ܰ א‬
ࡁḟࡀᡂ❧ࡍࡿࠋ
‫ܧ‬ሾȁܺ௡ െ ܺȁሿ ൌ Ͳ
௡՜҄
ܺ௡ ൌ ܺ ฻ ൝ ௡՜҄
ሼܺ௡ Ǣ ݊ ‫ܰ א‬ሽࡣࠊ୍ᵝྍ✚ศ
೵Ṇ᫬
ᐃ⩏ ȫǣ Ȑ ՜ ሾͲǡ ҄ሿࡀ
ሼ߬ ൑ ‫ݐ‬ሽǣ ሼ߱ ‫ߗ א‬Ǣ ߬ሺ߱ሻ ൑ ‫ݐ‬ሽ ‫ܨ א‬௧ ǡ ‫ ݐ‬൒ Ͳ
ࢆ‶ࡓࡍ᫬ࠊȫࢆሼ‫ܨ‬௧ ሽ െ ೵Ṇ᫬VWRSSLQJWLPH࡜࠸࠸ࠊ᝟ሗቑ኱⣔ሼ‫ܨ‬௧ ሽࡀ⮬᫂
࡛࠶ࡿ࡜ࡁ࡟ࡣࠊ༢࡟೵Ṇ᫬࡜࠸࠺ࠋࡲࡓ
‫ܨ‬ఛ ؔ ሼ‫ ؿ ܣ‬ȐǢ ‫ځܣ‬ሼ߬ ൑ ‫ݐ‬ሽ ‫ܨ א‬௧ ǡ ‫ ݐ‬൒ Ͳሽ
࡛ᐃ⩏ࡉࢀࡿ‫ܨ‬ఛ ࢆ߬ െ೵Ṇ᫬๓ྍ⟬ຍἲ᪘SUHȫDOJHEUD࡜࠸࠺ࠋ
ᐃ⌮ ߬ଵ ൑ ߬ଶ ֜ ‫ܨ‬ఛభ ‫ܨ ؿ‬ఛమ ‫ܨ‬ఛభ ‫ٿ‬ఛమ ൌ ‫ܨ‬ఛభ ‫ܨځ‬ఛమ ‫ܨ א ܣ‬ఛభ ‫ٿ‬ఛమ ֜ ‫ځܣ‬ሼ߬ଵ ൑ ߬ଶ ሽ ‫ܨ א‬ఛమ ‫ܨ‬ఛభ ‫ڀ‬ఛమ ൌ ߪሺ‫ܨ‬ఛభ ǡ ‫ܨ‬ఛమ ሻ
୲ା ؔ ‫ځ‬௨வ௧ ‫࡜ࡿࡍ࡜ ୲ܨ‬ḟࡀᡂ❧ࡍࡿ
ȫࡀ‫ܨ‬௧ା ̺೵Ṇ᫬฻ ሼ߬ ൏ ‫ݐ‬ሽ ‫ܨ א‬௧ ǡ ‫ ݐ‬൒ Ͳ
୲ା ؔ ሼ‫ ؿ ܣ‬ȐǢ ‫ځܣ‬ሼ߬ ൑ ‫ݐ‬ሽ ‫ܨ א‬௧ା ǡ ‫ ݐ‬൒ Ͳሽ
ൌ ሼ‫ ؿ ܣ‬ȐǢ ‫ځܣ‬ሼ߬ ൏ ‫ݐ‬ሽ ‫ܨ א‬௧ ǡ ‫ ݐ‬൒ Ͳሽ
ᐃ⌮ ሼ߬௡ Ǣ ݊ ‫ܰ א‬ሽࢆሼ‫ܨ‬௧ ሽ െ ೵Ṇ᫬ࡢิ࡜ࡍࡿ࡜ࠊḟࡀᡂ❧ࡍࡿ
߬௡ ՛ ȫ ֜ ߬ ࡣሼ‫ܨ‬௧ ሽ െ ೵Ṇ᫬
߬௡ ՝ ȫ ֜ ߬ ࡣሼ‫ܨ‬௧ା ሽ െ ೵Ṇ᫬࡛‫ܨ‬௧ା ൌ ‫҄ځ‬
௡ୀଵ ‫ܨ‬ఛ೙ ା 21
ド᫂㸸
߬௡ ՛ ȫ ֜ ሼ߬ ൑ ‫ݐ‬ሽ ൌ ‫҄ځ‬
௡ୀଵሼ߬௡ ൑ ‫ݐ‬ሽࡼࡾ᫂ࡽ࠿
߬௡ ՝ ȫ ֜ ሼ߬ ൏ ‫ݐ‬ሽ ൌ ‫҄ځ‬
௡ୀଵሼ߬௡ ൏ ‫ݐ‬ሽ ‫ܨ א‬௧ ‫ ׵‬ȫࡣሼ‫ܨ‬௧ା ሽ െ ೵Ṇ᫬
‫҄ځ א ܧ‬
‫ܨ‬
௡ୀଵ ఛ೙ ା
฻ ‫ ת ܧ‬ሼ߬௡ ൏ ‫ݐ‬ሽሻ ‫ܨ א‬௧ ǡ ݊ ‫߬ࡘ࠿ܰ א‬௡ ՝ ȫ ֜ ሼ߬ ൏ ‫ݐ‬ሽ ൌ‫҄׫‬
௡ୀଵ ሼ߬௡ ൏ ‫ݐ‬ሽ
҄
฻ ‫ ת ܧ‬ሼ߬ ൏ ‫ݐ‬ሽ ൌ‫׫‬௡ୀଵ ሺ‫ ת ܧ‬ሼ߬௡ ൏ ‫ݐ‬ሽሻ ‫ܨ א‬௧ ᐃ⌮ ܺࡀⓎᒎⓗྍ ࠊ߬ࡀ೵Ṇ᫬࡞ࡽࡤࠊ߱ ‫ ߗ א‬฽ ܺሺ߬ሻ ؔ ܺሺ߬ሺ߱ሻǡ ሺ߱ሻሻ ‫א‬
ܴࡣ‫ܨ‬ఛ ̺ྍ ࡜࡞ࡿࠋ
㞳ᩓ᫬㛫࣐ࣝࢳࣥࢤ࣮ࣝ
ᮏ⠇࡛ࡣࠊ᫬้ࡢ㞟ྜࢆܼ ା ࡜ࡍࡿࠋ༶ࡕ᫬้ࡢ㞟ྜࡀ㞳ᩓ࡛࠶ࡿ࡜ࡋ࡚ࠊ㞳
ᩓ᫬㛫ࡢ࣐ࣝࢳࣥࢤ࣮ࣝሼܺሺ‫ݐ‬ሻǢ ‫ ܼ א ݐ‬ା ሽ࡟ࡘ࠸࡚ᐃ⌮ࢆࡲ࡜ࡵࡿࠋ
ᐃ⌮ 'RRE ࡢᶓ᩿ᩘᐃ⌮'RRECVXSFRVVLQJWKHRUHP
ܺ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ ܼ א ݐ‬ା ሽࢆඃ࣐ࣝࢳࣥࢤ࣮ࣝ࡜ࡍࡿࠋ᫬้ܰࡲ࡛࡟ࢧࣥࣉࣝࣃࢫࡀሾܽǡ ܾሿ
ࢆୗ࠿ࡽୖ࡬࡜ᶓ᩿ࡍࡿᅇᩘࢆܷே ሺܺǢ ሾܽǡ ܾሿሻ࡜ࡍࡿࠋࡍ࡞ࢃࡕࠊܷே ሺܺǢ ሾܽǡ ܾሿሻࡣ
ܺሺ‫ݏ‬௜ ሻ ൏ ܽǡ ܺሺ‫ݐ‬௜ ሻ ൐ ܾǡ ݅ ൌ ͳǡ ‫ ڮ‬ǡ ݇Ǣ Ͳ ൑ ‫ݏ‬ଵ ൏ ‫ݐ‬ଵ ൏ ‫ ڮ‬൏ ‫ݏ‬௞ ൏ ‫ݐ‬௞ ൑ ܰ
࡜࡞ࡿ᭱኱ࡢ࡛݇࠶ࡿࠋࡇࡢ࡜ࡁࠊḟࡀᡂ❧ࡍࡿࠋ
‫ܧ‬ሾሺܺሺܰሻ െ ܽሻି ሿ
‫ܧ‬ሾܷே ሺܺǢ ሾܽǡ ܾሿሻሿ ൑
ܾെܽ
⣔ܺ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ ܼ א ݐ‬ା ሽࢆ୍ᵝྍ✚ศ࡞࣐ࣝࢳࣥࢤ࣮ࣝ࡜ࡋࠊ
ܷ҄ ሺܺǢ ሾܽǡ ܾሿሻ ؔ ܷே ሺܺǢ ሾܽǡ ܾሿሻ
ே՜҄
࡜ࡍࡿࠋࡇࡢ࡜ࡁࠊ
ሺܾ െ ܽሻ‫ܧ‬ሾܷ҄ ሺܺǢ ሾܽǡ ܾሿሻሿ ൑ ȁܽȁ ൅ ‫݌ݑݏ‬௡‫א‬௓ శ ‫ܧ‬ሾȁܺሺ݊ሻȁሿ ൏ ҄
࡜ࡍࡿࠋ
ド᫂㸸ᐃ⌮ ࡜୍ᵝྍ✚ศᛶࡼࡾ
ሺܾ െ ܽሻ‫ܧ‬ሾܷே ሺܺǢ ሾܽǡ ܾሿሻሿ ൑ ȁܽȁ ൅ ‫ܧ‬ሾȁܺሺܰሻȁሿ ൑ ȁܽȁ ൅ ‫݌ݑݏ‬௡‫א‬௓ శ ‫ܧ‬ሾȁܺሺ݊ሻȁሿ ൏ ҄
ࡼࡗ࡚ࠊ༢ㄪ཰᮰ᐃ⌮࡟ࡼࡾᚓࡿࠋ
ᐃ⌮ ࣐ࣝࢳࣥࢤ࣮ࣝ཰᮰ᐃ⌮ඃ࣐ࣝࢳࣥࢤ࣮ࣝܺ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ ܼ א ݐ‬ା ሽࡀྍ✚
ศࠊࡍ࡞ࢃࡕࠊ ‫݌ݑݏ‬௧‫א‬௓ శ ‫ܧ‬ሾȁܺሺ‫ݐ‬ሻȁሿ ൏ ҄ ࡛࠶ࢀࡤࠊ☜⋡㸯࡛᭷㝈࡞ᴟ㝈 ܺሺ҄ሻ ؔ
௧՜҄ ܺሺ‫ݐ‬ሻࡀᏑᅾࡍࡿࠋ
ド᫂㸸
ȃǣ ሼȰ ‫ א‬ȐǢ ሼܺሺ‫ݐ‬ǡ ߱ሻǢ ‫ ܼ א ݐ‬ା ሽࡀሾെ҄ǡ ൅҄ሿ࡛ᴟ㝈ࢆᣢࡓ࡞࠸ሽ
࡜࠾ࡃ࡜ࠊ
ȃ ൌ ሼȰ ‫ א‬ȐǢ ܺሺ‫ݐ‬ǡ ߱ሻ ൏ ܺሺ‫ݐ‬ǡ ߱ሻሽ
௧՜҄
௧՜҄
ൌ‫׫‬௔ǡ௕‫א‬ொǢ௔ழ௕ ሼȰ ‫ א‬ȐǢ ܺሺ‫ݐ‬ǡ ߱ሻ ൏ ܽ ൏ ܾ ൏ ܺሺ‫ݐ‬ǡ ߱ሻሽ
௧՜҄
௧՜҄
ࡇࡇ࡛ࠊ
ȃ௔ǡ௕ ൌ ሼȰ ‫ א‬ȐǢ ܺሺ‫ݐ‬ǡ ߱ሻ ൏ ܽ ൏ ܾ ൏ ܺሺ‫ݐ‬ǡ ߱ሻሽ
௧՜҄
௧՜҄
22
࡜ࡋࠊܷ҄ ሺܺǢ ሾܽǡ ܾሿሻࢆ⣔ ࡛ᐃ⩏ࡋࡓࡶࡢ࡜ࡍࡿ࡜ࠊ
ȃ௔ǡ௕ ‫ ؿ‬ሼȰ ‫ א‬ȐǢ ܷ҄ ሺܺǢ ሾܽǡ ܾሿሻ ൌ ҄ሽ
࡜࡞ࡿࠋ୍᪉ࠊ⣔ ࡼࡾ㸪ܲሺܷ҄ ሺܺǢ ሾܽǡ ܾሿሻ ൌ ҄ሻ ൌ Ͳ࡛࠶ࡿ࠿ࡽሺȃୟǡୠ ሻ ൌ Ͳ࡜
࡞ࡾࠊሺȃሻ ൌ Ͳࢆᚓࡿࠋࡉࡽ࡟ࠊ)DWRX ࡢ⿵㢟⿵㢟 ࡼࡾࠊ
‫ܧ‬ሾ ȁܺሺ‫ݐ‬ሻȁሿ ൌ ‫ܧ‬ሾ ȁܺሺ‫ݐ‬ሻȁሿ ൑ ‫ܧ‬ሾȁܺሺ‫ݐ‬ሻȁሿ ൑ ‫ܧ‬ሾȁܺሺ‫ݐ‬ሻȁሿ ൏ ҄
௧՜҄
௧՜҄
௧՜҄
௧‫א‬௓ శ
࡜࡞ࡿ࠿ࡽࠊ ሾܺሺ҄ሻሿ ൏ ҄DV࡜࡞ࡿࠋ
⣔ ୍ᵝྍ✚ศ࣐ࣝࢳࣥࢤ࣮ࣝࡣࠊ☜⋡ ࡛཰᮰ࡍࡿࠋ
஌ྍ✚ศ࣐ࣝࢳࣥࢤ࣮ࣝࡣࠊ☜⋡ ࡛཰᮰ࡍࡿࠋ
ṇ್࣐ࣝࢳࣥࢤ࣮ࣝࡣࠊ☜⋡ ࡛཰᮰ࡍࡿࠋ
ᐃ⌮ ܺ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ ܼ א ݐ‬ା ሽ ࡀࠊ୍ᵝྍ✚ศඃ࣐ࣝࢳࣥࢤ࣮ࣝ࡞ࡽࡤࠊ
௅భ
ܺሺ‫ݐ‬ሻ ՜ ܺሺ҄ሻሺ‫ ݐ‬՜ ҄ሻ࡜࡞ࡾࠊ
ܺሺ‫ݐ‬ሻ ൒ ‫ܧ‬ሾܺ҄ ȁ‫ܨ‬௧ ሿܽǤ ‫ݏ‬Ǥǡ
‫ܼ א ݐ‬ା
≉࡟ܺࡀ࣐ࣝࢳࣥࢤ࣮ࣝ࡞ࡽࡤࠊ➼ྕࡀᡂ❧ࡍࡿࠋ
ᐃ ⌮ /HY\'RRE ࡢ ୗ ᪉ ᐃ ⌮ ᡤ ୚ ࡢ ☜ ⋡ ✵ 㛫 ሺȐǡ ‫ܨ‬ǡ ܲሻ ࡟ ᑐ ࡋ ࡚ ࠊ
ሼ
௡ ǣ ݊ ‫ א‬െܰሽࢆ‫ࡢ ܨ‬㒊ศྍ⟬ຍἲ᪘ࡢ᪘࡛ࠊ
ିஶ ؔ‫҄ת‬
௞ୀଵ ି௞ ‫ି
ؿ ڮ ؿ‬ଶ ‫ି
ؿ‬ଵ ࡜࡞ࡿࡶࡢ࡜ࡍࡿࠋܺ ൌ ሼܺሺ݊ሻǢ ݊ ‫ א‬െܰሽࢆሼ
௡ ǣ ݊ ‫ א‬െܰሽ െඃ࣐ࣝࢳࣥࢤ࣮ࣝ࡜ࡍ
ࡿࠋࡍ࡞ࢃࡕࠊ
‫ܧ‬ሾܺሺ݊ሻȁ
௠ ሿ ൑ ܺሺ݉ሻܽǤ ‫ݏ‬Ǥ ǡ ݉ ൑ ݊ ൑ െͳǡ ‫ܧ‬ሾȁܺሺ݊ሻȁሿ ൏ ҄‫ א ݊׊‬െܰ
࡜ࡍࡿࠋࡇࡢ࡜ࡁࠊܺࡣࠊ୍ᵝྍ✚ศ࡛࠶ࡾࠊᴟ㝈ܺሺെ҄ሻ ؔ ௡՜ି҄ ܺሺ݊ሻࡀᴫ
཰᮰࠿ࡘᖹᆒ཰᮰ࡢព࿡࡛Ꮡᅾࡋࠊ
‫ܧ‬ሾܺሺ݊ሻȁ
ିஶ ሿ ൑ ܺሺെ҄ሻܽǤ ‫ݏ‬Ǥǡ
݊ ‫ א‬െܰ
ࡀᡂ❧ࡍࡿࠋࡲࡓࠊܺࡀ࣐ࣝࢳࣥࢤ࣮ࣝ࡞ࡽࡤࠊ᭱ᚋࡢ୙➼ᘧࡣ➼ྕࡀᡂ❧ࡍࡿࠋ
ᐃ⌮ 'RRE ࡢຎ࣐ࣝࢳࣥࢤ࣮ࣝ୙➼ᘧ ܺ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ ܼ א ݐ‬ା ሽࢆ㠀㈇್ຎ࣐
ࣝࢳࣥࢤ࣮ࣝ࡜ࡍࡿࠋࡇࡢ࡜ࡁࠊ௵ពࡢܿ ൐ Ͳ࡜݊ ‫ ܼ א‬ା ࡟ᑐࡋ࡚ḟ࡟୙➼ᘧࡀᡂ
❧ࡍࡿࠋ
ܿܲሺ ܺሺ݇ሻ ൒ ܿሻ ൑ ‫ܧ‬ሾܺሺ݊ሻͳሼ௦௨௣ೖರ೙ ௑ሺ௞ሻஹ௖ሽ ሿ ൑ ‫ܧ‬ሾܺሺ݊ሻሿ
௞ஸ௡
ᐃ⌮ 'RRE ศゎ
ܺ ൌ ሼܺሺ݊ሻǢ ݊ ‫ ܼ א‬ା ሽࡀܺሺ݊ሻ ‫ܮ א‬ଵ ሺȐǡ ‫ܨ‬ǡ ܲሻǡ ݊ ‫ ܼ א‬ା ࠊ࡜࡞ࡿ㐺ྜ㐣⛬࡜ࡍࡿࠋ
ࡇࡢ࡜ࡁࠊ‫ ܯ‬ൌ ሼ‫ܯ‬ሺ݊ሻǢ ݊ ‫ ܼ א‬ା ǡ ‫ܯ‬ሺͲሻ ൌ Ͳሽࢆ࣐ࣝࢳࣥࢤ࣮ࣝࠊ‫ ܣ‬ൌ
ሼ‫ܣ‬ሺ݊ሻǢ ݊ ‫ ܼ א‬ା ǡ ‫ܣ‬ሺͲሻ ൌ Ͳሽ ࢆྍண 㐣⛬࡜ࡋ࡚ࠊ
ܺሺ݊ሻ ൌ ܺሺͲሻ ൅ ‫ܯ‬ሺ݊ሻ ൅ ‫ܣ‬ሺ݊ሻǡ
݊ ‫ܼ א‬ା
෩ ࡜ྍண ࡜࠸࠺ࡼ࠺࡟ศゎ࡛ࡁࡿࠋࡲࡓࠊࡇࡢศゎࡣࠊ࣐ࣝࢳࣥࢤ࣮ࣝ‫ܯ‬
㐣⛬‫ܣ‬ሚࢆ
෩ ሺ݊ሻǡ ‫ܣ‬ሺ݊ሻ ൌ ‫ܣ‬ሚሺ݊ሻǡ ݊ ‫ ܼ א‬ା ሻ ൌ ͳ
ܲሺ‫ܯ‬ሺ݊ሻ ൌ ‫ܯ‬
23
࡜ࡋ࡚ࠊ
෩ ሺ݊ሻ ൅ ‫ܣ‬ሚሺ݊ሻǡ
ܺሺ݊ሻ ൌ ܺሺͲሻ ൅ ‫ܯ‬
݊ ‫ܼ א‬ା
࡜࡞ࡿ࡜࠸࠺ព࿡୍࡛ព࡛࠶ࡿࠋ
ܺࡀຎ࣐ࣝࢳࣥࢤ࣮ࣝ࡜࡞ࡿ࡜ࡣࠊ
ܲሺ‫ܣ‬ሺ݊ሻ ൑ ‫ܣ‬ሺ݊ ൅ ͳሻǡ ݊ ‫ ܼ א‬ା ሻ ൌ ͳ
࡜࡞ࡿ࡜ࡁ࡛࠶ࡾࠊࡲࡓࠊࡑࡢ᫬࡟㝈ࡽࢀࡿࠋ
ド᫂㸸
௡
‫ܣ‬ሺ݊ሻǣ ൌ ෍ ‫ܧ‬ሾܺሺ݇ሻ െ ܺሺ݇ െ ͳሻȁ‫ܨ‬௞ିଵ ሿ ܽǤ ‫ݏ‬Ǥǡ
݊ ‫ܰ א‬ǡ ‫ܣ‬ሺͲሻ ؔ Ͳ
௞ୀଵ
࡜ࡍࡿ࡜‫ྍࡣܣ‬ண ࠊ࠿ࡘࠊ
‫ܧ‬ሾሺܺሺ݊ሻ െ ‫ܣ‬ሺ݊ሻሻ െ ሺܺሺ݊ െ ͳሻ െ ‫ܣ‬ሺ݊ െ ͳሻሻȁ‫ܨ‬௞ିଵ ሿ ൌ Ͳ
࡜࡞ࡿࠋࡇࡇ࡛ࠊ‫ܯ‬ሺ݊ሻ ؔ ܺሺ݊ሻ െ ܺሺͲሻ െ ‫ܣ‬ሺ݊ሻ࡜ࡍࢀࡤࠊ‫࡜࣮ࣝࢤࣥࢳ࣐ࣝࡣ ܯ‬
࡞ࡾࠊ'RRE ࡢศゎࢆᚓࡿࠋࡲࡓࠊࡢຎ࣐ࣝࢳࣥࢤ࣮ࣝ࡟ࡘ࠸࡚ࡣࠊࡑࡢᐃ⩏
ࡼࡾ⮬࡛᫂࠶ࡿࠋ
⿵㢟 ܺ ൌ ሼܺሺ݊ሻǢ ݊ ‫ ܼ א‬ା ሽࢆඃ࣐ࣝࢳࣥࢤ࣮ࣝࡶࡋࡃࡣຎ࣐ࣝࢳࣥࢤ࣮ࣝ࡜ࡍ
ࡿ࡜ࠊ௵ពࡢ ൐ Ͳ࡟ᑐࡋ࡚ࠊḟࡀᡂ❧ࡍࡿࠋ
ܿܲሺ ȁܺሺ݇ሻ ȁ ൒ ͵ܿሻ ൑ Ͷ‫ܧ‬ሾȁܺሺͲሻȁሿ ൅ ͵‫ܧ‬ሾȁܺሺܰሻȁሿǡ ܰ ‫ ܼ א‬ା ௞ஸே
ᐃ⌮ /HY\ ࡢୖ᪉ᐃ⌮ ᡤ୚ࡢߦ ‫ܮ א‬ଵ ሺȐǡ ‫ܨ‬ǡ ܲሻ࡟ᑐࡋ࡚ࠊ‫ܯ‬ሺ݊ሻ ؔ
‫ܧ‬ሾߦȁ‫ܨ‬௡ ሿǡ ݊ ‫ ܼ א‬ା ࠊ࡜ࡍࡿ࡜‫ ؔ ܯ‬ሼ‫ܯ‬ሺ݊ሻǢ ݊ ‫ ܼ א‬ା ሽ ࡣࠊ୍ᵝྍ✚ศ࡞࣐ࣝࢳࣥࢤ࣮
࡛ࣝ
௅భ
‫ܯ‬ሺ݊ሻ ՜ ߟ ؔ ‫ܧ‬ሾߦȁ‫ܨ‬௡ ሿሺ݊ ՜ ҄ሻܽǤ ‫ݏ‬Ǥ ǡ ࠿ࡘ‫ܯ‬ሺ݊ሻ ՜ ߟሺ݊ ՜ ҄ሻ
ᐃ⌮ ߬ࢆ೵Ṇ᫬࡜ࡋࠊܺ ൌ ሼܺሺ‫ݐ‬ሻǢ ‫ ܼ א ݐ‬ା ሽࢆ࣐ࣝࢳࣥࢤ࣮ࣝ㸦ඃ࣐ࣝࢳࣥࢤ
࣮ࣝ㸧࡜ࡍࡿࠋࡇࡢ࡜ࡁ ሼܺሺ‫߬ ר ݐ‬ሻǢ ‫ ܼ א ݐ‬ା ሽࡣ࣐ࣝࢳࣥࢤ࣮ࣝ㸦ඃ࣐ࣝࢳࣥࢤ࣮
ࣝ㸧࡜࡞ࡿࠋ
ܺࡀ࣐ࣝࢳࣥࢤ࣮ࣝࡢሙྜ࡛ࡣ
௡
෍
௞ୀଵ
ͳሼ௞ஸఛሽ ሺܺሺ݇ሻ െ ‫ݔ‬ሺ݇ െ ͳሻሻ ൌ ܺሺ݊ ‫߬ ר‬ሻ െ ܺሺͲሻ
ሼͳሼ௡ஸఛሽ ൌ Ͳሽ ൌ ሼ߬ ൑ ݊ െ ͳሽ ‫ܨ א‬௡ିଵ ǡ ሼͳሼ௡ஸఛሽ ൌ ͳሽ ൌ ሼ߬ ൑ ݊ െ ͳሽ௖ ࡜࡞ࡿࡇ࡜࡟ὀពࡍࡿ࡜ࠊܺሺ‫߬ ר ݐ‬ሻࡣ‫ܨ‬௡ ྍ ࠿ࡘ
‫ܧ‬ሾܺሺ݊ ‫߬ ר‬ሻ െ ܺሺ݊ െ ͳ ‫߬ ר‬ሻȁ‫ܨ‬௡ିଵ ሿ ൌ Ͳ
‫ܧ ׵‬ሾܺሺ݊ ‫߬ ר‬ሻȁ‫ܨ‬௡ିଵ ሿ ൌ ܺሺ݊ െ ͳ ‫߬ ר‬ሻǡ ݊ ‫ܰ א‬
ᐃ⌮ ߬ ࢆ೵Ṇ᫬࡜ࡋࠊ ܺ ࢆඃ࣐ࣝࢳࣥࢤ࣮ࣝ࡜ࡍࡿࠋࡇࡢ࡜ࡁࠊḟࡢ
a
ܰࡀ࠶ࡗ࡚ࠊ߬ሺ߱ሻ ൑ ܰǡ ‫ א ߱׊‬Ȑ
‫ܴ א ܭ׌‬Ǣ ȁܺሺ݊ሻȁ ൑ ‫ܭ‬ǡ ‫׊‬ሺ݊ǡ ߱ሻ ‫ ܼ א‬ା ൈ Ȑ࠿ࡘ߬ ൏ ҄ܽǤ ‫ݏ‬Ǥ
‫ܧ‬ሾ߬ሿ ൏ ҄࠿ࡘ‫ܴ א ܭ׌‬Ǣ ȁܺሺ݊ሻ െ ܺሺ݊ െ ͳሻȁ ൑ ‫ܭ‬ǡ ‫׊‬ሺ݊ǡ ߱ሻ ‫ ܼ א‬ା ൈ Ȑ
24
ࡢ࠸ࡎࢀ࠿ࡀᡂࡾ❧ࡘ࡞ࡽࡤࠊܺሺ߬ሻࡣྍ✚ศ࡛࠶ࡾࠊ
‫ܧ‬ሾܺሺ߬ሻሿ ൑ ‫ܧ‬ሾܺሺͲሻሿ
ࡀᡂ❧ࡍࡿࠋࡲࡓࠊܺࡀ࣐ࣝࢳࣥࢤ࣮ࣝࡢሙྜ࡟ࡣࠊྠ᮲௳ࡢୗ࡛᭱ᚋࡢ୙➼ᘧ
ࡀ➼ྕ࡛ᡂ❧ࡍࡿࠋࡍ࡞ࢃࡕࠊ
‫ܧ‬ሾܺఛሿ ൌ ‫ܧ‬ሾܺ଴ ሿ
࡜࡞ࡿࠋ
⿵㢟 ܺࢆඃ࣐ࣝࢳࣥࢤ࣮ࣝ࡜ࡋࠊ߬ࢆ࠶ࡿܰ࡟ᑐࡋ࡚ࠊ߬ ൑ ܰ࡜࡞ࡿ೵Ṇ᫬
࡜ࡍࡿࠋࡇࡢ᫬ࠊܺሺ߬ሻ ‫ܮ א‬ଵ ሺȐǡ ‫ܨ‬ఛ ǡ ܲሻǡ࠿ࡘḟࡀᡂ❧ࡍࡿࠋ
‫ܧ‬ሾܺሺܰሻȁ‫ܨ‬ఛ ሿ ൑ ܺሺ߬ሻ
ド᫂㸸‫ܨ א ܧ‬ఛ ࡜ࡍࡿ࡜
ே
ே
‫ܧ‬ሾܺሺܰሻͳா ሿ ൌ ෍ ‫ܧ‬ሾܺሺܰሻͳா‫ת‬ሼఛୀ௡ሽ ሿ ൑ ෍ ‫ܧ‬ሾܺሺ݊ሻͳா‫ת‬ሼఛୀ௡ሽ ሿ ൌ ‫ܧ‬ሾܺሺ߬ሻͳா ሿ
௡ୀ଴
ே
௡ୀ଴
ȁܺሺ߬ሻȁ ൑ ෍ ȁܺሺ݊ሻȁ
௡ୀ଴
ࡼࡾࠊ‫ܧ‬ሾȁܺሺ߬ሻȁሿ ൏ ҄
ᐃ⌮ ௵ពᢳฟᐃ⌮ ߬ଵ ߬ଶ ࢆͲ ൑ ߬ଵ ൑ ߬ଶ ൑ ҄࡜࡞ࡿ೵Ṇ᫬࡜ࡍࡿࠋܺࡀ୍
ᵝྍ✚ศ࣐ࣝࢳࣥࢤ࣮ࣝ࡞ࡽࡤࠊḟࡀᡂ❧ࡍࡿࠋ
‫ܧ‬ሾܺሺ߬ଶ ሻȁ‫ܨ‬ఛభ ሿ ൑ ܺሺ߬ଵ ሻܽǤ ‫ݏ‬Ǥ
ド᫂㸸 ߬ࢆ௵ពࡢ೵Ṇ᫬࡜ࡍࡿࠋ⿵㢟 ࡜ᐃ⌮ ࡼࡾࠊ
ܺሺ߬ ‫݇ ר‬ሻ ൌ ‫ܧ‬ሾܺሺ݇ሻȁ‫ܨ‬ఛ‫ר‬௞ ሿ ൌ ‫ܧ‬ሾ‫ܧ‬ሾܺሺ҄ሻȁ‫ܨ‬௞ ሿȁ‫ܨ‬ఛ‫ר‬௞ ሿ ൌ ‫ܧ‬ሾܺሺ҄ሻȁ‫ܨ‬ఛ‫ר‬௞ ሿ
ᐃ⌮ ࡼࡾࠊ݇ ՜ ҄࡜ࡋ࡚
ܺሺ߬ሻ ൌ ‫ܧ‬ሾܺሺ҄ሻȁ
ሿǡ ؔ Ȫሺ‫׫‬௞ ‫ܨ‬ఛ‫ר‬௞ ሻ
ࡇࡇ࡛ࠊ‫ܨ א ܧ‬ఛ ࡜ࡍࡿ࡜ࠊ‫ ת ܧ‬ሼ߬ ൑ ݇ሽ ‫ܨ א‬ఛ‫ר‬௞ ࠋࡋࡓࡀࡗ࡚ࠊ‫ ת ܧ‬ሼ߬ ൏ ҄ሽ ‫ א‬ä࡜
࡞ࡾࠊ
‫ܧ‬ሾܺሺ߬ሻͳா‫ת‬ሼఛழ҄ሽ ൌ ሾܺሺ҄ሻͳா‫ת‬ሼఛழ҄ሽ ሿ
ࡋࡓࡀࡗ࡚ࠊ
‫ܧ‬ሾܺሺ߬ሻͳா‫ת‬ሼఛୀ҄ሽ ൌ ሾܺሺ҄ሻͳா‫ת‬ሼఛୀ҄ሽ ሿ
௨ୖࡼࡾࠊ ‫ܧ‬ሾܺሺ߬ሻͳா ൌ ሾܺሺ҄ሻͳா ሿ
ࡇࢀࡼࡾࠊ‫ܨ א ܨ‬ఛభ ࡜ࡍࡿ࡜
‫ܧ‬ሾܺሺ߬ଶ ሻ െ ܺሺ߬ଵ ሻͳி ͳሺఛభǡఛమሻ ሿ ൌ Ͳ
࡜࡞ࡿࡢ࡛ࠊド᫂ࡀ࡛ࡁࡿࠋ
ᐃ⌮ ࣐ࣝࢳࣥࢤ࣮ࣝ୰ᚰᴟ㝈ᐃ⌮
㸦㸧X 1 , X 2 ,... ࡀ࣐ࣝࢳࣥࢤ࣮ࣝ࡜ࡣ E X n ,i | Fn ,i 1 X n ,i 1 ࡛࠶ࡿࡢ࡛࣐ࣝࢳࣥࢤ
࣮ࣝᕪศ Yn ,i
X n ,i X n ,i 1࡛࠶ࡾࠊE Yn,i Fn,i 1
0 ࠋ
㸦㸧஧ḟ࣮࣓ࣔࣥࢺ E(Yn2,i ) f ࠋ
25
f
E ª¬Yn2,i Fn ,i 1 º¼ ࡜ࡍࡿ࡞ࡽࡤ ¦ V n ,k 2 n㼡㼡㼡㼡㼡㼡㼞
o f pV 2
㸦㸧᮲௳௜ࡁศᩓ V n2,i
k 1
ࠋ
f
ª
º n of
㸦㸧 ¦ E «Yn2,k 1 Y tH »
o 0࡜ࡍࡿࠋ
^
`
n ,k
¬
¼
k 1
f
ࡇࡢ᫬ࠊ
N (0, V 2 ).ࡇࡢ᮲௳ࡣࣜࣥࢹ࣋ࣝࢡ᮲௳࡜࿧ࡪࠋ
ศᕸ཰᮰
¦ Yn,k o
k 1
☜⋡㐣⛬
ࣈࣛ࢘ࣥ㐠ື
ᐃ⩏ Bt ࡣᶆ‽ࣈࣛ࢘ࣥ㐠ື࡛࠶ࡿ࡜ࡢ᮲௳㸸
㸦㸧 B0 0 ྛȰ࡟ᑐࡋࠊ t o Bt (Z ) ࡣ㐃⥆
㸦㸧 t ! s t 0 ࡢ᫬ Bt Bs㹼N 0, t s լ Fs
Vࠉ
^Bu : u d s` ᐃ⌮ࣈࣛ࢘ࣥ㐠ື ^ Bt , t t 0` ࡣḟࡢᛶ㉁ࢆࡶࡘ
࢞࢘ࢫ⣔ ^ Bt ` ࡣ࢞࢘ࢫ⣔࡛࠶ࡿࡘࡲࡾ௵ពࡢ⥺ᙧ⤖ྜ
a1 Bt1 ..... an Btn , ai R, ti t 0 ࡣṇつศᕸ࡟ᚑ࠺☜⋡ኚᩘ࡛࠶ࡿ
ᖹᆒ E > Bt @ 0 E > Bs Bt @ min ^s, t` ඹศᩓ Cov( Bs , Bt )
ึᮇ್ B0 1 ࢧࣥࣉࣝࣃࢫࡢ㐃⥆ᛶ t o Bt ࡣ☜⋡࡛㐃⥆
⊂❧ቑศ 0 d t1 t2 .... tn ࡢ࡜ࡁ
Bt0 , Bt1 , Bt2 Bt1 , Bt3 Bt2 ,...., Btn Btn1 ࡣ⊂❧࡛࠶ࡿ
ᐃ⌮ ࣈࣛ࢘ࣥ㐠ືࡣ࣐ࣝࢥࣇ㐣⛬࡛࠶ࡿ
௵ពࡢ t ! s t 0 ࡟ᑐࡋࠊ E Bti Fs Bsi , i 1, 2,.... ^
E Bti Bsi Bt j Bsj Fs
`
EG ij t s , i 1, 2.... ࡀᡂ❧ࡍࡿࠋ
ᐃ⌮ /«Y\ࡢᐃ⌮㸦㸧
M t 㸸㐃⥆ ^ Ft ` ࡟ࡘ࠸࡚࣐ࣝࢳࣥࢤ࣮ࣝ M 0
0ࡀ M
M t ࡣ ^ Ft ` ࡟ࡘ࠸࡚ࡢࣈࣛ࢘ࣥ㐠ື࡜࡞ࡿࠋ
t
t , t ! 0 ࢆ‶ࡓࡍ᫬
㸦㹲㸼㹱࡛࠶ࡿࠋ M t2 M t ࡀ࣐ࣝࢳࣥࢤ࣮ࣝ࡟࡞ࡿࡶࡢࠋ E M t2 M
t
Fs M t2 M
t
E M t2 M s2 Fs E( M
t
M
s
Fs ) ࠋ㸧
☜⋡✚ศ࡜ఀ⸨ࡢබᘧ
26
୍⯡࡟ࣈࣛ࢘ࣥ㐠ື ^ Bt ` ࡟㐺ྜࡋࡓ☜⋡㐣⛬ ^G t ` ࡟ᑐࡋ࡚ఀ⸨ᆺ☜⋡✚
ศࡀᐃ⩏ࡉࢀࡿ I t t
³ G s dB s
0
^I t ` ࡶࡲࡓࣈࣛ࢘ࣥ㐠ື ^B ` ࡟㐺ྜࡋࡓ☜⋡㐣⛬࡟࡞ࡿ
t
ࡋࡓࡀࡗ࡚☜⋡ᚤศ᪉⛬ᘧࡣࣈࣛ࢘ࣥ㐠ື ^ Bt ` ࡟㐺ྜࡋࡓ☜⋡㐣⛬ ^ X t ` ࡀ
x0 ³ f s, X s ds ³ g s, X s dBs X t 0
0
t
t
ࢆ‶ࡓࡍࡶࡢࢆồࡵࡼ࡜࠸࠺ၥ㢟࡞ࡢ࡛࠶ࡿ☜⋡㐣⛬ ^ X t ` ࡀ X t ᙧࢆࡶ
ࡘ࡜ࡁఀ⸨㐣⛬,WƷRSURFHVV࡜࠸࠺
ᚤศᙧࢆ⏝࠸࡚ dX Fds GdB ࡜␎グࡍࡿࡇ࡜ࡶከ࠸ఀ⸨㐣⛬ X t ࢆ㛵ᩘ
f t , X t ࡶࡲࡓఀ⸨㐣⛬࡟࡞ࡿࡋ
f (t, x) ࡜ྜᡂࡋ࡚࡛ࡁࡿ☜⋡㐣⛬ Y t ࡓࡀࡗ࡚
Y t 0
0
t
t
y0 ³ F1 s ds ³ G1 s dBs ࡲࡓࡣ dY
F1dt G1dB ࡢᙧࢆࡶࡘࡇࡇ࡛)V࡜*Vࡢලయᙧࢆ୚࠼ࡿࡢࡀఀ⸨ࡢබᘧ࡛࠶ࡿ
ఀ⸨ࡢබᘧኚᩘ㛵ᩘIW[ࡀW࡟㛵ࡋ࡚ᅇ㐃⥆ⓗᚤศྍ⬟[࡟
ᐃ⌮
㛵ࡋ࡚ᅇ㐃⥆ⓗᚤศྍ⬟࡛࠶ࢀࡤ
wf
wf
1 w2 f
2
㹢Y
t , X t dt t , X t dx t , X t dx 2 wt
wx
2 wx
2
ࡇࡇ࡛ dX Fdt GdB, dX 㸻G 2 dt ᚑࡗ࡚
1 w2 f
wf
wf
2
t, X t t, X t F t t, X t G t 2 2 wx
wt
wx
wf
F㸰㸻 t , X t G t wt
ࡀᚓࡽࢀࡿࠋ
࡞࠾ఀ⸨ࡢබᘧࡢᮏ㉁ࡣḟࡢ✚බᘧ࡟࠶ࡿఀ⸨࣮ࣝࣝ
F㸯㸻
dt 2
dB dt dt dB 0, dB 2
dt %HVVHO 㐣⛬
ߚ ൌ ሺߚଵ ǡ ߚଶ ǡ ‫ ڮ‬ǡ ߚ௡ ሻࡣ㹬ḟࣈࣛ࢘ࣥ㐠ື࡛࠶ࡾࠊ݊ ‫⋡☜ࡿ࠶ࠋܰ א‬㐣⛬5ࡣ
ܴ௧ ൌ ȁȁߚ௧ ȁȁLHܴ௧ଶ ൌ σ௡௜ୀଵሺߚ௜ ሻଶ ሺ‫ݐ‬ሻ࡜ᐃ⩏ࡉࢀࡿࠋఀ⸨ࡢබᘧࡼࡾ
ܴ݀௧ଶ ൌ σ௡௜ୀଵ ʹߚ௜ ሺ‫ݐ‬ሻ݀ߚ௜ ሺ‫ݐ‬ሻ ൅ ݊݀‫ݐ‬
௵ពW!ܲሺܴ௧ ൌ Ͳሻ ൌ Ͳ☜⋡㐣⛬ :ࡀḟࡢᘧ࡛ᐃ⩏ࡉࢀࡿ㸸
௡
ܹ݀௧ ൌ
ͳ
෍ ߚ௜ ሺ‫ݐ‬ሻ ݀ߚ௜ ሺ‫ݐ‬ሻ
ܴ௧
௜ୀଵ
: ࡣᐇᩘ್௜ࡁࡢࣈࣛ࢘ࣥ㐠ື࡛ࠊ☜⋡㐣⛬5ࡣḟࡢᘧ࡟‶ࡓࡍ㸸
݀ሺܴ௧ଶ ሻ ൌ ʹܴ௧ ܹ݀௧ ൅ ݊݀‫ݐ‬
27
ࡇࡇ࡛ࡣߩ௧ ൌ ܴ௧ଶ ࡜タᐃࡍࡿࠋ
݀ߩ௧ ൌ ʹඥߩ௧ ܹ݀௧ ൅ ݊݀‫ݐ‬
௡ିଵ ௗ௧
ࢆᚓࡿࠋᚑࡗ࡚ࠊ5ࡣ %HVVHO㐣⛬࡜࿧
ఀ⸨ࡢබᘧࢆ⏝࠸࡚ܴ݀௧ ൌ ܹ݀௧ ൅
ଶ ோ೟
ࡤࢀࡿࠋߩࡣ㹬஌ %HVVHO㐣⛬࡜࿧ࡤࢀࡿࠋ
ᐃ⩏㹲ྛߜ ൒ Ͳ࡜‫ ݔ‬൒ Ͳ࡟ࡘ࠸࡚ࠊ☜❧㐣⛬=㹲ࡣึᮇ್‫ࢆ ݔ‬ᣢࡕࠊ‫ܳܵܧܤ‬ఋ ሺ‫ݔ‬㸧
࡜グධࡉࢀࡿ㸸
௧
ܼ௧ ൌ ‫ ݔ‬൅ ʹ ‫׬‬଴ ඥܼ௦ ݀ߚ௦ ൅ ߜ‫ݐ‬
țࡣ =㹲ࡢḟඖ࡛࠶ࡿࠋ
ᐃ⌮ ߜ ൐ Ͳ࡟ࡘ࠸࡚\ ࡢᐦᗘࢆ᭷ࡍࡿ‫ܳܵܧܤ‬ఋ ࡢ༙⩌ࡣ
ഌ
ͳ ‫ ݕ‬మ
൅
ඥ‫ݕݔ‬
ఋ ሺ‫ݔ‬ǡ
‫ݍ‬௧
ቇ ǡ‫ ݐ‬൐ Ͳǡ ‫ ݔ‬൐ Ͳ
‫ݕ‬ሻ ൌ ቀ ቁ ൬െ
൰ ‫ܫ‬ఔ ቆ
ʹ
ʹ ‫ݔ‬
‫ݐ‬
Yࡀߜ ࡟ᑐᛂࡍࡿࡑࡋ࡚‫ܫ‬జ ࡀY࡟ࡘ࠸࡚ࡢ%HVVHO㛵ᩘࠋ
[ ࡟࠾࠸࡚\ࡢᐦᗘࢆ
ഃ
ഃ
‫ݕ‬
‫ݍ‬௧ఋ ሺͲǡ ‫ݕ‬ሻ ൌ ሺʹ‫ݐ‬ሻିమ ǻሺߜȀʹሻିଵ ‫ ݕ‬మିଵ ቀെ ቁ Ǥ
ʹ‫ݐ‬
ఋ ሺ‫ݔ‬ǡ
͛
ଶ
ଶ
࡛⾲ࡍࠋࡉࡽ࡟‫ ݔ‬ൌ ‫ ݔ‬ǡ ‫ ͛ݕ‬ൌ ‫ ݕ‬ኚᩘኚ᥮࡛‫ݍ‬௧
‫ݕ‬ሻࡣḟࡢᘧ࡟࡞ࡿ㸸
ଶ
ଶ
జ
‫ݕ‬
൅
‫ݕݔ‬
‫݌‬௧ఋ ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ‫ି ݐ‬ଵ ቀ ቁ ቆെ
ቇ ‫ܫ‬ఔ ቀ ቁ ǡ ݂‫ ݐݎ݋‬൐ Ͳǡ ‫ ݔ‬൐ Ͳ
ʹ
‫ݔ‬
‫ݐ‬
ࡑࡋ࡚
‫ݕ‬ଶ
ఋ ሺͲǡ
ିఔ ିሺఔାଵሻ
ିଵ ଶఔାଵ
‫݌‬௧
‫ݕ‬ሻ ൌ ʹ ‫ݐ‬
ǻሺߥ ൅ ͳሻ ‫ݕ‬
ቆെ ቇǤ
ʹ‫ݐ‬
28
➨஧❶ ༢୍ኚ㔞ศᯞ㐣⛬࡟ࡘ࠸࡚
ศᯞ㐣⛬ࡢẕ㛵ᩘ࡟ࡘ࠸࡚
㠀㈇ᩚᩘ =㹬࡟࠾࠸࡚ࡣศᯞ㐣⛬ࡀ࣐ࣝࢥࣇ㐃㙐࡛࠶ࡿࠋ
0,1,2...`࡛࠶ࡿ᫬ศᯞ㐣⛬ࡢ☜⋡㛵ᩘࡣ㹹S kN 㹻Sk t 0,
^Z㸸㹬㸻
n
¦S
k
1
࡛ᐃ⩏ࡉࢀࡿࠋ
P i 㸪j P ^Z n 1
j | Zn
i
i t 1 ࠉj t 0,
° p ࠉif
ࠉࠉ ࠉ
ࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ
i` ® j
2.1
i 0 ࠉj t 0.
°¯G 0 j ࠉif
pki : k 0,1, 2,...
ศᯞ㐣⛬ࡣ࠶ࡿ㞟ᅋࡢྛඖ⣲ࡢ⦾Ṫ㐣⛬ࢆ⾲ࡍ࡜ᛮࢃࢀ࡚ࡶⰋ࠸ࠋึᮇ್ࡣ =
࡛ࠊྛඖ⣲ࡀ⊂❧࡛☜⋡㛵ᩘ㹹 pk 㹻ࡢཎ๎ࡢୗ࡛ࣛࣥࢲ࣒ࡢᩘ࡛ḟୡ௦ࡢᏊᏞ
ࢆసࡿࠋࡇࡢࡼ࠺࡟సࡽࢀࡓᏊᏞࡢ࿴ࡀ =㸦➨୍ୡ௦㸧࡟࡞ࡿࠋ== ࡼࡾ = ࢆ㐀
ࡗ࡚⥆ࡅࡿࠋ୍ࡘࡢඖ⣲ࡣྠࡌࠕ࠾ぶࠖࡼࡾసࡽࢀࡓࠕᏊᏞࠖ࡜ࡶྠ୍ୡ௦ࡢ
௚ࡢඖ⣲࡜ࡶ⊂❧࡛࠶ࡿࠋ➨㹬ୡ௦ࡢඖ⣲ࡢᩘ㔞ࡣ☜⋡ኚᩘ =㹬ࠋ㸦㸧ࡼࡾほ
=Q㸻 ࡞ࡽࡤ =Q㸻 ࡟࡞ࡿࠋࡇ࠺ࡋ࡚ࠊ௵ពୡ௦࡟࠾࠸࡚ࡶᩘ㔞 ࡢព
ࡿ࡜ࠊ=
࿡ࡣࡇࡢศᯞ㐣⛬ࡀ඲⁛࡟࡞ࡿࠋ
ศᯞ㐣⛬࡟ࡘ࠸࡚ࡢ◊✲ࡀ㛗࠸Ṕྐࡢ࠶ࡿࡶࡢ࡛࠶ࡿࠋ◊✲⪅ࡓࡕࡢண᝿㏻
ࡾࠊࡇࡢࣔࢹࣝࡣ≀⌮ࡸ⏕≀࡞࡝ࡓࡃࡉࢇࡢศ㔝࡟ᛂ⏝࡛ࡁࡿࠋࡇࡢၥ㢟ࡣ ᖺ )UDQFLV*DOWRQ࡟ࡼࡗ࡚ึࡵ࡚ᑟධࡉࢀࡓࠋ᭱ึࡣ࢖ࢠࣜࢫ㈗᪘ࡢⱑᏐ඲⁛
࡟㛵ࡍࡿ◊✲࡛࠶ࡗࡓࠋࡑࡋ࡚ 5HYHUHQG+HQU\:DWVRQ ࡟ࡼࡗ࡚࠺ࡲࡃゎỴࡉ
ࢀࡓࠋㄽᩥ *DOWRQDQG:DWVRQ࡟㍕ࡗ࡚࠶ࡿࡓࡵࠊศᯞ㐣⛬ࡣ *DOWRQDQG
:DWVRQ 㐣⛬࡜࿧ࡤࢀࡿࠋศᯞ㐣⛬ࡢṔྐ࡟ࡘ࠸࡚ࡣ '.HQGDOOࢆཧ↷ࡋ
ࡓ࡯࠺ࡀⰋ࠸ࠋึᮇࡢᩥ⊩┠㘓ࢆ▱ࡿࡓࡵ 7(+DUULVࢆཧ↷ࡋ࡚ࡶⰋ࠸ࠋ
ศᯞ㐣⛬ࡢၥ㢟࡟㛵ࡍࡿ◊✲࡛ࡣࠊึᮇ್࡟≉࡟ὀពࢆᘬ࠿ࢀࡿࡓࡵࠊ
^Z , n
0,1, 2.... ࢆึᮇ್ L ࡢศᯞ㐣⛬ࢆ⾲♧ࡍࡿࠋከࡃࡢሙྜࡣ =㸻㸯࡜ࡍࡿ
ࡓࡵࠊ Z
(1)
n
i
n
`
Zn ࡛᭩ࡃࠋ
ᩘᏛ࡟࠾࠸࡚ࠊẕ㛵ᩘ㸦ࡰ࠿ࢇࡍ࠺ࠊⱥJHQHUDWLQJIXQFWLRQ㸧ࡣࠊ㸦⮬↛
ᩘ࡛ῧᏐ௜ࡅࡽࢀࡓ㸧ᩘิ^DQ`࡟㛵ࡍࡿ᝟ሗࢆෆໟࡋࡓಀᩘࢆᣢࡘࠊᙧᘧ෢⣭
ᩘ࡛࠶ࡿࠋẕ㛵ᩘࡣࠊ୍⯡⥺ᆺᅇᖐၥ㢟ࡢゎỴࡢࡓࡵ࡟ࢻ࣭ࣔ࢔ࣈࣝ࡟ࡼࡗ࡚
ᖺ࡟ึࡵ࡚⏝࠸ࡽࢀࡓࠋ」ᩘࡢ⮬↛ᩘ࡛ῧᏐ௜ࡅࡽࢀࡿᩘࡢ㓄ิ㸦ከ㔜ᩘ
ิ㸧ࡢ᝟ሗࢆྲྀࡾ㎸ࢇࡔከኚᩘ෢⣭ᩘࢆྠᵝ࡟⪃࠼ࡿࡇ࡜ࡶ࡛ࡁࡿࠋ
ẕ㛵ᩘ࡟ࡣࠊ㏻ᖖᆺẕ㛵ᩘRUGLQDU\JHQHUDWLQJIXQFWLRQࠊᣦᩘᆺẕ㛵ᩘ
࣋
H[SRQHQWLDOJHQHUDWLQJIXQFWLRQࠊࣛࣥ࣋ࣝࢺ⣭ᩘ/DPEHUWVHULHVࠊ࣋
ࣝ⣭ᩘ%HOOVHULHVࠊࢹ࢕ࣜࢡࣞ⣭ᩘ'LULFKOHWVHULHV࡞࡝ᵝࠎ࡞ࡶࡢ
ࡀ࠶ࡿࠋࡇࢀࡽ࡟ࡘ࠸࡚ࡣᐃ⩏࡜౛ࢆᚋ㏙ࡍࡿࠋཎ⌮ⓗ࡟ࡣ࠶ࡽࡺࡿิ࡟ࡘ࠸
࡚ࡑࢀࡒࢀࡢ✀㢮ࡢẕ㛵ᩘࡀᏑᅾࡍࡿ㸦ࡓࡔࡋࠊࣛࣥ࣋ࣝࢺ⣭ᩘ࡜ࢹ࢕ࣜࢡࣞ
ᆺࡣῧᏐࢆ࠿ࡽጞࡵࡿࡇ࡜ࡀᚲせ㸧ࡀࠊᢅ࠸᫆ࡉ࡟ࡘ࠸࡚ࡣࡑࢀࡒࢀࡢ✀㢮
࡛┦ᙜ␗࡞ࡿ࠿ࡶࡋࢀ࡞࠸ࠋ࡝ࡢẕ㛵ᩘࡀ᭱ࡶ᭷ຠ࠿ࡣࠊࡑࡢิࡢᛶ㉁࡜ゎࡃ
29
࡭ࡁၥ㢟ࡢヲ⣽࡟౫Ꮡࡍࡿࠋ
ศᯞ㐣⛬ࢆศᯒࡍࡿ࡜ࡁࡢ୍ࡘ㔜せ࡞ᡭẁ࡛࠶ࡿ㏻ᖖᆺẕ㛵ᩘ㸦*HQHUDWLQJ
)XQFWLRQV㸧࡜࠸࠺ࡢࡣࠊᘧ࡛ᐃ⩏ࡉࢀࡿࠋ
f
f 㹱㸻¦ pk s㹩, s d 1 㸦㸧
k 0
㸦㸧ᘧࢆ⧞ࡾ㏉ࡋ࡚ࡳࡿ࡜
f0 s s, f1 s f > f n ( s)@ f ( s ), f n 1 s ࡞ࡢ࡛
¦ P(1, j )s
j
f ( s);
j
¦ P(i, j )s > f (s)@ , i t 1.
i
j
㸦㸧
j
⌮⏤ࢆ⡆༢࡛ㄝ᫂ࡍࡿ࡜
㸦㸧ࡼࡾ ¦ P 1, j s j
¦p s
j
j
㸦㸧࡟ᑐࡋ࡚ l1 l2 ...li
¦ P i, j s ¦ ¦
j
j l1 l2 ...li
㹨
j࡜ࡍࡿ࡜
P l1 P l2 ...P li s l1 l2 ...li
¦ P l s ¦ P l s ...¦ P l s
l1
1
l1
2
f s ࠋ
j
j
l2
l2
i
li
li
ª¬ f s º¼ , i t 1
i
ࡇࡇ࡛ࡣ 3㹬LMࢆ㹬ᅇ㑄⛣☜⋡࡟ࡋ࡚3㹬㸯M㸻3=Q M_= ࠋ
ࢥࣝࣔࢦࣟࣇ୙➼ᘧࡼࡾ
¦P
n 1
(1, j ) s j
j
¦¦ P (1, k ) P(k , j)s ¦ P (1, k )¦ P(k , j)s
j
n
j
j
n
k
k
j
¦ Pn (1, k ) > f (s)@
k
k
¦ P (1, j )s
n
j
j
f ( n ) ( s ) ࠉ࡜ࡍࡿ࡜f n 1 ( s )
f n > f ( s)@. ࡇࢀ࡛ࡣ୍ࡘࡢ㔜せ࡞බᘧ㸦㸧ࢆド᫂ࡉࢀࡓ㸸
f n s f n s 㸦㸧
30
㸦㸧࡜㸦㸧ᘧࡼࡾ
f
¦ P i, j s
n
j 0
j
¬ª f n s ¼º . i
ศᯞ㐣⛬ࡢ࣓ࣔࣥࢺࡣୗࡢᘧ࡛⾲ࡍࠋ
EZ1 ¦ P (1, j ) j f ' (1) m
ࡉࡽ࡟
EZn
¦ P (1, j ) j
n
f n'1 (1) f ' (1) ... ª¬ f ' (1) º¼
f n' (1)
ࡲࡓࠊḟࡢᘧࢆ⏝࠸ࠊf n''1 (1)
f n'' (1)
n
mn
2
f '' (1) ª¬ f n' (1) º¼ f ' (1) f n'' (1), f n'' (1)ࢆồࡵࡽࢀࡿࠋ
f '' (1) ª¬ m 2 n 2 m 2 n 3 ... m n 1 º¼
ࡇࡢᨾ࡟V 2㸻YDULDQFH=࡜ࡍࡿ࡜ࠊ
1
var Zn
V 2 m n 1 (m n 1)
m z1
°
if
m 1
®
m 1
°
nV 2
¯
ࠉ
㸦㸧
ࣃ࣮࣡ࢩ࣮ࣜࢬศᕸ
ࣃ࣮࣡ࢩ࣮ࣜࢬศᕸࡣ࡭ࡁ⣭ᩘࡼࡾᵓᡂࡉࢀࠊ
㒊ศ㞟ྜ 1 ୖࡢ㞳ᩓศᕸ࡛࠶ࡿࠋ
a (a0 , a1 , a2 ...) ࡀ㠀㈇ᐇᩘิ࡜᝿ᐃࡍࡿࠋᡃࠎࡣಀᩘࡢࢩ࣮ࢣࣥࢫ࡜ࡋ࡚ᣢࡘ
࡭ࡁ⣭ᩘ࡟⯆࿡ࢆᣢࡗ࡚࠸ࡿࠋ᭱ึ㹬ḟࡢ㒊ศ࿴ࢆ᭩࠸࡚࠾ࡃ㸸
g n T ࡑࡋ࡚ࣃ࣮࣡ࢩ࣮ࣜࢬ㹥ࡀ g T g T n
¦a T
k
k
,T R 㸦㸧
k 0
lim gn T ,T R ࡜ᐃ⩏ࡉࢀࡿ㸸
nof
f
¦a T
n
n
,T R 㸦㸧
n 0
ȟ㸻 ࡢ᫬ࠊࡑࡢࢩ࣮ࣜࢬࡀ཰᮰ࡍࡿࡇ࡜ࢆὀពࡍࡿࠋࡑࡋ࡚ g 0 a0 ࡇࡢሙ
ྜࡢ࡯࠿ࠊr > 0, f ࡛ T r ࡢ࡜ࡁࢩ࣮ࣜࢬࡀ⤯ᑐ཰᮰ࡑࡋ࡚ T ! r ࡢ࡜ࡁⓎᩓ
ࡍࡿࡼ࠺࡞㹰ࡀᚲࡎᏑᅾࡍࡿࠋ㹰ࡀ཰᮰ࡢ༙ᚄ࡛࠶ࡾࠊ㹰㸼 ࡜௬ᐃࡍࡿࠋ㹰㸺
࡛҄࠶ࢀࡤࡇࡢࢩ࣮ࣜࢬࡀ㹰࡛⤯ᑐ཰᮰࠶ࡿ࠸ࡣⓎᩓࡍࡿࠋ㸫㹰࡛ࡇࡢࢩ࣮ࣜ
ࢬࡀ⤯ᑐ཰᮰࠶ࡿ࠸ࡣ᮲௳཰᮰࠶ࡿ࠸ࡣⓎᩓࡍࡿࠋࡇࢀ࠿ࡽ T > 0, r ࡢ✵㛫࡟ไ
㝈ࡍࡿࠋࡇࡢ✵㛫ࡀࣃ࣓࣮ࣛࢺ✵㛫࡛࠶ࡿࠋU㸺҄ࡲࡓȟ㸻㹰ࡢሙྜḟࡢ⤖ᯝ
ࡶᡂ❧ࡍࡿ࠿ࡶࡋࢀ࡞࠸ࡀㄝ᫂ࡀ」㞧࡛୙ᚲせ࡛࠶ࡿࠋ
☜⋡ኚᩘ 1 ࡣ㛵ᩘ㹥㸦ࣃ࣓࣮ࣛࢱ T > 0, r ࢩ࣮ࢣࣥࢫ a (a0 , a1 , a2 ...) ࢆᣢࡘ㸧࡟
㛵㐃ࡘࡅࡽࢀࡓࣃ࣮࣡ࢩ࣮ࣜࢬศᕸࢆ᭷ࡍࡿࠋ1 ࡀ㞳ᩓᆺ☜⋡ᐦᗘ㛵ᩘࢆᣢࡘ㸸
anT n
㸦㸧
, n 0,1, 2.... fT n g T ࡑࡋ࡚ศᕸ㛵ᩘ FT n ¦
n
f k ࡛࠶ࡿࡓࡵࠊ1ࡢศᕸ㛵ᩘ
k 0 T
31
g n T ,n
g T FT n 0,1, 2.... 㸦㸧
Jȟࢆ฼⏝ࡋ࡚ 1ࡢ☜⋡ẕ㛵ᩘࢆ⡆༢࡟సࢀࡿ㸸
f
1 f
t 0, r / T ࡟ᑐࡋ࡚ࠉP t ¦ t n fT n an (tT ) n
¦
g T n 0
n 0
g tT ࠉࠉࡼࡗ࡚ g T 1ࡢ☜⋡ẕ㛵ᩘ 3 ࡣḟࡢᘧ࡛⾲ࡍ㸸
g tT r
, ࠉt P t E t N T
g T ᚑࡗ࡚
g c tT Pc t T , Pcc t g T 㸦㸧
g cc tT 2
T ࡀ᫂ࡽ࠿࡛࠶ࡿࠋ
g T ☜⋡ẕ㛵ᩘࡢᛶ㉁࡟ࡼࡾ E ( N )
Pc 1 ,V ( N )
EN Pcc 1 Pc 1 ª¬ Pc 1 º¼ ࠋࡼࡗ࡚
2
T g c T g T 㸦㸧
V ( N ) T 2 g cc T / g T T g c T / g T [T g c T / g T ]2 ࠋ
ࡲࡓࠊE ( N )࡜V ( N ) ࡀḟࡢ㛵ಀ࡟‶ࡓࡍࡇ࡜ࢆὀពࡋࠊE ( N )࡜V ( N ) ࢆࡉࡽ࡟ᐃ
⩏ࡍࡿࡇ࡜ࡀ࡛ࡁࡿࠋ
E ( N ) ࡣȟ࡟ࡘ࠸࡚ᚤศࡍࡿࠋ
wE ( N ) / wT
T g cc T / g T g c T / g T T g c T / g 2 T ࡀ࠶ࡿࡓࡵ
2
T wE ( N ) / wT㸻V ( N ) ࠉࠉࡼࡗ࡚
1
§ w
·
V ( N )㸻¨
log T ¸ © wE ( N )
¹
㸦㸧
32
⛣Ẹ㡯ࡢ࠶ࡿศᯞ㐣⛬ࡢ㏲ḟ☜⋡ẚ᳨ᐃࡢ㔜せ࡞⤖ㄽ
ࣔࢹࣝ࡜≉ᛶ್ࡢ᥎ᐃ
z0
1, z n
zn1
¦[
n ,i
Yn 㸦㸧
i 1
ศᯞ㐣⛬࡛ࡣࠊ z0 ࡛᭱ึࡢ㞟ᅋࡢಶయᩘࢆ⾲ࢃࡍࠋ zn ࡣ➨㹬ୡ௦ࡢ㞟ᅋࡢಶయ
ᩘࢆ⾲ࢃࡍࠋ㞟ᅋࡢྛࠎࡢಶయࡣ௚ࡢࡶࡢ࡜ࡣ⊂❧࡟Ꮚࢆ⏕ࡳࡔࡍࠋࡓ࡜࠼ࡤࠊ
➨㹬㸫㸯ୡ௦➨ L ␒┠ࡢಶయࡀ [ k ,i ಶࡢ᪂ࡋ࠸ಶయࢆ⏕ࡳࡔࡍࠋゝ࠸᪉ࢆኚ࠼ࢀ
ࡤࠊ [ k ,i ࡣ Z k 1 ୡ௦ࡢྛᡂဨࡢᏊᏞᩘ࡛࠶ࡿࠋ Z k 1 ୡ௦ྛᡂဨࡢᏊᏞࡢ඲యࡣ
zn1
¦[
n,i
࡛⾲♧ࡋࠊᏊᏞ㡯࡜࿧ࡪࠋ Yn ࡀ⛣Ẹ㡯࡛࠶ࡾࠊ➨㹬ୡ௦ࡢእ᮶ࡢᡂဨ࡛࠶
i 1
ࡿࠋࡇࡢ Z n 1 ୡ௦ᏊᏞ㡯࡜➨㹬ୡ௦ࡢእ᮶㡯ࢆྜࢃࡏ࡚ࠊ➨㹬ୡ௦ࡢ㞟ᅋࡢ඲
య Z n ࡟࡞ࡿࠋࡇࡢ Z n ᩚᩘ್ࡢ☜⋡ኚᩘࡢิ࡛࣐ࣝࢥࣇ㐃㙐ࢆࡘࡃࡿࠋ[ ni ࡀᏊᏞ
ᩘ࡜ࡋ࡚㠀㈇࡛࠶ࡿࠋ
Yn ࡀ⛣Ẹ㡯࡜ࡋ࡚ṇ࡛࠶ࢀࡤእ࠿ࡽ㞟ᅋࡢ୰࡟᮶ࡿሙྜࠊ㞽࡛࠶ࡿ࡜ࡁ࡟㞟ᅋ
ࡢ୰࡜እ஺ὶࡀ࡞࠸ሙྜࠊ㈇࡛࠶ࢀࡤ㞟ᅋࡢᡂဨࡀ㞟ᅋࡢእ࡟ฟ࡚࠸ࡃሙྜࠊ
ࡑࢀࡒࢀࢆ⾲ࢃࡍࠋ
Z Q
ࡇࡢࣔࢹ࡛ࣝࡣࠊほ ࡉࢀࡿࡶࡢ ZQࠊYn ほ ࡉࢀ࡞࠸ࡶࡢ ¦ [ ni ࡜௬ᐃࡍࡿࠋ
i 1
ᅗ ࡣᏊᏞࠊ⛣Ẹࠊ㞟ᅋ඲యࡢ㛵ಀࢆ⾲ࢃࡍࠋ㯮࠸ⅬࡀᏊᏞࠊⓑ࠸Ⅼࡀ⛣Ẹࠊ
⦪⥺ୖࡢ඲㒊Ⅼࢆྜࢃࡏ࡚ୡ௦ࡢ඲య࡟࡞ࡿࠋ
㸱ୡ௦
㸰ୡ௦
㸯ୡ௦
᫬㛫㍈
㸮ୡ௦
ࡏୡ௦
ᅗ 2-1
┤ほⓗ࡟௨ୗࡢ౛࡛ヲࡋࡃㄝ᫂ࡍࡿࠋ
33
౛㸯 ࢿࢬ࣑ㅮ
ࡡࡎࡳㅮࡣࠊࣆ࣑ࣛࢵࢻᆺࡢ㝵ᒙ⤌⧊ࢆᙧᡂࡋࠊୖࡢ㝵ᒙ࡟࠸ࡿ⪅ࡀୗࡢ⪅࠿
ࡽ㔠ࢆ྾࠸ୖࡆࡿ࡜࠸࠺ᙧ࡛ࠊ㓄ᙜࡢࡳࢆ┠ⓗ࡜ࡋ࡚άືࡋ࡚࠸ࡿ⤌⧊࡛࠶ࡿࠋ
ࡓ࡜࠼ࡤࠊ㸯 ୓෇࡛఍ဨ࡟࡞ࡿࠋ᪂ࡋ࠸఍ဨࢆ࡝ࢇ࡝ࢇ⤂௓ࡍࡿࠋ᪂఍ဨࡀ
ேቑ࠼ࡿࡈ࡜࡟ ୓෇ࡢ⤂௓ᩱࢆࡶࡽ࠼ࡿࠋ⤂௓࡟ࡼࡿ᪂఍ဨࡢ᪉ࡶࡑࡢᚋ
᪂ࡋ࠸఍ဨࢆ⤂௓ࡋࠊ ேࡈ࡜࡟⤂௓ᩱ ୓෇ࢆࡶࡽ࠼ࡿࠋࡇࡢ᪉ἲ࡛ㅮࡢ୺
ദ⪅ࡣ㯲ࡗ࡚࠸࡚ࡶ఍ဨࡀቑ࠼࡚⾜ࡁࠊ኱ඈࡅ࡛ࡁࡿࠋึᮇ࡟఍ဨ࡟࡞ࡗࡓே
ࡶ࠿࡞ࡾඈ࠿ࢀࡿࠋ⨾࿡ࡋ࠸ඈࡅヰ࡞ࡢ࡛ࠊ఍ဨᩘࡀ⇿Ⓨⓗ࡟ቑ࠼ࡿࠋࡕࡻ࠺
࡝㰡ࡢᏊᏞࡀ⇿Ⓨⓗ࡟ቑ࠼࡚⾜ࡃᵝᏊ࡟ఝ࡚࠸ࡿࡢ࡛ࠊࡡࡎࡳㅮ࡜ゝࢃࢀࡿࠋ
࡛ࡶே㛫ࡢᩘ࡟㝈ࡾࡀ࠶ࡿ௨ୖࠊ᪂ධ఍ဨࡣᚲࡎ㢌ᡴࡕ࡟࡞ࡗ࡚ࠊ᭱ᚋࡢ᪉࡟
఍ဨ࡟࡞ࡗࡓேࡓࡕࡣ኱ࡁࡃᦆࡋࡗ࡚ࡋࡲ࠺ࠋࡇࢀࡣἲᚊ࡛⚗Ṇࡉࢀࡿࡶࡢ࡛
࠶ࡿࠋࡇࡢ౛࡛ࡣ୍ேᑐࡋ࡚᪂⤂௓఍ဨᩘࡀ [ ni ࡛࠶ࡿࠋ➨㸯௦ࡢ఍ဨ z1 ࢆ᪂
z
ࡋ࠸ேࢆ⤂௓ࡋ࡚ࠊ➨஧௦ࡢ఍ဨ z2 ¦
[1i ࡜࡞ࡿࠋࡇࡢ౛࡛ࡣ⛣Ẹ㡯ࡀ࡞࠸ࠋ
1
i
౛㸰 ᪂ᆺ࢖ࣥࣇ࢚ࣝࣥࢨࡢὶ⾜
᪥ᮏᅜෆࡢᝈ⪅ᩘࡣ Z n ࡜ࡍࢀࡤࠊ་⒪ᶵ㛵ࡢሗ࿌ࡼࡾ⤫ィࢆྲྀࡿࡇ࡜ࡀ࡛ࡁࡿࠋ
୍ேࡢᝈ⪅ࡀ [ ni ே࡟⛣ࡍࠋ࢖ࣥࣇ࢚ࣝࣥࢨࡢὶ⾜࡟కࡗ࡚ࠊ᪂ࡋ࠸ᝈ⪅ࡶ࡝ࢇ
ZL
࡝ࢇቑ࠼ࡿࠋࡇࡢ⛣ࡉࢀࡓேࡓࡕࡣࠕᏊᏞࠖ࡜࿧ࡧࠊ ¦ [ ni ࡛⾲ࡍࠋࡇࡢᏊᏞࡢ
i 1
ᩘࡣほ ࡛ࡁ࡞࠸ࠋᅜෆ࡛ࡣ࡞ࡃࠊᾏእࡼࡾ࢖ࣥࣇ࢚ࣝࣥࢨࢆឤᰁࡋ࡚᪥ᮏ࡟
᮶ࡓேࡣ⛣Ẹ㸦 Yn 㸧࡛࠶ࡿࠋ Yn ࡣฟධᅜࡢグ㘓࡞࡝ࡢࢹ࣮ࢱࡼࡾほ ྍ⬟࡛࠶
ࡿࠋࡇࡢ౛࡜ྠࡌࡼ࠺࡟ఏᰁ⑓ࡢὶ⾜ࡀ⛣Ẹ㡯ࡢ࠶ࡿศᯞ㐣⛬ࣔࢹࣝࢆ⏝࠸࡚
ㄝ᫂ࡋ࡚ࡶࡼ࠸ࠋ
᭱ᑬ᥎ᐃࡢࢩ࣑࣮ࣗࣞࢩࣙࣥ࡜⤖ᯝ
࣏࢔ࢯࣥศᕸࡼࡾࡢ᭱ᑬ᥎ᐃ࡜ࢩ࣑࣮ࣗࣞࢩࣙࣥ
[ niࡣᮇᚅ್ Pࡢ࣏࢔ࢯࣥศᕸ࡟ᚑ࠺^ [ n , i `ࡣ⊂❧࡛࠶ࡿࠋ
[ni㹼Po(P )
e P P [ni
L ͐͐Q ͐͐
[ ni !
Yniࡣᮇᚅ್ Oࡢ࣏࢔ࢯࣥศᕸ࡟ᚑ࠺^ Y n , i `ࡣ⊂❧࡛࠶ࡿࠋ
Yni㹼Po(O )
e O O yni
L ͐͐Q ͐͐ ࡉࡽ࡟^ [ n , i `࡜^ Y n , i `ࡣ⊂❧࡛࠶
yni !
ࡿࠋ
᭱ᑬ᥎ᐃ㔞ࡣ
34
Pˆ
n
n
i 1
i 1
¦ Zi ¦ Yi
n
¦Z
n
,Oˆ
¦Y
i
i 1
n
Y 㸦㸧
i 1
i 1
ȣࡢ್ࡣ୕ࡘࡢྍ⬟ࡀ࠶ࡿࠋȣ㸺㸦VXEFULWLFDO㸧ȣ㸻㸦FULWLFDO㸧࡜ȣ㸼
㸦VXSHUFULWLFDO㸧
ࠋ࢖ࣥࣇ࢚ࣝࣥࢨࡢὶ⾜࡟ࡣࠊȣࡣ = ࡢᮇᚅ್࡜ࡋ࡚ࠊᝈ⪅
ࡣᖹᆒⓗ࡟ఱே࡟⛣ࡍ࠿ࠊࡑࡢᏊᏞࡢつᶍࢆ⾲ࢃࡍࠋȣ㸺㸯ࡢ᫬࡟ࠊᏊᏞࡢつ
ᶍࡀࡕ࠸ࡉ࠸ࡃ࡞ࡿ࠿ࡽࠊᝈ⪅ࡢつᶍࢆ⦰ῶࡋ࡚࠸ࡿࡇ࡜ࢆ⾲♧ࡋࠊὶ⾜ࡀ࠸
ࡘ࠿⤊ࢃࡿ࡜⪃࠼ࡽࢀࡿࠋࡉࡽ࡟ࠊ Z n ࡢᮇᚅ್ࢆィ⟬ࡍࡿ࡜
E(Z n )㸻 E(E(Z n Z Q ))㸻 E(Z Q P +Yn )
=E(Z n 1 )P + O
O PO P 2 O 㻸 P n 1O
§ 1 P n · 㸦㸧
¸
© 1 P ¹
O¨
ࡇࢀࡼࡾࠊȣ㸼㸯ࡢ᫬࡟ࠊ Z n ࡢᮇᚅ್ࡀ࠿࡞ࡾ኱ࡁࡃ࡞ࡿ࡜ぢ࠼ࡿࠋᝈ⪅ࡢつ
ᶍࡣ࡝ࢇ࡝ࢇᣑ኱ࡋࠊࣃࣥࢹ࣑ࢵࢡ㸦ឤᰁ⇿Ⓨ㸧ࡢྍ⬟ᛶࡀ㧗࠸࡜⪃࠼ࡽࢀࡿࠋ
࣏࢔ࢯࣥศᕸࡼࡾࡢࢩ࣑࣮ࣗࣞࢩࣙࣥࡢ⤖ᯝࢆぢࡿ㸸
㹫㸸ୡ௦㸹㹬㸸ࢩ࣑࣮ࣗࣞࢩࣙࣥᅇᩘ
ᅗ ࡣ P Q ȣ㸻Ȣ ࡜タᐃࡍࡿ࡜ࡁ Pˆ ࡜ Oˆ ࡢࣄࢫࢺࢢ࣒࡛ࣛ
࠶ࡿࠋ
ᅗ ࡣ P Q ȣ㸻Ȣ㸻㸴࡜タᐃࡍࡿ᫬ Pˆ ࡜ Oˆ ࡢࣄࢫࢺࢢ࣒ࣛ
࡛࠶ࡿࠋ
ᗄఱศᕸࡼࡾࡢ᭱ᑬ᥎ᐃ࡜ࢩ࣑࣮ࣗࣞࢩࣙࣥ
[ niࡣᡂຌ☜⋡㹮ࡢᗄఱศᕸ࡟ᚑ࠺^ [ n , i `ࡣ⊂❧࡛࠶ࡿࠋ
ࠋ
[
[ ni㹼 Geo ( p ) (1 p㸧
p L ͐Q ͐
Yniࡣᡂຌ☜⋡Tࡢᗄఱศᕸ࡟ᚑ࠺^ Y n , i `ࡣ⊂❧࡛࠶ࡿࠋ
ni
Yni
<ni㹼Geo(q) 㸦q㸧
q L ͐͐Q ͐ ࡉࡽ࡟^ [ n , i `࡜^ Y n , i `ࡣ⊂❧࡛࠶
ࡿࠋ
Z niࡢᮇᚅ್ P
1 p Y ࡢᮇᚅ್ O
n
p
1 q
q
᭱ᑬ᥎ᐃ㔞ࡣ
35
n
1 pˆ
pˆ
Pˆ
¦
i 1
n
Z i ¦ Yi
i 1
n
¦Z
n
,Oˆ
¦Y
1 qˆ
qˆ
i 1
n
i
Y i 1
i 1
ᗄఱศᕸࡼࡾࡢࢩ࣑࣮ࣗࣞࢩࣙࣥࡢ⤖ᯝࢆぢࡿ㸸
㹫ୡ௦㹬ࢩ࣑࣮ࣗࣞࢩࣙࣥᅇᩘ
ᅗ ࡣ P Q 㹮㸻ࠊT ࡜タᐃࡍࡿ࡜ࡁ Pˆ ࡜ Oˆ ࡢࣄࢫࢺࢢࣛ
࣒࡛࠶ࡿࠋࡇࡢ᫬ȣࡢ⌮ㄽ್ࡣ Ȣࡢ⌮ㄽ್ࡣ㸷࡛࠶ࡿࠋ
ᅗ ࡣ P Q 㸳㹮㸻ࠊT ࡜タᐃࡍࡿ᫬ Pˆ ࡜ Oˆ ࡢࣄࢫࢺࢢ࣒ࣛ
࡛࠶ࡿࠋࡇࡢ᫬ࠊȣࡢ⌮ㄽ್ࡣ ࡛ࠊȢࡢ⌮ㄽ್ࡣ㸷࡛࠶ࡿࠋ
Histogram of lamdahat
4
0
6
0
F
re
q
u
e
n
c
y
6
0
0
0
2
0
2
0
4
0
F
re
q
u
e
n
c
y
8
0
8
0
1
0
0
1
0
0
1
2
0
Histogram of muhat
0.68
0.69
0.70
0.71
0.72
4.8
muhat
4.9
5.0
5.1
lamdahat
5.2
ᅗ 㸫
36
Histogram of lamdahat
60
0
0
20
20
40
Frequency
80
60
40
Frequency
100
120
80
140
Histogram of muhat
1.00496
1.00502
5.85
5.95
muhat
6.05
6.15
lamdahat
ᅗ㸫
Histogram of lamdahat
4
0
F
re
q
u
e
n
cy
6
0
2
0
4
0
0
2
0
0
F
re
q
u
e
n
cy
8
0
6
0
1
0
0
8
0
Histogram of muhat
0.910
0.920
0.930
8.5
muhat
9.0
9.5
10.0
lamdahat
ᅗ 㸫
37
40
30
0
10
20
Frequency
30
20
0
10
Frequency
40
50
Histogram of lamdahat
50
Histogram of muhat
0.965
0.975
0.985
0.995
7.5
8.0
muhat
8.5
9.0
9.5
lamdahat
ᅗ㸫
㏲ḟ☜⋡ẚ᳨ᐃ
Z1 , Z 2 ,㻸 Z n ࡀ୍ḟ⊂❧ྠศᕸ࡟ᚑ࠺࡟࡞ࡽࡤࠊ㏲ḟ☜⋡᳨ᐃ㸦6HTXHQWLDO
3UREDELOLW\ 5DWLR 7HVW␎⛠ 6357㸧࡛ࡣ㸪୍⯡ࡢ᳨ᐃࡢࡼ࠺࡟ᶆᮏᩘࢆ࠶ࡽ
࠿ࡌࡵᅛᐃࡍࡿࡇ࡜ࢆࡏࡎ࢜ࣥࣛ࢖ࣥࢹ࣮ࢱ࡟ᑐᛂྍ⬟࡛࠶ࡿࠋᶆᮏࢆᚓࡿࡓ
ࡧ࡟㸪᳨ᐃ௬ㄝ㹆 ࡜ᑐ❧௬ㄝ㹆 ࡢᑬᗘࢆ⟬ฟࡋ㸪ࡑࡢẚ / ࡀࢢࣞ࢖ࢰ࣮ࣥ㸦᥇
ᢥ㝈⏺࡜Რ༷㝈⏺ࡢ㛫㸧࡟࠶ࡿ㝈ࡾุ᩿ࢆಖ␃ࡋࠊࢧࣥࣉࣜࣥࢢࢆ⥆ࡅࡿࠋࡇ
ࡢ⠊ᅖࢆ㉸࠼ࡓ࡜ࡁ࡟᳨ᐃࢆ⤊஢ࡍࡿࠋ
ᮏㄽᩥ࡛ࡣࠊ
ࡇࡢ 6357 ᳨ᐃࢆ฼⏝ࡍࡿࠋ
೵Ṇ᫬้ T
H0 : P
°
½°
f z f z 㻸 f1 zn > a , b @¾ min ®t t 1: log 1 1 1 2
f 2 z1 f 2 z2 㻸 f 2 z2 °¯
°¿
P0 VS +1 : P
㸦㸧
P1 ࡢ᳨ᐃࡢ୰ࠊ᳨ᐃ௬ㄝ H 0 ࡀ┿࡞ࡢ࡟ᑐ❧௬ㄝ H1 ࡜ุ
᩿ࡉࢀࡿሙྜࡣࡔ➨୍✀ࡢㄗࡾࡀⓎ⏕ࡋࠊ☜⋡ࡀȘ࡛࠶ࡿࠋᑐ❧௬ㄝ H1 ࡀ┿࡞
ࡢ࡟᳨ᐃ௬ㄝ H 0 ࡜ุ᩿ࡉࢀࡿሙྜࡣࡔ➨஧✀ࡢㄗࡾࡀⓎ⏕ࡋࠊ☜⋡ࡀ㸦㸯ș㸧
࡛࠶ࡿࠋȘࡣ᳨ᐃࡢࢧ࢖ࢬ㸦VL]H㸧࡛࠶ࡿࡢ࡛ࠊ㸮࡜㏆࡙ࡅࡤ㏆࠸࡯࡝ᮃࡲࡋ
࠸ࠋșࡣ᳨ᐃຊ㸦SRZHU㸧࡛࠶ࡿࡢ࡛ࠊ㸯࡜᥋㏆ࡍࢀࡤࡍࡿ࡯࡝Ⰻ࠸ࠋ
38
$:$/'㸦㸧㸦ཧ⪃ᩥ⊩㸯㸧ࡢㄽᩥࡼࡾȘࠊș࡜ DࠊE ࡢ㛵ಀࡣୗグࡢᘧ࡛ィ
⟬࡛ࡁࡿ㸸
a
log
1 E
E
b log
1D
D
㸦㸧
㏲ḟ☜⋡ẚ᳨ᐃࡣᖜᗈ࠸ศ㔝࡛฼⏝ࡉࢀ࡚࠸ࡿࠋᐇ౛ࡣᆅ㟈ࡢ⿕ᐖ᝟ሗฎ⌮ࠊ
㟁Ꮚ᝟ሗ㏻ಙࢩࢫࢸ࣒㐺ᛂⓗヰ⪅㆑ู>ཧ㸴@࡞࡝࡛࠶ࡿࠋᮏㄽᩥࡢࣔࢹ࡛ࣝࡣࠊ
㏲ḟ☜⋡ẚ᳨ᐃࡀ࢖ࣥࣇ࢚ࣝࣥࢨࡢὶ⾜࡜ᢚไࡢㄢ㢟࡟㐠⏝࡛ࡁࡿࠋࡓ࡜࠼ࡤࠊ
࢖ࣥࣇ࢚ࣝࣥࢨὶ⾜୰ࠊᝈ⪅ࡀቑ࠼࡚⥆ࡅࡿ≧ἣࡼࡾࠊࢹ࣮ࢱࡀ࢜ࣥࣛ࢖࡛ࣥ
࠶ࡿࠋ࢖ࣥࣇ࢚ࣝࣥࢨὶ⾜ࢆᢚไࡍࡿࡓࡵ࡟࣡ࢡࢳࣥࢆᢞධࡍࡿࠋࡑࡋࡓࡽ࣡
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ᑐ❧௬ㄝୗ࡛ࢩ࣑࣮ࣗࣞࢩࣙࣥࡢ⌮ㄽⓗ࡞ศᯒ
+㸯㸸P ǼQ ࡢୗ࡛㸦㸧ࡼࡾ
'
O
㹢U s ( 2 )ds dBs , U0 0 ࡜࡞ࡿࡢ࡛, Ȣ㸻㸯ࠊȪ㸻 ࡜タᐃࡋࡓ࡜ࡁࠊ
V
V Us
U㹱ࡣț㸻 ḟඖࠊࢻࣜࣇࢺ '
V
ࠊึᮇ್ ࡢ࣋ࢵࢭࣝ㐣⛬࡛࠶ࡿࠋV ࡢ᫬ࠊ
V §
'· V
'
§T ·
ࠋ
E1 ¨ c ¸㸻E1 U1 㸻 E1 ¨ U1 ¸㸻 E㸦
1 U1㸧
O ©
V¹ O
O
©c¹
ࢻࣜࣇࢻ௜ E1 U1 ࡢ್࡟ࡘ࠸࡚ࡣࠊ
/LQHWVN\ࡼࡾィ⟬᪉ἲࡀ⤂௓ࡉࢀ࡚࠸
ࡿࠋ」㞧࡞ᩘᏛ⌮ㄽ࡛ド᫂ࡉࢀࡓࡓࡵࠊᮏㄽᩥ࡛ࡣ⌮ㄽ㒊ศࡢド᫂ࢆ┬␎ࡍࡿࠋ
ࢩ࣑࣮ࣗࣞࢩࣙࣥࡢ┠ⓗࡣᑐ❧௬ㄝࡢᇶ࡛㹡ࡢ್ࢆㄪᩚࡍࡿࡇ࡜࡟ࡼࡗ࡚೵Ṇ
᫬้ࡢ᮲௳࡟‶ࡓࡍୖࢧࣥࣉࣜࣥࢢࡢ኱ࡁࡉࢆࢥࣥࢺ࣮ࣟࣝࡍࡿࠋࢧࣥࣉࣜࣥ
ࢢࡢ኱ࡁࡉࡢつᶍࡀỴࡵࢀࡤࠊ+ ࡢᇶ࡛㏲ḟ㸦ȪQ㸧ࡢᡭἲ࡟ࡼࡾ஘ᩘ࡟ᑐࡋ࡚
ィ⟬್࡜ࣔࢹࣝᑐࡋ࡚⌮ㄽ್ࡣ୍⮴ࡍࡿ࠿࡝࠺࠿࡜ほᐹࡋࡓ࠸ࠋ
ࢩ࣑࣮ࣗࣞࢩࣙࣥࡢ⤖ᯝ
ࡇࡢ⠇࡛ࡣࢩ࣑࣮ࣗࣞࢩࣙࣥࡢ⤖ᯝࢆ㍕ࡗ࡚࠶ࡿࠋ
ᖐ↓௬ㄝࢆ᳨ドࡍࡿࡓࡵ࡟⤫ィゎᯒࢯࣇࢺ 5 ࢆ⏝࠸ࡓࠋࣉࣟࢢ࣒ࣛࢆ᭩ࡃ㝿ࠊ
ḟࡢࢧ࢖ࢻࢆࡼࡃཧ⪃ࡉࡏ࡚࠸ࡓࡔ࠸ࡓ㸸
KWWSFVHQDURDIIUFJRMSWDNH]DZDUWLSVUKWPOࠋ
6WRSSLQJ7LPHࡢ⌮ㄽ್࡜ࢩ࣑࣮ࣗࣞࢩࣙࣥィ⟬್ࡀ࡯ࡰ୍⮴ࡢ⤖ㄽࡀ࡛ࡁ࡚
࠸ࡿࠋ
㸦⾲ ࢆཧ↷㸧
c mˆ Tc -㸯 ࡀ + ࡢᇶ࡛ᶆ‽ṇつศᕸ࡟ᚑ࠺ࡇ࡜ࢆ᳨ドࡍࡿࡓࡵṇつᛶ࡟᭷ຠ࡞
.Rlmogorov-Smirnov ᳨ᐃ㸦NVWHVW㸧ࢆ⾜ࡗ࡚࠸ࡓࠋ5 ࡢ୰࡛ࡣ S ್࡛⾲ࡍࠋS ࡢ
್ࡣ኱ࡁ࠸࡯࡝ࠊṇつศᕸࡀ☜ᐃ࡛ࡁࡿࠋ୍⯡ⓗ ࡜ẚ࡭ࡿࡶࡢ࡛࠶ࡿࠋ್
࡜ࣄࢫࢺࢢ࣒ࣛࢆྜࢃࡏ࡚ศᯒࡍࡿ࡜ࠊ c mˆ Tc -㸯 ࡢṇつᛶࢆࡼࡃ☜ㄆ࡛ࡁࡓࠋ
58
⾲㹐ࡼࡾᖐ↓௬ㄝࡀ┿ࡢ᫬
P ┿ࡢȪ ୡ௦
ࢩ࣑࣮ࣗࣞࢩࣙࣥᅇᩘ Q
7F
F 7F
Ȣ
㸦೵Ṇ᫬้㸧67
㹮YDOXH
c mˆ Tc -㸯㹎LFWXUH 3LFWXUH 3LFWXUH 3LFWXUH 3LFWXUH 3LFWXUH
⾲㹫 ┿ࡢȪ ୡ௦
Ǽ
Ȣ
䠬㼕㼏㼠㼡㼞㼑㻣㻌
㻼㼕㼏㼠㼡㼞㼑㻤
㻼㼕㼏㼠㼡㼞㼑㻥
㻼㼕㼏㼠㼡㼞㼑㻝㻜
ࢩ࣑࣮ࣗࣞࢩࣙࣥᅇᩘ
Q
7㹡
㸦೵Ṇ᫬้㸧67
㹮YDOXH
c mˆ T㹡 -㸯㻼㼕㼏㼠㼡㼞㼑㻝㻝㻌㻼㼕㼏㼠㼡㼞㼑㻝㻞
59
SLFWXUH
80
60
0
20
40
Frequency
100
120
140
Histogram of B
-2
0
2
4
B
SLFWXUH
60
40
20
0
Frequency
80
100
Histogram of B
-3
-2
-1
0
1
2
3
B
60
SLFWXUH
100
0
50
Frequency
150
Histogram of B
-4
-2
0
2
4
B
SLFWXUH
60
40
0
20
Frequency
80
100
Histogram of B
-3
-2
-1
0
1
2
3
4
B
61
SLFWXUH
100
0
50
Frequency
150
Histogram of B
-2
0
2
4
B
SLFWXUH
30
20
10
0
Frequency
40
50
Histogram of B
-3
-2
-1
0
1
2
3
B
62
3LFWXUH
100
0
50
Frequency
150
200
Histogram of B
-4
-2
0
2
4
B
3LFWUXUH
100
50
0
Frequency
150
200
Histogram of B
-4
-2
0
2
B
63
3LFWXUH
200
0
100
Frequency
300
Histogram of B
-2
0
2
4
B
3LFWXUH
100
50
0
Frequency
150
Histogram of B
-4
-3
-2
-1
0
1
2
3
B
64
3LFWXUH
40
0
20
Frequency
60
80
Histogram of B
-4
-3
-2
-1
0
1
2
B
3LFWXUH
200
100
0
Frequency
300
Histogram of B
-4
-2
0
2
4
B
65
➨ᅄ❶ 6357 ᳨ᐃ㸦6HTXHQWLDO3UREDELOLW\5DWLR7HVW㸧
㏲ḟ☜⋡ẚ᳨ᐃ㸦6357㸧ࡣ $EUDKDP :DOG ᖺࡼࡗ࡚㛤Ⓨࡋࡓ≉ᐃࡢ㏲ḟ௬
ㄝ᳨ᐃ࡛࠶ࡿࠋ1H\PDQ ࡜ 3HDUVRQ ࡢ ⤖ᯝࡣࠊ㏲ḟゎᯒၥ㢟࡜ࡋ࡚෌ᐃᘧ໬
ࡋࡓࠋࡇࡢ⤖ᯝࡣ :DOG ࡟ᙳ㡪ࢆ୚࠼ࡓࠋ
ࡶ࡜ࡶ࡜〇㐀ᴗࡢศ㔝࡛ࡢရ㉁⟶⌮ࡢ◊✲࡛౑⏝ࡍࡿࡓࡵ࡟㛤Ⓨࡉࢀࡓࡀࠊ⌧
ᅾᵝࠎ࡞ศ㔝࡛౑⏝ࡉࢀ࡚࠸ࡿࠋ౛࠼ࡤᆅ㟈㆙ሗࢩࢫࢸ࣒ࡢ◊✲ࡸ་⸆ရ┦㛵
ࡢࢸࢫࢺ࡞࡝ࠋ೵Ṇ᫬้ࡢタᐃࡼࡾࠊᕼᮃࡍࡿ᮲௳࡟⮳ࡿࡲ࡛ࢧࣥࣉࣜࣥࢢࢆ
⥆ࡁࠊᕼᮃ᮲௳࡟‶㊊ࡉࡏࡿ୍␒᪩࠸᫬Ⅼ࡛ࢧࣥࣉࣜࣥࢢࡀṆࡲࡿࠋࡇࡢࡼ࠺
࡞≉ᚩࢆᣢࡕࠊ㆙ሗᛶ࡜ண ᛶࡀ௜ࡃᐇ⏝ᛶࡢ㧗࠸ゎᯒ᪉ἲ࡛࠶ࡿࠋ
:DOG ࡢ 6357 ᳨ᐃ
:DOG ࡢ 6357 ᳨ᐃࡢព࿡
㼃㼍㼘㼐 ࡢ㏲ḟ☜⋡ẚ᳨ᐃ㸦6357㸧ࡣࠊ⌧ᅾ 6,6$ ࡛၏୍ࡢ࣋࢖ࢬ⤫ィⓗฎ⌮࡛࠶ࡿࠋ
࣋࢖ࢬ⤫ィ࡟ᚲせ࡞ࡢࡣࠊ஦๓ࡢࢹ࣮ࢱ཰㞟ࢆᐇ⾜ࡍࡿࡇ࡜࡜ࠊࣔࢹࣝࡢୗ࡛
ᡂᯝࡢᮇᚅࡣ㠀ᖖ࡟ヲ⣽࡟ㄝ᫂ࡍࡿࡇ࡜࡛࠶ࡿࠋ:DOG ࡢ 㻿㻼㻾㼀 ࡟࠾࠸࡚ࠊ≉ᐃ
ࡢ᮲௳ࡢୗࠊࢹ࣮ࢱ཰㞟ࡢỴᐃ᫬࡟‶ࡓࡉࢀ࡚࠸ࡿሙྜ࡟ࡣࠊࢹ࣮ࢱ཰㞟ࠊ཰
㞟ࡋࡓࢹࢱࡢゎ㔘ࢆ⥅⥆࡟㛵ࡋ࡚ྲྀࡽࢀࡿࠋ
㏲ḟゎᯒࡣࠊᐇ㦂ࡲࡓࡣ཰㞟ࡉࢀࡓࢧࣥࣉࣜࣥࢢᩘࡣࠊᐇ㦂㛤ጞ᫬࡟ᅛᐃࡉࢀ
ࡓྂ඾ⓗ࡞௬ㄝ᳨ᐃ࡜ࡣ␗࡞ࡾࡲࡍࠋྂ඾ⓗ࡞௬ㄝ᳨ᐃ࡛ࡣࠊࢹ࣮ࢱࡢ཰㞟ࠊ
ࢹ࣮ࢱࡢゎᯒࢆ⪃៖ࡏࡎ࡟ᐇ⾜ࡉࢀࡿࠋࡍ࡭࡚ࡢࢹ࣮ࢱࡀ཰㞟ࡉࢀࡓᚋ࡟ศᯒ
ࡀ⾜ࢃࢀࠊ⤖ㄽࡀᘬࡁฟࡉࢀࡿࠋࡋ࠿ࡋ࡞ࡀࡽࠊࢩ࣮ࢣࣥࢩࣕࣝศᯒ࡛ࡍ࡭࡚
ࡢࢣ࣮ࢫࡀ཰㞟ࡉࢀࡓᚋ࡟┤᥋ศᯒࡉࢀࠊࡑࡢ᫬Ⅼࡲ࡛࡟཰㞟ࡉࢀࡓࢹ࣮ࢱࡣࠊ
ࡑࡢᚋࠊ᪂ࡓ࡟཰㞟ࡉࢀࡓࢣ࣮ࢫ࠿ࡽᚓࡽࢀࡓ᪂ࡓ࡞᝟ሗࢆ⤌ࡳ㎸ࢇࡔ≉ᐃࡢ
ᩘ್࡜ẚ㍑ࡉࢀ࡚࠸ࡿࠋࡇࡢ࢔ࣉ࣮ࣟࢳࡣࠊࢹ࣮ࢱ཰㞟୰࡟⤖ㄽࢆฟࡍࡇ࡜ࢆ
ྍ⬟࡟ࡋࠊྂ඾௬ㄝ᳨ᐃࡢሙྜࡢࡼ࠺࡟ࠊ᭱⤊ⓗ࡞⤖ㄽࡣࠊ࠾ࡑࡽࡃࠊ࠿࡞ࡾ
᪩࠸ẁ㝵࡛฿㐩ࡍࡿࡇ࡜ࡀ࡛ࡁࡿࠋࢹ࣮ࢱ཰㞟ࡀᑡ࡞࠸౛࡜ࡑࢀ௨๓ࡢỴᐃ஦
㡯ࡢᚋ࡟⤊஢ࡍࡿࡇ࡜ࡀ࡛ࡁࡿࡼ࠺࡟ࠊே㛫ࡢ⏕ά࡜୙ᖾࠊ㔠⼥㈓⵳ࡢ㠃࡛⠇
⣙ࡣࠊ࠿࡞ࡾࡢྍ⬟ᛶࡀ࠶ࡿࠋࡇࡢࡼ࠺࡞౛ࢆ⪃࠼ࡿ࡜㏲ḟゎᯒࡢ฼Ⅼࡣࢃ࠿
ࡾࡸࡍ࠸ࠋ
≧ἣࢆㄝ᫂ࡍࡿࡓࡵ࡟ࡲࡎᇶ‽ࢆタᐃࡋ࡞ࡅࢀࡤ࡞ࡽ࡞࠸ࠋ࠶ࡿᇶ‽ࡢୗ࡛࠶
ࢀࡤࠕᑡ࡞࠸ࠖࠊ࠶ࡿᇶ‽ࡢୖ࡛࠶ࢀࡤࠕከ࠸ࠖ࡜ุ᩿ࡀྍ⬟࡛࠶ࡿࠋ୍ࡘࡣࠊ
ࢹ࣮ࢱ཰㞟๓࡟ࠊࡇࢀࡽࡢᇶ‽್ࢆᐃ⩏ࡍࡿᚲせࡀ࠶ࡿࠋ㏻ᖖ࡛࠶ࢀࡤṇࡢᩘ
ࡢࣃ࣮ࢭࣥࢺࢆ୚࠼ࡿࠋ୍ࡘ┠ࡢࣃ࣮ࢭࣥࢺ௨ୖࡢሙྜࡣᨵၿ㸦ከ࠸㸧ࡉࢀࡓࠋ஧
ࡘ┠ࡢࣃ࣮ࢭࣥࢺ௨ୗࡢ๭ྜࡣຎ໬㸦ᑡ࡞࠸㸧ࡋࡓ࡜ࡋ࡚ࡢ≧ἣࢆุ᩿ࡍࡿࠋ
ᐃ⩏ࡉࢀ㜈್࡟⮳ࡿࡲ࡛ࢧࣥࣉࣜࣥࢢࢆ⥆ࡅ࡚㏣ຍࡍࡿࠋ཰㞟ࡉࢀࡓࢧࣥࣉࣝ
ࢆࡋࡁ࠸್࡟㉸࠼ࡿ᫬Ⅼ࡛ࢹ࣮ࢱ཰㞟ࡀ⤊஢ࡉࢀࡿࠋ
ࡇࡢ஧ࡘࡢࣃ࣮ࢭࣥࢺ್ࡀᚲせ࡜ࡉࢀࠊ⤫ィⓗ࡛ࡣ᳨ฟຊ ș ࡜᭷ពỈ‽ Ș ࡜
࿧ࡤࢀࡿࠋ᳨ฟຊ࡜ࡢࡣࠊ┿ࡢ୰࡟┦㐪Ꮡᅾࡍࡿሙྜࠊ᳨ฟຊ ș ࡢ☜⋡࡛┦
66
㐪ࢹ࣮ࢱࢆほᐹࡍࡿᶵ఍࡛࠶ࡿࠋࡓ࡜࠼ࡤ᳨ฟຊࢆ 㸣࡟タᐃࡉࢀ࡚࠸ࡿࠋࢧ
ࣥࣉࣝ୰ࡿ࡟ᐇ㝿࡟Ꮡᅾࡍࡿᕪ␗ࢆⓎぢࡍࡿࡀ 㸣࡛࠶ࡿࠋȘࡣḟ஦㇟ࡢ☜⋡
ࢆ⾲ࡍ㸸┦㐪ࢆⓎぢࡋࡓ࡜ᛮ࠺ࡢ᳨࡛ᐃࢆ⤊஢ࡉࡏࡓࡢ࡟ࠊᐇ㝿࡟ࡣࡇࡢ┦㐪
ࡀᏑᅾࡋ࡚࠸࡞࠸ࠋȘ ࡣ 㸣ࢆタᐃࡉࢀࡓ᫬ࠊࢧࣥࣉࣜࣥࢢࡣ 㸣ࡢ☜⋡࡛ㄗࡗ
࡚೵Ṇࡉࢀ࡚࠸ࡓ࡜ࡢព࿡࡛࠶ࡿࠋ
୍⯡ⓗ࡛ࡣ᳨ᐃຊ࡜᭷ពỈ‽ࡢタᐃࡣᗈࡃཷࡅධࢀࡽࢀ࡚࠸ࡿ㏻ᖖࡢ್࡟ࡍࡿࠋ
᳨ᐃࡢෆᐜ࡞࡝⪃࠼࡚ࠊኚ᭦ࡶྍ⬟࡛࠶ࡿࠋ
ࡋࡁ࠸್ࡢỴᐃ
:DOG ࡢ⤖ᯝࡼࡾࠊD ࡜㹠ࡣḟࡢ㛵ಀࡀ࠶ࡿࠋDE ࡣṇ࡜㈇ࡢ㜈್࡛࠶ࡾࠊࡇࡇ࡛
Șࡣ➨୍✀ࡢ㐣ㄗ☜⋡șࡣ➨஧✀ࡢ㐣ㄗ☜⋡࡛࠶ࡿࠋ
§ E ·
§ 1 E ·
a log ¨
¸ , b log ¨
¸.
© 1-D ¹
© D ¹
ලయⓗ࡟ぢࡿ࡜6357 ᳨ᐃࡢᖐ↓௬ㄝ࡜ᑐ❧௬ㄝࡣḟࡢࡶࡢ࡜࡞ࡿ㸸
‫ܪ‬଴ ǣ ș ൌ ș଴ ܸܵ‫ܪ‬ଵ ǣ ș ൌ șଵ ḟࡢᑬᗘ㛵ᩘࢆ⏝࠸࡚ 6357 ᳨ᐃࢆ⾜࠺ࠋ
௡ ݂ ሺ‫ ݔ‬ሻ
ଵ ௜
Ȧ୧ ൌ ෑ
ǡ ݊ ൌ ͳǡʹǡ ‫ ڮ‬㸦ͶǤͳ㸧
௜ୀଵ ݂଴ ሺ‫ݔ‬௜ ሻ
6357 ᳨ᐃࡢ┠ⓗࡣࡼࡾ᪩࠸ẁ㝵࡛௬ㄝࡢṇ☜ᛶࢆุ᩿ࡍࡿࡓࡵࠊDE ࡢ᮲௳ࡢ
ࡶ࡜࡛ࠊE ࡀ኱ࡁ࠸࡯࡝ࠊ
D ࡀᑠࡉ࠸࡯࡝ࢧࣥࣉࣜࣥࢢࡢ೵Ṇ᫬้ࡀ㐜ࡃ࡞ࡗ࡚ࠊ
ࡼࡾཝᐦ࡞᳨ᐃࡀ࡛ࡁࡿࠋࡇࡢẁ㝵࡛ᡤᮃࡢ᳨ฟຊࠊㄗሗ⋡ࢆᥦ౪ࡍࡿࡓࡵ࡟ࠊ
ࡋࡁ࠸್ࢆタᐃࡋࡼ࠺ࠋ᳨ฟ⋡ܲ஽ ᳨ᐃຊߚ࡜ᑐᛂࡑࡋ࡚ㄗሗ⋡ܲி஺ ᭷ពỈ
‽Ș࡜ᑐᛂࠋ
⾲グࢆ⡆␎໬ࡍࡿࡓࡵ࡟‫ ؔ ݔ‬ሺ‫ݔ‬ଵ ǡ ‫ ڮ‬ǡ ‫ݔ‬௡ ሻࡑࡋ࡚݂௝ ሺ‫ݔ‬ሻ ؔ σ௡௜ୀଵ ݂ሺ‫ݔ‬௜ ሻǡ 㸦݆ ൌ Ͳǡͳ㸧 ௙భ ሺ௫ሻ
݂଴ ሺ‫ݔ‬ሻ݀‫ݔ‬
భ ௙బ ሺ௫ሻ
ߚ ൌ ܲ஽ ൌ ‫׬‬ோ ݂ଵ ሺ‫ݔ‬ሻ݀‫׬ ݔ‬ோ
భ
ൌ ‫׬‬ோ Ȧ ௡ ሺ‫ݔ‬ሻ݂଴ ሺ‫ݔ‬ሻ݀‫ ݔ‬㹵 ൌ ߙȦ௡ ሺ‫ݔ‬ሻ
భ
ఉ
Ȧ௡ ሺ‫ݔ‬ሻ ORJ ൒ ܾ࡜ࡢ஦ᐇࢆ฼⏝ࡋ࡚ࠊḟࡢᘧࡀᑟࡃࡇ࡜ࡀ࡛ࡁࡿࠋ
ఈ
ͳ െ ߙ ൌ ͳ െ ܲி஺ ൌ ͳ െ න ݂଴ ሺ‫ݔ‬ሻ ݀‫ ݔ‬ൌ න
ோభ
ோబ
݂଴ ሺ‫ݔ‬ሻ
݂ ሺ‫ݔ‬ሻ ݀‫ ݔ‬ൌ න Ȧିଵ
௡ ሺ‫ݔ‬ሻ݂ଵ ሺ‫ݔ‬ሻ ݀‫ݔ‬
݂ଵ ሺ‫ݔ‬ሻ ଵ
ோబ
ଵିఈ
࡞ࡢ࡛Ȧିଵ
൒ ܽିଵ ࠋ 㸦㸧
௡ ሺ‫ݔ‬ሻ ൌ
ଵିఉ
ࡇࢀࡽࡢᘧࡣࠊ⚾ࡓࡕ࡟㐩ᡂࡍࡿࡓࡵ࡟ᚲせ࡞ࡋࡁ࠸್ࡢୖࡢቃ⏺ࢆ୚࠼ࡿࠋ
ܲ஽ ሺߚሻ࡜ܲி஺ ሺߙሻࡣḟࡢ㛵ಀ࡟‶ࡓࡍ
ͳെߚ
ߚ
ܽ ൑
ǡ ܾ ൒ ͳെߙ
ߙ
67
ܽ ൌ
ଵିఉ
ଵିఈ
ఉ
ࡑࡋ࡚ܾ ൌ ࡜ࡍࡿࠋࡇࢀࡽࡢࡋࡁ࠸್ࡣࠊヨ㦂ࡢㄗࡾ☜⋡ࡣࠊ
ఈ
Ș࡜șࡢ㑅ᢥ࡟ࡼࡗ࡚ᣦᐃࡉࢀࡓࡼ࠺࡟ࠊᑡ࡞ࡃ࡜ࡶ㝈ࡾᑠࡉࡃࡍࡿࡇ࡜ࢆಖ
ドࡍࡿࡀࠊ⤖ᯝࡣ࠶ࡲࡾ࡟ࡶಖᏲⓗ࡛࠶ࡿྍ⬟ᛶࡀ࠶ࡿࠋࡇࢀࡼࡾ༢⣧࡞㜈್
ἲ࡛೵Ṇつ๎ࡀ⾲♧ࡉࢀࡿ㸸
ή ൏ Ȧ ௡ ൏ ܾ㸸ࢧࣥࣉࣜࣥࢢࢆ⥅⥆
ή Ȧ ௡ ൒ ǣଵ ࢆ㑅ᢥ
ή Ȧ ௡ ൑ ǣ଴ ࢆ㑅ᢥ
ࡘࡲࡾࠊ㐺ษ࡟ࡋࡁ࠸್ࢆタᐃࡍࡿࡓࡵ࡟ࡣࠊȘșࢆ஦๓࡟Ỵᐃࡋ࡞ࡅࢀࡤ࡞
ࡽ࡞࠸ࠋᩘ್ࡣࠊ࢔ࣉࣜࢣ࣮ࢩࣙࣥ࡟౫Ꮡࡍࡿ࡛࠶ࢁ࠺ࠋ㞳ᩓⓗ࡞ሙྜ࡟ࡣࠊ
ࢧࣥࣉࣝ㛫ࡢ㜈್ࢆ஺ᕪࡍࡿࡇ࡜ࡀ࡛ࡁࡿࠋࡋࡓࡀࡗ࡚ࠊㄗᕪࢆࢥࣥࢺ࣮ࣟࣝ
ࡍࡿࡓࡵࠊࡼࡾ✚ᴟⓗ࡟ࡋࡁ࠸್ࢆタᐃࡍࡿ࠿ࡶࡋࢀ࡞࠸ࠋࡶࡕࢁࢇࠊཝᐦ࡞
ቃ⏺ࡣ㐃⥆ⓗ࡞ሙྜ࡟౑⏝ࡍࡿࡇ࡜ࡀ࡛ࡁࡿࠋ
ศᯞ㐣⛬ࡢ 6357 ᳨ᐃ
H0 : m 1
°
®
' ࡜タᐃࡍࡿࠋ
°¯ H 1: m 1 n
ศᯞ㐣⛬࡟࠾࠸࡚ 6357 ᳨ᐃ࡜ࡣࢧࣥࣉࣜࣥࢢࡀḟࡢᘧ࡟‶ࡓࡍ࡜ࡁ೵Ṇࡍࡿ㸸
Tac,b inf ^n t 1: log / cn a, b ` 㸦㸧
ࡇࡇ࡛ࡣ / cn ࡀᑐᩘᑬᗘẚ࡛࠶ࡿࠋ
/ cn
P1 z1 , y1 㻸 zn , yn zn 1 P0 z1 y1 㻸 zn , yn zn1
P z , y

P z , y
n
i 1
1
i
i
0
i
i
zi 1 zi 1 . 㸦㸧
log /Tc c d a ࡢ࡜ࡁH 0ࢆ㑅ᢥࡍࡿࠋ
a ,b
c
log /T c t b ࡢ࡜ࡁH1ࢆ㑅ᢥࡍࡿࠋ
a ,b
D ࡜ E ࡀࢧ࢖ࢬ㸦VL]H㸧࡜ࣃ࣮࣡㸦SRZHU㸧࡛࠶ࡿࡢ࡛ࠊᣦᐃỈ‽࡟ᚑ࠺ࠋ
㸦౛࠼
ࡤ ࡜ ࡞࡝㸧ࠋD ࡜ E ࡢ್ࡣ᳨ᐃࡢ᳨ฟຊࢆ⾲ࡍ㔜せ࡞ᩘ್࡛࠶ࡿࠋ᳨ฟຊ
ࡣࡇࡢ⾲ࡢࡼ࠺࡟⌮ゎࡋ࡚࠾ࡃࠋ
ᐇ㝿࡟ᡂࡾ❧ࡗ࡚࠸ࡿࡢࡣ
ᖐ↓௬ㄝ +㸮
ᑐ❧௬ㄝ +㸯
᳨
+
ṇࡋ࠸
➨஧✀㢮ㄗࡾ
ᐃ
㸯㸫Ș
ș
⤖
+
➨୍✀㢮ㄗࡾ
ṇࡋ࠸
ᯝ
Ș
㸯㸫ș
ࡲࡎศᯞ㐣⛬࡟࠾࠸࡚ᑐᩘᑬᗘẚࢆぢ࡚ࡳࡼ࠺ࠋ➨୕❶࡛ࡣ㸦㸧ᘧ࡛ᑐᩘ
68
ᑬᗘẚࢆ୚࠼ࡽࢀࡓࠋ㔜せ࡞ࡢ࡛ࠊࡶ࠺୍ᗘ᭩࠸࡚࠾ࡃࠋ
ln log Ln࡜ࡍࡿࠋ
ln
¦ ^log a
n
zi 1 zi yi
i 1
lnc
`
( zi yi㸧
log T zi 1 log A(T ) log byi yiI log B I ࠉࠉ㸦4.5㸧
log / cn .
n [ct ]࡛᭩ࡁ᥮࠼ࡿ࡜
l[cct ]
'
cV
o
Z 1 2 [ ct ]
'
cV
[ ct ]
2
¦ Yi i 1
'Ot '
Xt ࠉ 2 2
V
V
2V
'
2
2
³
t
'2
2c 2V 2
¦Z
㸩o(
i 1
i 1
1
) ࠉࠉࠉࠉࠉࠉࠉࠉ㸦4.6㸧
c2
ࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ㸦4.7㸧
X u du
0
[ ct ]
ᚑࡗ࡚ศᯞ㐣⛬࡟ᑐࡍࡿ 6357 ᳨ᐃࡢ೵Ṇ᫬้ࡣḟࡢᘧ࡛ᐃ⩏ࡉࢀࡿࠋ
inf ^n : lnc a, b ` inf ^ct : l[cct ] a, b `
Tac,b
୧㎶࡟ t ࡛๭ࡿ࡜
Tac,b
c
inf ^t : l[cct ] a, b `
ࠉࠉ
o Ta ,b

'
{ inf ®t : ࠉ 2
V
¯
2
t
½
'
O
t
'
X t ࠉ 2 2 ³0 X u du (a, b) ¾ ࠉࠉࠉࠉࠉ㸦4.8㸧
2V
V
¿
ࡢ &,5 㐣⛬ࡼࡾ dX t 㸦O -'X t )dt V X t dW ࠋᚑࡗ࡚ X t ࡣḟ஧ࡘࡢ⾲⌧ࡀ࠶ࡿࠋ
H 0 : m 1, '
ࠉࠉX t
0
t
t
0
0
³ Odu V ³
X u dWu ࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ㸦4.9㸧
H1 : m 1 ' / c, ' ! 0
ࠉࠉX t
t
t
0
0
³ Odu ³ 'X
u
du V ³
t
0
X u dWu ࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ㸦4.10㸧
ࠉ
ࡇࡢ᫬௬ㄝࡈ࡜࡟6357ࡢ⾲⌧ࡀ␗࡞ࡿࠋ
H 0 : m 1, '
0

'§ '
Ta ,b { inf ®t 㸸 ¨
V © 2V
¯
H 1 : m 1 ' / c, ' ! 0
ࠉࠉTa ,b
³
ࠉࠉTa ,b

'§ '
Ta ,b { inf ®t 㸸 ¨ V © 2V
¯
t
0
X u du ³
½
·
X u dWu ¸ (a, b) ¾ ࠉࠉࠉ㸦4.11㸧
¹
¿
t
0
³
t
0
X u du ³
t
0
½
·
ࠉ㸧
X u dWu ¸ (a, b) ¾ ࠉࠉ㸦4.12
¹
¿
69
ࡇࡇ࡛ࡣࡶ࠺୍ᗘ ''6 ᐃ⌮ࢆ౑ࡗ࡚ࠊ᫬㛫ኚ᭦ࢆࡍࡿࠋࡑ࠺ࡍࡿ࡜6357 ࡢ⾲⌧
ࢆࣈࣛ࢘ࣥ㐠ື࡛⾲ࡏࡿࡇ࡜ࡀぢࡘࡅࡓࠋ㹂㹂㹑ᐃ⌮ࡢ᮲௳ࡣ➨୕❶࡜ྠᵝ࡟
タᐃࡍࡿࠋ
M 㸻 t X s dW ½
s°
° t ³0
°
t
°
DDS࡟ࡼࡗ࡚ ® M t ³ X s ds ¾࡟ᑐࡋ࡚Bs M U s࡜ࡋ࡚,
0
°
°
1
°U s㸻 M s
°
¯
¿
'
P㸻㸼0࡜ࡍࡿ࡜
2V

ª a b º½
(4.13)
H 0 : W a,b { inf ® s t 0 : P s Bs « , » ¾ ࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ
¬ 2 P 2P ¼ ¿
¯

ª a b º½
(4.14)
H1 : W a,b { inf ® s t 0 : P s Bs « , » ¾ ࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ
¬ 2 P 2P ¼ ¿
¯
^
^
`
`

b ½
➨✀ㄗࡾD 㸻PH 0 lTcc t b o P ® PW a,b B t
¾ 㸦 4.15㸧
a ,b
W a ,b
2P ¿
¯

a ½
➨✀ㄗࡾE 㸻PH1 lTcc d a o P ® PW a,b BW d
¾ ࠉ㸦 4.16㸧
a ,b
a ,b
2P ¿
¯
௨ୖࡼࡾWald ࡢ᪉ἲ࡟ࡼࡾ㜈್ a, b ࡢ್ࢆỴࡵࡿࡇ࡜ࡀṇᙜ໬ࡉࢀࡿࠋ
a
§ E
log ¨
© 1D
·
¸,b
¹
§1 E ·
log ¨
¸
© D ¹
௨ୖࡢィ⟬᪉ἲࢆ฼⏝ࡋ࡚ࠊࢩ࣑࣮ࣗࣞࢩࣙࣥࢆᐇ⾜ࡋ࡚ࡳࡓࠋ஦๓࡟Ș࡜ș
ࡢ್ࢆタᐃࡋࡓࠋ
㸦㸧࡜㸦㸧࡛ィ⟬ࡉࢀࡓ್࡜ࡃࡽ࡭ࡓ⤖ᯝࡣⰋ࠸࡜⪃
࠼ࡿࠋ೵Ṇ᫬้ࡢ⤖ᯝࡣ⌮ㄽ್ࡢィ⟬ࡀ୙༑ศ࡛ࠊࢩ࣑࣮ࣗࣞࢩ್࡛ࣙࣥࡀࡼ
࠸࡜ホ౯࡛ࡁ࡞࠸ࠋ
70
ศᯞ㐣⛬࡟ᑐࡍࡿ 6357 ᳨ᐃࡢࢩ࣑࣮ࣗࣞࢩࣙࣥ
⾲ + ࡀ┿ࠊ
Ș
㹫㸻㸯
ș
➨ ✀㢮ࡢㄗࡾ + ࡀ┿ࠊ
㹫㸻
Ș
ș
➨ ✀㢮ࡢㄗࡾ 71
➨஬❶ ⛣Ẹ㡯ࡢ࠶ࡿ ኚ㔞ศᯞ㐣⛬
㸳 ࣔࢹࣝ࡟ࡘ࠸࡚ࡢゎㄝ
ࣔࢹࣝ࡜┤ឤࡢព࿡
ྛኚ㔞ࡣ☜⋡✵㛫㸦Ȑ$3㸧࡛ᐃ⩏ࡉࢀࡿࠋ =155ࢆࡑࢀࡒࢀ㠀㈇ᩚᩘࠊ
ṇᩚᩘࠊᐇᩘࠊ㠀㈇ᐇᩘࢆ⾲ࡍࠋ
NMѮ=ࡑࡋ࡚ LOѮ^`ࡢ࡜ࡁ➨ L ✀㢮≀ࡢ➨㹩ୡ௦ࡢᩘࡣ㹖㹩L ࡛⾲♧ࡉࢀ
ࡿࠋࡇࡢᩘࡣ➨ N㸯ୡ௦ࡢ≀ࡢྛᡂဨࡢᏊᏞ࡜እ࠿ࡽ㣕ࡧ㎸ࢇࡔᡂဨ㸦⛣Ẹ㸧
ࡢ࿴࡛Ỵࡵࡿࠋࡇࡇ࡛ࡣࠊL 㸰✀㢮ࡢ≀ࡀ࠶ࡾࠊࢱ࢖ࣉ ࡜ࢱ࢖ࣉ ࡜࿧ࡪ
ࡢሙྜࢆ⪃࠼࡚ࡳࡿࠋ
㸦㸧
ª X k ,1 º X k 1,1 ª [ k , j ,1,1 º X k 1,2 ª [ k , j ,2,1 º ª H k ,1 º
» ¦ «[
» «H »
« X » ¦ «[
¬ k ,2 ¼ j 1 ¬ k , j ,1,2 ¼ j 1 ¬ k , j ,2,2 ¼ ¬ k ,2 ¼
ヲࡋࡃゎㄝࡍࡿ࡜ࢱ࢖ࣉ㸯ࡢࡶࡢ N ୡ௦ࡢᩘ ;N ࡣ୕ࡘ㒊ศࡢ࿴࡛Ỵࡵࡽࢀࡿ㸸
N ୡ௦࡛ࢱ࢖ࣉ㸯ࡢࡶࡢ㸦;N㸫㸯㸧ࡼࡾ⏕ࡲࢀࡓ N ୡ௦ࢱ࢖ࣉ㸯ࡢᏊᏞ
㸦ȥ㹩M㸧
㸹ղN ୡ௦࡛ࢱ࢖ࣉ㸰ࡢࡶࡢ㸦;N㸫㸯㸧ࡼࡾ⏕ࡲࢀࡓ N ୡ௦ࢱ࢖ࣉ
㸯ࡢᏊᏞȥ㹩M㸹ճ➨㹩ୡ௦࡛እ࠿ࡽ㣕ࡧ㎸ࢇࡔࢱ࢖ࣉ㸯ࡢࡶࡢ㸦Ȝ㹩㸧
ࠋ
M ࡣྛ㞟ᅋࡢ➨㹨␒┠ࡢᡂဨ࡜⾲♧ࡋ࡚࠾ࡃࠋ
࠸ࡃࡘࡢ౛࡛ᣲࡆ࡚࠾ࡃࠋ
౛㸯㸸࠶ࡿⰼࡣⓑ㸦ࢱ࢖ࣉ㸯㸧࡜㉥㸦ࢱ࢖ࣉ 㸧஧Ⰽࡀ࠶ࡿࠋ୍ᰴ࡟࠾ⰼࡢⰍࡣ
ྠ୍࡛࠶ࡿࡀࠊ✀ࡀ࡛ࡁࡓࡽࠊⓑ࡜㉥࡝ࡕࡽ࡛ࡶ⏕ࡲࢀࡿࠋⓑⰍࡢࠕẕぶࠖ
㸦ȥ㹩㸫㸯M㸧࡛࠶ࡿࡀࠊⓑࡢࠕᏊ౪ࠖ ᰴ㸦ȥ㹩M㸻㸧࡜㉥ࡢࠕᏊ౪ࠖ㸯ᰴ㸦ȥ
㹩M㸻㸯㸧ࡀ㐀ࢀࡿࠋᙜ↛㉥Ⰽࡢ࠾ⰼࡶ࡝ࡕࡽࡢⰍࡢᏊ౪ࡀ⏕ࡲࢀࡿࠋ⏕≀ヨ
㦂୍࡛ᰴ࡜୍ᰴศࡅ࡚ࠊᏊᏞᩘࢆᩘ࠼ࡽࢀࡿ⎔ቃࢆసࡿࠋࡑࡋ࡚ྛୡ௦ⓑⰼ࡜
㉥ⰼࢆᩘ࠼ࡿ㸦;㹩 ࡜ ;㹩㸰㸧
ࠋഅࠎⰍࡢࣂࣛࣥࢫࢆ⪃࠼࡚ࠊ᪂ࡋ࠸࠾ⰼࢆ⿵඘ࡍ
ࡿࡇ࡜ࡀ࠶ࡾࠊࡇࢀࡣ⛣ẸࡢȜ㹩 ࡜Ȝ㹩 ࡛࠶ࡿࠋ
౛㸰㸬஧ࡘࡢ㖟⾜࡬㡸㔠ࡍࡿࠋ㹩ࡢᮇ㛫ࡣ㸯᭶㛫࡟ࡍࡿࠋ㸯᭶㸯᪥㖟⾜ $ ࡜㖟
⾜ % ࡑࢀࡒࢀ࠶ࡿ㔠㢠ࡢ㡸࠿ࡾࡍࡿࠋ㸯᭶ᮎඖ㔠ࢆᘬࡁฟࡋ࡚ࠊ฼ᜥࡢࡳࢆ෌
㡸㔠࡟ࡍࡿࠋࡓࡔ฼ᜥࡣ௒ࡲ࡛ࡢ㖟⾜࡟࠾ࡃࡇ࡜ࡶ࡛ࡁࡿࡋࠊ୍㒊௚ࡢ㖟⾜࡟
ධࢀࡿࡇ࡜ࡶྍ⬟࡛࠶ࡿࠋࣛࣥࢲ࣒࡛Ỵࡵࡿ࡛ࡼ࠸ࠋࡇࡢࡼ࠺࡟᭶ᮎ࡛ඖ㔠ࢆ
ᘬࡁฟࡋ࡚ࠊ฼ᜥࡼࡾ฼ᜥࢆ⏘⏕ࡍࡿࡇ࡜ࢆ⪃࠼ࡿ࡜ࠊ ኚ㔞ศᯞ㐣⛬࡟ᚑ࠺࡛
ࡣ࡞࠸ࡔࢁ࠺ࠋ࡝ࡕࡽ࠿ࡢ㡸㔠ࡀ㸮࡟࡞ࡿ࡜ࠊࡑࡕࡽࡢ㖟⾜࡬ඖ㔠ࢆᢞධࡋࠊ
ࡉࡽ࡟฼ᜥࢆฟࡿࠋࡇࡢ㒊ศࡣࠕ⛣Ẹ㡯࡛ࠖ࠶ࡿࠋࡇࡢࣔࢹ࡛ࣝྛᮇᮎ $ 㖟⾜
࡜ % 㖟⾜ࡢ㡸㔠ṧ㧗ࢆ⪃ᐹ࡛ࡁࡿ࡜ᛮ࠺ࠋ
⌧ᐇ࡛ࠊࡶࡋ฼ᜥࡢ෌㡸㔠ࡍࡿศ๭ࡾࡣࣛࣥࢲ࣒࡛ࡣ࡞ࡃࠊ࠾ᐈᵝࡢពᛮ࡛Ỵ
ࡵࡽࢀࡿ࡜ࠊ㡸㔠ࡢ⾜ࡁ᪉ἲࡣ฼Ꮚ⋡ࡸ㖟⾜ࡢࢧ࣮ࣅࢫ࡞࡝࡜㛵ಀࡀ࠶ࡿ࡛ࡣ
࡞࠸࠿࡜⪃࠼ࡿࠋ࡞ࡢ࡛ࠊᮏࣔࢹࣝࢆ฼⏝ࡋ࡚ࠊ฼Ꮚ⋡ኚືࡸࢧ࣮ࣅࢫᨵၿᚋ
72
ࡢ㡸㔠ṧ㧗ࡀヨ⟬ࡀྍ⬟࡛ࡣ࡞࠸࠿࡜⪃࠼࡚࠾ࡃࠋ
;
Ȝ1,1
;
;
;
;
;
;
;
;
;
ᅗ㸳㸫㸯 ࣔࢹࣝࡢㄝ᫂
グྕࡢព࿡
ࣔࢹ࡛ࣝࡣ ^X 0㸪[ k , j ,i ,H k : k , j N , i 1㸪
2` ࡀ⊂❧࡛࠶ࡿ㸸
ࠉࠉࠉࠉX k
ࠉࠉࠉ[ k , j ,i
ࠉࠉࠉࠉH k
ª X k ,1 º
« X » ࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ㸦5.2㸧
¬ k ,2 ¼
ª [ k , j ,i ,1 º
«[
» ࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ㸦5.3㸧
¬ k , j ,i ,2 ¼
ª H k ,1 º
«H » ࠉࠉࠉࠉࠉࠉࠉࠉ㸦5.4㸧
¬ k ,2 ¼
2
2
E [1㸪1,i 㸧㸺fL E ( H1 )㸺f ࡑࡋ࡚ࠊ 㸦
ୗࡢグྕ࡟ࡘ࠸࡚ㄝ᫂ࡍࡿࠋ
73
m[i
E [1,1,i R2 , m[
V [i
Var [1,1,i R 2*2 , V [
mH
E (H1 ) R2 , VH
> m[1
m[ 2 @ R2*2
1
V [1 V [ 2 R 2*2 2
var(H1 ) R 2*2
NѮ=࡛࠶ࡿ࡜ࡁ )N Ȫ;;͐;.㸧࡜ࡍࡿࠋ ᘧࡼࡾ
E &㹩 |㹄㹩-1 X k 1,1m[1 X k 1,2 m[2 mH m[ &㹩-1 mH
ࡋࡓࡀࡗ࡚E &㹩 E &㹩 m[ E &㹩-1 mH
m[ E &㹩-1 mH
m㸦
[ m[ E &㹩-2 mH ) mH
m[ 2 E &㹩-2 m[ mH mH
m[ 2 (m[ E &㹩-3 mH ) m[ mH mH
m[ 3 E &㹩-3 m[ 2 mH m[ mH mH
㻸㻸
k 1
j
ࠉࠉࠉࠉࠉࠉE &㹩㸻m㹩
[ E & 0 + ¦ m[ mH ࠉࠉࠉࠉࠉ
ࠉࠉࠉࠉࠉࠉࠉ㸦5.5㸧ࠉࠉࠉࠉࠉ
j 0
FULWLFDOLW\ SDUDPHWHU࡟ࡘ࠸࡚
᫂ࡽ࠿࡟ࠊ m[ ࡣࡇࡢ㐃㙐ຠᯝࡢ୰࡛ࡣ㔜せ࡞ᙺ๭ࢆᢸ࠺ࠋ
ᮏㄽᩥ࡛ࡣḟࡢࡼ࠺࡞ m[ ࢆᣢࡕࠊᑐ⛠ᆺࡢ⛣Ẹ㡯ࡢ࠶ࡿ ኚ㔞ศᯞ㐣⛬ࢆ⪃
࠼ࡿ㸸
ªD E º
ࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉm[㸻 «
» ࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ㸦5.6㸧
¬E D ¼
ࡉࡽ࡟ࠊȘ㸼 ࠿ࡘș㸼 ࡢሙྜࡢࡳὀ┠ࡍࡿࠋ
ࡇࡢᇶ࡛⮫⏺ࣂ࣓࣮ࣛࢱ㸦FULWLFDOLW\SDUDPHWHU㸧ࡣȨ Șș࡜ᐃ⩏ࡉࢀࡿ
࡜ࠊ⛣Ẹ㡯ࡢ࠶ࡿ㸰ኚ㔞ศᯞ㐣⛬ࡀḟࡢࡼ࠺࡞୕ࡘࡢ≧ែࡀ࠶ࡿ㸸
⮫⏺ᮍ‶㸦VXEFULWLFDO㸧Ȩ㸺㸯
⮫⏺㸦FULWLFDO㸧Ȩ㸻㸯
⮫⏺㉸㐣VXSHUFULWLFDOȨ㸼㸯
౛㸯ࢆ⏝࠸࡚┤ほⓗ࡟ㄝ᫂ࡍࡿࠋⓑⰼࡀྠⰍࡢⓑⰼࢆ⏕ࡴᮇᚅ್ࡣȘࠊ␗Ⰽࡢ
㉥ⰼࢆ⏕ࡴᮇᚅ್ࡣș㉥ⰼࡀྠⰍࡢ㉥ⰼࢆ⏕ࡴᮇᚅ್ࡣȘࠊ␗Ⰽࡢⓑⰼࢆ⏕
ࡴᮇᚅ್ࡣș࡛࠶ࡿࠋȘ㸩ș㸻Ȩ㸺㸯࡛࠶ࡿ࡜ࡣࠊྛⰍࡢⰼࡣ௒ࡢୡ௦ࡼࡾᑡ
࡞ࡃᏊᏞࢆ⏕ࡴࡢ࡛ࠊ⛣Ẹ࡛᳜ධࡢⰼࢆ⪃࠼࡞ࡅࢀࡤࠊḟୡ௦ࡢⰼࡢ⥲ᩘࢆῶ
ࡗ࡚࠸ࡃഴྥ࡛࠶ࡿࠋ≀✀ࢆ⥔ᣢࡍࡿࡓࡵࠊேᕤ࡛᪂ࡋ࠸࠾ⰼࡢ᳜ධࡀᚲせ࡛
࠶ࡿࡔࢁ࠺ࠋȘ㸩ș㸻Ȩ㸻㸯࡛࠶ࡿ࡜ࡢࡇ࡜ࡣࠊ௒ࡢୡ௦ࡢⰼࡢつᶍࡀ࣮࢟ࣉ
࡛ࡁࡿࡼ࠺࡞㏿ᗘ࡛ḟୡ௦ࡢᏊᏞࢆ⏕ࡴࠋ⛣Ẹ࡛ᑡࡋቑ㔞ྍ⬟࡛࠶ࡿࠋȘ㸩ș
Ȩ㸼㸯ࡢ࡛࠶ࡿ࡜ࡁࠊᏊᏞᩘࡣぶࡢᩘࢆ㉸࠼ࡿࡼ࠺࡞ቑ࠼᪉࡛ࠊ࠸ࡎࢀࡀ⇿Ⓨ
ⓗ࡞ᩘ࡟࡞ࡗ࡚ࠊ␗ᖖ஦ែࢆࢥࣥࢺ࣮ࣟࣝࡍࡿࡓࡵࠊ஦๓㆙ሗࢩࢫࢸ࣒ࡀᚲせ
74
ࡔࢁ࠺ࠋ
ึᮇ್࡟ࡘ࠸࡚ࡣ⡆༢໬࡟ࡍࡿࡓࡵࠊ⛣Ẹ㡯ࡢ࠶ࡿ ኚ㔞ศᯞ㐣⛬ࡢึᮇ್
ࡣ ; ࡜ᐃ⩏ࡉࢀࡿࡇ࡜ࡀከ࠸ࠋ
㸦ゝࢃࢀࡿ㸮ࢫࢱ࣮ࢺ㸧
ᮏㄽᩥ࡛ࡣࠊ m[ z 0 ࡑࡋ࡚ &㹩 0 ࡜タᐃࡍࡿࠋ
⛣Ẹ㡯ࡢ࠶ࡿ ኚ㔞ศᯞ㐣⛬ࡢ᥎ᐃ
&/6 ࡼࡾ⛣Ẹ㡯ࡢ࠶ࡿ㸰ኚ㔞ศᯞ㐣⛬ࡢ᥎ᐃ
ࡇࡇ࡛ࡣᮏㄽᩥࡢࣔࢹࣝ࡟ᑐࡍࡿ &/6 ᥎ᐃࢆ⾜࠺ࠋྛኚ㔞ࡢ≉ᛶ್ࢆồࡵࡿ
ୖࠊ⌮ㄽⓗ࡞ศᕸࢆィ⟬ࡍࡿࠋ
E X k | Fk 1 m[ X k 1 mH ࡼࡾ
M k㸻X k -E (X k Fk 1 )=X k m[ X k 1 mH , k N . 㸦㸧
୍ࡘࡢᗎิࢆᑟධࡍࡿ㸸
8㹩 ;N! 㸯7;N ;N;N 㸦㸧
ª㸯º
NѰ=1㸻 « » 㹐2 ¬㸯¼
ࡇࡢᘧࡼࡾࠊḟࡢ⤖ᯝࡀ࠶ࡿ㸸
ࡍ࡭࡚ࡢ NѰ=࡟ᑐࡋ࡚ 8㹩Ӌ㸮ࡑࡋ࡚ 8N㸻8N㸫㸯㹫Ȝ!0㹩! NѰ1ࠋ
⌮⏤ࡣḟ࡛࠶ࡿ㸸
ࡀ㹫ȥࡢᅛ᭷࣋ࢡࢺ࡛ࣝ࠶ࡾࠊᅛ᭷್ࡀ㸯࡛࠶ࡿࠋ
Ȩ Șș㸻㸯ࡢព࿡ࡣࠊ
࡞ࡢ࡛
1㸪m[ X
k , k 1
㸻1T m[ X k ,k 1㸻1T X k , k 1㸻8 k 1 ª㸯º
ࡉࡽ࡟ࠊ u㻑㸻 « » 㹐2 ¬-㸯¼
9㹩 ī;N! ī7;N ;N;NNѰ1࡜ࡍࡿ࡜
9㹩㸦Șș㸧9㹩ī㹫Ȝ!ī0㹩!
⌮⏤ࡣḟ࡛࠶ࡿ
Ī ࡣࡀ㹫ȥࡢᅛ᭷࣋ࢡࢺ࡛ࣝ࠶ࡾࠊᅛ᭷್ࡀȘș࡛࠶ࡿࠋ࡞ࡢ࡛
㻑T
㻑T
u㻑㸪
[ mX k 1 㸻u m[ X k 1㸻㸦Șș㸧u X k 1 . ;N㸻㸦8㹩9㹩㸧;N㸻㸦8㹩9㹩㸧NѰ= ᘧ࡛⾲ࡍࣔࢹࣝ࡟ᑐࡋࠊ&/6 ᥎ᐃࢆ⾜ࡗ࡚ࠊ⮫⏺ࣃ࣓࣮ࣛࢱࢆぢࡘࡅࡿࠋ
᪂ࡋ࠸ࣃ࣓࣮ࣛࢱț㸻Șșࢆᐃ⩏ࡍࡿࠋࡑࡋ࡚
Ș㸻㸦Ȩț㸧ș㸻㸦Ȩ̿ț㸧ࠋ
ᘧࡣḟࡢᘧ࡟᭩ࡁ᥮࠼ࡿ㸸
75
M k㸻X k -E (X k Fk 1 )=X k m[ N k 1 mH , k N . ࡉࡽ࡟ 㸯ª U G U -G º
X k㸻 «
X k 1 +M k 1 +mH , k N . 2 ¬ U -G U G »¼
&/6 ἲ࡛᥎ᐃࡍࡿ࡜ࡁࠊࡲࡎ ;͐;Q ࢆྵࡴḟࡢᘧࢆ᭱ᑠ໬࡟ࡍࡿ㸸
n
¦
k 1
1 ªU G
Xk «
2 ¬ U -G
2
U -G º
X mH
U G »¼ k 1
¦ n 1, xk mH 1 xk 1
° Un 㸻 k 1
,
2
n
°
1,
x
¦k 1
k 1
°
᥎ᐃ್
®
n
㻑 xk 1
° ¦ k 1 u㻑㸪xk mH u,
.
°G n
2
n
㻑
u
,
x
°¯
¦k 1
k 1
ࡇࡇ࡛ࡣ
1 ª«1º» R 2 , u㻑 ª« 1 º» R 2࡛࠶ࡾࠊ
¬1¼
¬ 1¼
n
2

½
H n ® x1 ,㻷 xn ( R 2 ) n : ¦ 1 xk 1 ! 0 ¾
k 1
¯
¿
n

½
2
H㻑n ® x1 ,㻷 xn ( R 2 ) n : ¦ u㻑 xk 1 ! 0 ¾
k 1
¯
¿
;㸻 ࡣ 5 ✵㛫㞽࣋ࢡࢺࣝࠋ
㸯
㸯
U㸻D +E ࠊG㸻D EࡼࡾD㸻 U +G ࠊE㸻 U G ࠋ
2
2
ªDˆ㹬 º 㸯ª㸯㸯º ª Uˆ㹬 º
« ˆ »㸻 «
» « ˆ » 㸯
㸯
2
E
¬
¼ ¬G㹬 ¼
¬ 㹬¼
ḟࡢㄽᩥ࡛ࡣࡉࡽ࡟㔜せ࡞⤖ㄽࡀ࠶ࡿࠋ
$V\PSWRWLF EHKDYLRU RI &/6 HVWLPDWRUV IRU W\SH GRXEO\ V\PPHWULF
FULWLFDO*DOWRQ:DWVRQSURFHVVHVZLWKLPPLJUDWLRQ
%\0DUWRQ,VSDQ\.ULVWRI.RUPHQGL*\XOD3DS
ᐃ⌮㸯
76
1
³ Y d Y 1, mH t t

n( U n 1)
o0
D
³
1
0
If V[ 1 1
Yt 2 dt
ࠉn o fࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ㸦5.19㸧
,
0 ࠉࠉࡑࡋ࡚
§ 3 V[ 1 1
D
n3/2 Uˆ n 1 o N ¨ 0,
2
¨
1, mH
©
f V[ u㻑 u㻑 ! 0 ࠉࠉࡑࡋ࡚
·
¸
¸
¹
ࠉn o fࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ㸦5.20㸧
1
ª n1/2 Dˆ n D º D
y dW㻑t ª 1 º
³
0 t
« 1/2
» o DE 1
« 1» ࠉn o f ࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉࠉ㸦5.21㸧
« n Eˆn E »
y
dt
¬
¼
³0 t ¬ ¼
(W㻑 ) ࡀᶆ‽ࣈࣛ࢘ࣥ㐠ື࡛࠶ࡾࠊ
(W ) ࡜⊂❧࡛࠶ࡿࠋ
t tR
t tR
ª n1/2 Dˆ n D º D § 3 V[ u㻑 u㻑
« 1/2
» o N ¨ 0,
ˆ
¨¨ 4 E u㻑, H 2
«n En E »
1
¬
¼
©
·
¸ª 1 º
¸¸ «¬ 1»¼
¹
n o fࠉࠉࠉࠉࠉࠉࠉࠉࠉ㸦5.22㸧
&/6 ࡼࡾ⛣Ẹ㡯ࡢ࠶ࡿ㸰ኚ㔞ศᯞ㐣⛬ࡢ᥎ᐃࡢࢩ࣑࣮ࣗࣞࢩࣙࣥ
࣏࢔ࢯࣥศᕸࢆ฼⏝ࡋࠊ ࡢ⌮ㄽࢆࢩ࣑࣮ࣗࣞࢩ᳨࡛ࣙࣥ㦂ࡍࡿ
ª [ k , j ,i ,1 º
ª H k ,1 º
,
[ k , j ,i «
H
» k «H » ࢆほ ࡉࢀࡿ㒊ศ࡜⪃࠼ࡿࠋ
¬ k ,2 ¼
¬[ k , j ,i ,2 ¼
[ k , j ,i ࡜ H k ࡑࢀࡒࢀᮇᚅ್ m[ ࡜ mH ࡢ࣏࢔ࢯࣥศᕸ࡟ᚑ࠺ࠋ
[ k , j ,i ,1㹼P0 m[ 1 [ k , j ,1,1㹼P0 D , [ k , j ,2,1㹼P0 E H k ,1㹼P0 H k 1 [ k , j ,i ,2㹼P0 m[ 2 [ k , j ,1,2㹼P0 E , [ k , j ,2,1㹼P0 D H k ,2㹼P0 H k 2 㹶k 1,1
㹶k 1,1
㹨㸻㸯
㹨㸻㸯
X㹩㸻 ¦ [ k , j ,1 ¦ [ k , j ,2 H k ࡀᮇᚅ್ X k 1m[1 X k 2 m[2 mH ࡢ࣏࢔ࢯࣥศᕸ࡟ᚑ
࠺ࡇ࡜ࡀࢃ࠿ࡿ㸸
X k ,1㹼P0 X k 1,1D X k 1,2 E H k 1 X k ,2㹼P0 X k 1,1E X k 1,2D H k 2 ⤫ィࢯࣇࢺ 5 ࢆ฼⏝ࡋࢩ࣑࣮ࣗࣞࢩࣙࣥࢆ⾜࠺ࠋ
77
P
Q
mH1 mH 2 Ș ș Ȩ
ˏ ➨ 㸯
ᅇ
➨ ᅇ
➨ ᅇ
ț
Dˆ n Eˆn ᩘ⌮ࣔࢹࣝࡣࠊ⮬↛⛉Ꮫ࡟࠾࠸࡚ࡢࡳ࡛ࡣ࡞ࡃࠊ♫఍⛉Ꮫࡸேᩥ⛉Ꮫ࡟࠾࠸࡚
ࡶ⏝࠸ࡽࢀࡿࠋᩘ⌮ࣔࢹࣝࡀ⏝࠸ࡽࢀ࡚࠸ࡿศ㔝ࢆ⥙⨶ࡍࡿࡇ࡜ࡣ㞴ࡋ࠸࡜⪃
࠼ࡽࢀࡿࡀࠊ౛࠼ࡤࠊ≀⌮ᏛࠊᕤᏛࠊ⏕≀Ꮫࠊ⤒῭Ꮫࠊ♫఍Ꮫࠊᚰ⌮Ꮫࠊィ⟬
ᶵ⛉Ꮫࠊ⏕ែᏛࠊ⚄⤒⛉ᏛࠊศᏊ⏕≀Ꮫࠊ⏕≀⤫ィᏛࠊච␿Ꮫࠊᆅ⌫≀⌮Ꮫࠊ
ኳᩥᏛࠊ㟁Ẽᅇ㊰ࠊᶵᲔᕤᏛࠊ⯟✵ᕤᏛࠊẼ㇟ᏛࠊゝㄒᏛࠊィ㔞ᩥ⊩Ꮫࠊఏᰁ
⑓ឤᰁண ࠊ࣮࢜࣌ࣞࢩࣙࣥࢬ࣭ࣜࢧ࣮ࢳ࡞࡝ࡀ࠶ࡿࠋ
㏆ᖺࡣࢥࣥࣆ࣮ࣗࢱࡢᛶ⬟ࡢྥୖ࡟ࡼࡾࠊ」㞧࡞ᩘ⌮ࣔࢹ࡛ࣝࡶࡑࡢࡩࡿࡲ࠸
ࢆࢩ࣑࣮ࣗࣞࢩࣙࣥ࡟ࡼࡗ࡚ぢࡿࡇ࡜ࡀ࡛ࡁࡿࡓࡵࠊᵝࠎ࡞ศ㔝࡛⏝࠸ࡽࢀࡿ
ࡼ࠺࡟࡞ࡗ࡚࠸ࡿࠋᩘ⌮ࣔࢹࣝࡣࠊᑐ㇟࡜ࡍࡿ⌧㇟ࡸࠊᐃᘧ໬ࡢᢳ㇟ᗘ࡞࡝࡟
ࡼࡗ࡚ᵝࠎ࡞ࡶࡢࡀ࠶ࡿࠋ
ᑐ㇟࡜࡞ࡿ⌧㇟ࡀ኱つᶍ࡛ேᡭ࡟ࡼࡿゎᯒࡀᅔ㞴ࠊ࠶ࡿ࠸ࡣࢼࣅ࢚ࢫࢺ࣮ࢡࢫ
᪉⛬ᘧࡢࡼ࠺࡟ࣔࢹࣝࡢゎࢆゎᯒⓗ࡟ᚓࡽࢀ࡞࠸ሙྜࡣࠊࢥࣥࣆ࣮ࣗࢱ࡟ࡼࡿ
ࢩ࣑࣮ࣗࣞࢩࣙࣥ࡟ࡼࡗ࡚ゎࢆồࡵࡿࠋ௦⾲ⓗ࡞࢔ࣝࢦࣜࢬ࣒࡜ࡋ࡚ࠊ࢜࢖ࣛ
࣮ἲࠊࣝࣥࢤ㸻ࢡࢵࢱἲࠊ᭷㝈せ⣲ἲࠊࣔࣥࢸ࢝ࣝࣟἲ➼ࡀ࠶ࡿࠋࢥࣥࣆ࣮ࣗ
ࢱࡢᛶ⬟ྥୖ࡟ࡼࡗ࡚ࠊᢅ࠼ࡿᩘ⌮ࣔࢹࣝࡢᖜࡀ኱ኚᗈࡲࡗࡓࠋୖ㏙ࡋࡓࡼ࠺
࡟ࠊᩘ⌮ࣔࢹࣝࢆᵓ⠏ࡍࡿࡇ࡜࡟ࡼࡗ࡚ᚓࡽࢀࡿࡇ࡜ࡣࠊࡲࡎࡣ⌧㇟ࡢ⌮ゎࡀ
࠶ࡆࡽࢀࡿࠋࡲࡓࠊᩘᏛⓗ࡟⾲⌧ࡍࡿࡇ࡜࡟ࡼࡗ࡚ࠊᢅ࠸ࡀᐜ᫆࡟࡞ࡾࠊᩘᏛ
ࡢ▱ぢࢆά⏝ࡍࡿࡇ࡜ࡀ࡛ࡁࡿࠋ
78
⤊ࢃࡾ࡟
ᮏㄽᩥࢆࡲ࡜ࡵࡿ࡟࠶ࡓࡾ㸪ᶓ὾ᅜ❧኱ᏛᩍᤵỌ஭ᆂ஧ඛ⏕࡟ࡣ㸪◊✲඲⯡
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