Minato
August 06, 2023
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# 勉強会資料 / “Asymptotic Statistics” Section 2.1

A.W. van der Vaart著 “Asymptotic Statistics”の勉強会資料を公開します。

2.1節 Basic Theory
2.5 Lemma (Helly's lemma)

August 06, 2023

## Transcript

1. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ୈ 3 ճ

ษڧձ A.W. van der Vaart ”Asymptotic Statistics” Section 2.1 ょ゙が〣ิ୊〝ろ゚ぢや〣ෆ౳ࣜ Minato 2022 ೥ 10 ݄ 1 ೔ 1 / 43
2. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ࠓճ〣ษڧձ〣಺༰ ه๏〣४උ ໨࣍

1 〤」〶〠 ࠓճ〣ษڧձ〣಺༰ ه๏〣४උ 2 ょ゙が〣ิ୊ (p.9) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺ ょ゙が〣ิ୊ 3 ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) 1 / 43
3. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ࠓճ〣ษڧձ〣಺༰ ه๏〣४උ 〤」〶〠

2 / 43 ࠓճ〣಺༰ Chapter 2ɿStochastic Convergence [1] Section 2.1ɿBasic Theory (p.8, 9) ิ୊ 2.5ɿょ゙が〣ิ୊ ゆ゜り゜や〣ఆཧ (ఆཧ 2.4) 〣ূ໌〣ཁ఺〝〟぀ิ୊〣ূ໌ ྫ 2.6ɿろ゚ぢや〣ෆ౳ࣜぇ༻⿶〔֬཰ม਺ྻ〣Ұ༷ۓີੑ〣ٞ࿦ A.W. van der Vaart ”Asymptotic Statistics”
4. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ࠓճ〣ษڧձ〣಺༰ ه๏〣४උ ه๏〣४උ

Table: ओ〟ه๏ N ࣗવ਺શମ〣ू߹ R ࣮਺શମ〣ू߹ Q ༗ཧ਺શମ〣ू߹ Rk ू߹ {x = (x1 , . . . , xk )⊤ : xi ∈ R (i = 1, . . . , k)} Qk ू߹ {x = (x1 , . . . , xk )⊤ : xi ∈ Q (i = 1, . . . , k)} R≥0 ू߹ {x ∈ R : x ≥ 0} E[·] ظ଴஋ V[·] ෼ࢄ F(a+) F 〣 x = a 〠⿼々぀ӈଆۃݶ F(a−) F 〣 x = a 〠⿼々぀ࠨଆۃݶ Xn ⇝ X Xn ⿿ X 〠෼෍ऩଋ『぀ 3 / 43
5. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ࠓճ〣ษڧձ〣಺༰ ه๏〣४උ よぜぷ゚ಉ࢜〣౳߸ɾෆ౳߸〣ఆٛ

kɿ2 Ҏ্〣ࣗવ਺ Rk 〣ݩ x = (x1 , . . . , xk )⊤, y = (y1 , . . . , yk )⊤ 〠ର「ɺ x ≤ y def ⇐⇒ xi ≤ yi (∀i = 1, . . . , k) x = y def ⇐⇒ xi = yi (∀i = 1, . . . , k) x ̸= y def ⇐⇒ ∃j s.t. xj ̸= yj x < y def ⇐⇒ x ≤ y ⿾〙 x ̸= y x < y ⇔ xi ≤ yi (∀i = 1, . . . , k) ⿾〙 ∃j s.t. xj < yj ಉ༷〠ɺx ≥ y, x > y 〷ఆٛ『぀ɻ 4 / 43
6. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ໨࣍ 1 〤」〶〠 ࠓճ〣ษڧձ〣಺༰ ه๏〣४උ 2 ょ゙が〣ิ୊ (p.9) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺ ょ゙が〣ิ୊ 3 ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) 5 / 43
7. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〠〙⿶〛 (p.9 લ൒) ゆ゜り゜や〣ఆཧ (ఆཧ 2.4, p.8) 〣ূ໌〣ཁ఺〝〟぀〣〤ょ゙が〣 ิ୊ (Helly’s lemma) 〜⿴぀ɻ ょ゙が〣ิ୊〤ɺ〞え〟෼෍ؔ਺ྻ〠ର「〛〷෼෍ؔ਺ɺ⿴぀⿶〤 defective 〟෼෍ؔ਺〠ऑऩଋ『぀෦෼ྻ⿿ଘࡏ『぀〈〝ぇओு『 ぀ิ୊〜⿴぀ɻ ょ゙が〣ิ୊〣આ໌〠ೖ぀લ〠ɺ ෼෍ؔ਺〣ੑ࣭ defective 〟෼෍ؔ਺〣ఆٛ ෦෼ྻ〣ఆٛ〹ੑ࣭ 〠〙⿶〛⿼《〾⿶『぀ɻ 6 / 43
8. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ෼෍ؔ਺〣ੑ࣭ ෼෍ؔ਺〣ੑ࣭ Rk ্〣෼෍ؔ਺ F 〠ؔ「〛ɺ࣍〣ੑ࣭⿿੒〿ཱ〙ɿ 1 ඇݮগؔ਺〜⿴぀ɻ 2 ֤఺〜ӈ࿈ଓ〜⿴぀ɻ 3 (ଟ࣍ݩ〣৔߹) ۣܗ〣࣭ྔ (masses of cells) ⿿ඇෛ〜⿴぀ a ྫ⿺〥ɺk = 2 〣〝　ɺa ≤ b ぇຬ〔『೚ҙ〣 a, b ∈ R2 〠ର「ɺ F(b) + F(a) − F(a1 , b2 ) − F(a2 , b1 ) ≥ 0 ⿿੒〿ཱ〙ɻ 4 x → ∞ 〣〝　 F(x) → 1.b 5 x → −∞ 〣〝　 F(x) → 0.c aょ゙が〣ิ୊〣࠷ऴஈམ〣ূ໌”In the higher-dimensional case, ...”ぇࢀরɻ1 ࣍ݩ 〜⿴ぁ〥 a ≤ b ⇒ F(b) − F(a) ≥ 0 ぇҙຯ『぀⿿ɺ〈ぁ〤ඇݮগੑ〽〿੒〿ཱ〙ɻ b lim x1,...,xk→∞ F(x1, . . . , xk) = 1 ぇҙຯ『぀ɻ c lim xi→−∞ F(x1, . . . , xk) = 0 ⿿શ〛〣 i = 1, . . . , k ぇҙຯ『぀ɻ 7 / 43
9. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ defective 〟෼෍ؔ਺ Def: defective 〟෼෍ؔ਺ ෼෍ؔ਺ F 〣ੑ࣭ (1)ʙ(5) 〣⿸〖ɺ(1)ʙ(3) ⿿੒〿ཱ〙⿿ɺ(4) 〝 (5) ⿿੒〿ཱ〔』ɺ x → ∞ 〠⿼⿶〛 1 ະຬ〣ۃݶ஋ぇ〷〙 x → −∞ 〜 0 〽〿େ　⿶਺〣ۃݶ஋ぇ〷〙 〜⿴぀〽⿸〟ؔ਺〣〈〝ぇɺdefective 〟෼෍ؔ਺ (defective distribution function) 〝⿶⿸ɻ (1) ඇݮগੑɾ(2) ӈ࿈ଓੑɾ(3) ۣܗ〣࣭ྔ⿿ඇෛ〝⿶⿸෼෍ؔ਺ 〣ੑ࣭ぇຬ〔『Ұํ〜ɺx → ∞, x → −∞ 〣ۃݶ〠ؔ『぀෼෍ؔ ਺〣ੑ࣭ぇຬ〔《〟⿶〽⿸〟ؔ਺ぇɺ”defective”〝දݱ「〛⿶぀ defective 〟෼෍ؔ਺〠ର「ɺ(4)ɾ(5) (ۃݶ〣ੑ࣭) 〷ຬ〔『〽⿸ 〟ؔ਺〣〈〝ぇ proper 〟෼෍ؔ਺〝⿶⿸ɻ 8 / 43
10. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ਺ྻ〣෦෼ྻ Def: ਺ྻ〣෦෼ྻ {an }∞ n=1 ぇ਺ྻ〝「ɺφ : N → N ぇॱংぇอ〙୯ࣹɺ『〟い〖 n < m ⇒ φ(n) < φ(m) 〜⿴぀〝『぀ɻ〈〣〝　ɺ{aφ(n) }∞ n=1 ぇ {an }∞ n=1 〣෦෼ྻ (subsequence) 〝⿶⿸ɻ ෦෼ྻ〤ݩ〣਺ྻ⿾〾ແݶݸ〣߲ぇऔ〿ग़『〈〝〠〽〿〜　〔਺ ྻ〜⿴぀⿿ɺ〒〣߲〣ॱং〤ݩ〣ॱং〝มい〾〟⿶ɻ『〟い〖ɺ φ(1) < φ(2) < · · · < φ(n) < . . . . ্ه〣෦෼ྻ〣ఆٛ〤ɺࣸ૾ぇ༻⿶〔෦෼ྻ〣ఆٛ〜⿴぀ɻຊจ 〜〤ɺࣸ૾ぇཅ〠ॻ⿾』ɺ෦෼ྻぇ {anj }∞ j=1 〝දه「〛⿶぀〔〶ɺ ຊと゘ぐへ〜〷جຊత〠〈〣දه〠ଇ〘〛ূ໌ぇਐ〶぀ɻ 9 / 43
11. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ਺ྻ〣෦෼ྻ〣ੑ࣭ Th: ऩଋ『぀਺ྻ〣෦෼ྻ〷ಉ」ۃݶ஋ぇ〷〙 [2] ਺ྻ {an } 〠⿼⿶〛ɺ lim n→∞ an = a 〜⿴぀〝『぀ɻ ਺ྻ {an } 〣೚ҙ〣෦෼ྻ {aφ(n) } 〠ର「ɺ lim n→∞ aφ(n) = a. ෦෼ྻ〣֓೦ぇ༻⿶〛ɺ࣮਺〣࿈ଓੑ〠ؔ『぀ެཧ⿾〾る゚びきぽ-ゞ ぐご゚てゔぷ゘と〣ఆཧ⿿ಋ⿾ぁ぀ɻ Th: る゚びきぽ-ゞぐご゚てゔぷ゘と〣ఆཧ [2] ਺ྻ {an } ⊂ R ⿿༗ք〣〝　ɺऩଋ෦෼ྻぇ〷〙ɻ 10 / 43
12. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊ ょ゙が〣ิ୊ (Lemma2.5, p.9) {Fn }∞ n=1 ぇ Rk ্〣ྦྷੵ෼෍ؔ਺〣ྻ〝『぀ɻ⿴぀෦෼ྻ {Fnj }∞ j=1 ⿿ ଘࡏ「〛ɺ⿴぀ (defective 〠〟〿⿸぀) ෼෍ؔ਺ F 〠ର「ɺF 〣೚ҙ〣 ࿈ଓ఺ x ∈ Rk 〠〙⿶〛ɺ lim j→∞ Fnj (x) = F(x). 11 / 43
13. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ゆ゜や゜や〣ఆཧ〝ゞぐご゚てゔぷ゘と〣ఆཧ〣ؔ࿈ ゆ゜り゜や〣ఆཧ (Theorem2.5, p.8) Xn ぇ Rk ্〣֬཰ม਺〝『぀ɻ 1 Xn ⿿⿴぀֬཰ม਺ X 〠෼෍ऩଋ『぀〝　ɺ{Xn : n ∈ N} 〤Ұ༷ ۓີ〜⿴぀ɻ 2 Xn ⿿Ұ༷ۓີ〟〾〥ɺ⿴぀֬཰ม਺ X ⿿ଘࡏ「〛ɺ Xnj ⇝ X(j → ∞). Th: る゚びきぽ-ゞぐご゚てゔぷ゘と〣ఆཧ [2] ਺ྻ {an } ⊂ R ⿿༗ք〣〝　ɺऩଋ෦෼ྻぇ〷〙ɻ ゆ゜り゜や〣ఆཧ (2) 〤ɺる゚びきぽ-ゞぐご゚てゔぷ゘と〣ఆཧ 〣֬཰ม਺൛〝ղऍ〜　぀ɻ ゆ゜り゜や〣ఆཧ (2) ぇূ໌『぀〔〶〣ݤ〝〟぀ิ୊⿿ɺょ゙が〣 ิ୊〜⿴぀ɻ 12 / 43
14. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ํ਑ ෦෼ྻ {Fnj }∞ n=1 ぇఆٛ「ɺؔ਺ F ぇߏ੒『぀ɻ ؔ਺ F ぇ⿶　〟〿ߟ⿺぀〣〤೉「⿶〣〜ɺ࣍〣खॱ〜ূ໌『぀ɻ 1 る゚びきぽ-ゞぐご゚てゔぷ゘と〣ఆཧぇ܁〿ฦ「ద༻『぀〈〝〜ɺ ೚ҙ〣 q ∈ Qk 〠ର「ɺFn j (q) ⿿ऩଋ『぀〽⿸〟෦෼ྻ {Fnj }∞ n=1 ぇߏ੒『぀ɻFn j (q) 〣ऩଋઌぇ G(q) 〝『぀ɻ 2 G 〤 Qk ্〠⿼⿶〛〣〴ఆٛ《ぁ〔ؔ਺〜⿴぀ɻ〈ぁぇ〷〝〠ɺ Rk ্〣ؔ਺ F ぇఆٛ『぀ɿ F(x) = inf G(q) : q ∈ Qk, q > x 3 F ⿿ඇݮগ〟ӈ࿈ଓؔ਺〜⿴぀〈〝ぇࣔ『ɻ 4 F 〣೚ҙ〣࿈ଓ఺ x ∈ Rk 〠〙⿶〛ɺ lim j→∞ Fnj (x) = F(x) ⿿੒〿ཱ 〙〈〝ぇࣔ『ɻ 5 ଟ࣍ݩ〣〝　〠ɺF ⿿ defective 〟෼෍ؔ਺〜⿴぀〔〶〠ຬ〔『〮 　ੑ࣭ぇࣔ『ɻ 13 / 43
15. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ ू߹ Qk 〣ݩぇ q1 , q2 , · · · ∈ Qk 〝൪߸ぇৼ぀ɻ1 ɹ ਺ྻ {Fn (q1 )}∞ n=1 〤༗ք〕⿾〾ɺる゚びきぽ-ゞぐご゚てゔぷ゘と〣ఆ ཧ〽〿ऩଋ෦෼ྻぇ〷〙ɻ〒〣〽⿸〟ఴࣈू߹ぇ n1 j ∞ j=1 〝「ɺऩଋ ઌぇ G(q1 ) 〝『぀ɻ『〟い〖ɺ G(q1 ) = lim j→∞ Fn1 j (q1 ). ࣍〠ɺ n1 j ∞ j=1 〣த⿾〾 {Fn (q2 )}∞ n=1 〣ऩଋ෦෼ྻ n2 j ∞ j=1 ぇऔ〿ग़ 「ɺऩଋઌぇ G(q2 ) 〝⿼。ɻ ऩଋ෦෼ྻ〣औ〿ํ⿾〾 n1 j ∞ j=1 ⊃ n2 j ∞ j=1 〜⿴〿ɺ G(q2 ) = lim j→∞ Fn2 j (q2 ). 1Qk 〤Մࢉແݶू߹〜⿴぀⿾〾ࣗવ਺શମ〣ू߹ N 〭〣શ୯ࣹ⿿ଘࡏ『぀〔〶ɺ〈〣 〽⿸〟ૢ࡞⿿Մೳ〜⿴぀ 14 / 43
16. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ ҎԼɺಉ༷〣खଓ　ぇଓ々぀〈〝〜ɺ n1 j ∞ j=1 ⊃ n2 j ∞ j=1 ⊃ · · · ⊃ ni j ∞ j=1 ⊃ . . . (1) ぇಘ぀ɻ 〈〈〜ɺ৽〔〟ఴࣈू߹ nj ∞ j=1 ぇɺ֤ j ∈ N 〠ର「 nj = nj j ∈ nj j ∞ j=1 〝〟぀〽⿸ɺର֯ઢ্〠਺ྻぇऔ〿ग़『ɻ Table: {nj } 〣औ〿ํ〣ぐゐがで {n} 1 2 3 4 5 6 7 . . . {n1 j } n1 1 (= n1 ) n1 2 n1 3 n1 4 n1 5 . . . {n2 j } n2 1 n2 2 (= n2 ) n2 3 . . . {n3 j } n3 1 n3 2 n3 3 (= n3 ) . . . ද⿾〾ɺ{nj } 〣औ〿ํ〤ݩ〣਺ྻ〣ॱংぇอ࣋「〛⿶぀〔〶ɺ {Fnj (q)}∞ j=1 〤 {Fn (q)}∞ n=1 〣෦෼ྻ〜⿴぀〈〝⿿い⿾぀ɻ 15 / 43
17. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ 〈〣〽⿸〠 nj ∞ j=1 ぇબ〫〈〝〠〽〿ɺ ೚ҙ〣 i ∈ N 〠ର「ɺlim j→∞ Fn j (qi ) = G(qi ). (2) ⿿੒〿ཱ〙ɻ (2) 〣ূ໌ ೚ҙ〣 i ∈ N 〠ର「ɺFni j (qi ) 〤 G(qi ) 〠ऩଋ『぀ɻ〈〈〜ɺࣜ (1) ⿼〽 〨 nj ∞ j=1 〣औ〿ํ⿾〾ɺ {nj }∞ j=i ⊂ ni j ∞ j=1 〜⿴぀ɻऩଋ『぀਺ྻ〣෦෼ྻ〤ಉ」ۃݶ஋〠ऩଋ『぀〣〜ɺFn j (qi ) 〤 G(qi ) 〠ऩଋ『぀ (஫ɿj < i 〜 nj ∈ ni j ∞ j=1 〜〟。〛〷ऩଋઌ〣ٞ ࿦〠〤Өڹ「〟⿶)ɻ 16 / 43
18. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ ֤ n 〠ର「ɺྦྷੵ෼෍ؔ਺ Fn 〤ඇݮগ〕⿾〾ɺ q ≤ q′ ⇒ G(q) ≤ G(q′). (3) (3) 〣ূ໌ q ≤ q′ 〣〝　ɺྦྷੵ෼෍ؔ਺ Fn j 〣ඇݮগੑ⿾〾ɺ೚ҙ〣 j 〜 Fn j (q) ≤ Fn j (q′) 〜⿴぀ɻ〳〔ɺࣜ (2) 〽〿ɺ lim j→∞ Fn j (q) = G(q), lim j→∞ Fn j (q′) = G(q′). ۃݶ஋〣େখؔ܎〷อ〔ぁ぀⿾〾ɺG(q) ≤ G(q′). 17 / 43
19. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ G(q) 〤 Qk ্〠⿼⿶〛〣〴ఆٛ《ぁ〛⿶぀ɻ 〒〈〜ɺRk ্〣ؔ਺ F ぇ F(x) = inf q>x G(q) = inf G(q) : q ∈ Qk, q > x 〝ఆ〶぀ɻ〈〣〝　ɺF 〤ඇݮগؔ਺〜⿴぀ɻ F ⿿ඇݮগؔ਺〜⿴぀〈〝〣ূ໌ x ≤ y 〣〝　ɺ {G(q) : q ∈ Qk, q > x} ⊃ {G(q) : q ∈ Qk, q > y} ⿿੒〿ཱ〙ɻ〽〘〛ɺԼݶ〣ੑ࣭ 2 〽〿 F(x) ≤ F(y) 〜⿴぀⿾〾ɺF 〤ඇݮগؔ਺ɻ 2Ұൠ〠ɺू߹ A, B 〠ର「〛 A ⊃ B 〟〾〥 inf A ≤ inf B 〜⿴぀ɻ 18 / 43
20. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ 《〾〠ɺF 〤֤఺ x ∈ Rk 〠ର「ӈ࿈ଓ〜⿴぀ɻ〟】〟〾ɺε > 0 ぇ೚ ҙ〠〝〘〔〝　ɺ⿴぀ q > x ⿿ଘࡏ「〛ɺG(q) − F(x) < ε ⿿੒〿ཱ〖ɺ 〈〣〈〝⿾〾 x ≤ y ≤ q ぇຬ〔『೚ҙ〣 y 〜 F(y) − F(x) < ε ⿿੒〿ཱ 〙⿾〾〜⿴぀ɻ F ⿿֤఺ x ∈ Rk 〠ର「ӈ࿈ଓ〜⿴぀〈〝〣ূ໌ (1/3) ೚ҙ〣 x ∈ Rk 〠ର「ɺF ⿿఺ x ্〜ӈ࿈ଓؔ਺〜⿴぀〈〝ぇࣔ『ɻ〒 〣〔〶〠〤ɺҎԼぇࣔ【〥ྑ⿶ɿ ∀ε > 0, ∃q ∈ Qk s.t. q > x ⿾〙 [x < ∀y < q ⇒ F(y) − F(x) < ε]. (4) දهぇ؆ܿ〠『぀〔〶〠ɺAx = {G(q) : q ∈ Qk, q > x} 〝⿼　ɺ F(x) = inf Ax 〝ද『〈〝〝『぀ɻ 19 / 43
21. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ F ⿿֤఺ x ∈ Rk 〠ର「ӈ࿈ଓ〜⿴぀〈〝〣ূ໌ (2/3) ೚ҙ〣 ε > 0 ぇݻఆ『぀ɻF(x) = inf Ax 〽〿ɺԼݶ〣ఆٛ⿾〾ɺ ∃s ∈ Ax s.t. F(x) + ε > s. (5) s ∈ Ax 〽〿ɺ ∃q ∈ Qk s.t. q > x ⿾〙 s = G(q). (6) ࣜ (5) 〝ࣜ (6) 〽〿ɺ G(q) − F(x) < ε. 〈〈〜ɺx < y < q ぇຬ〔『೚ҙ〣఺ y 〠ର「ɺ F(y) = inf{G(q′) : q′ ∈ Qk, q′ > y} ≤ G(q). 20 / 43
22. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ F ⿿֤఺ x ∈ Rk 〠ର「ӈ࿈ଓ〜⿴぀〈〝〣ূ໌ (3/3) 〈ぁ〽〿ɺ F(y) − F(x) ≤ G(q) − F(x) < ε. ্ࣜ⿾〾ࣜ (4) ⿿੒〿ཱ〙⿾〾ɺF ⿿೚ҙ〣఺ x ∈ Rk 〜ӈ࿈ଓ〜⿴぀ 〈〝⿿ࣔ《ぁ〔ɻ 〟⿼ɺຊจ〜〤”x ≤ y ≤ q”ぇຬ〔『೚ҙ〣 y 〜 F(y) − F(x) < ε 〝ɺ ౳߸ぇೖぁ〔ܗ〜ॻ⿾ぁ〛⿶぀ɻy = x 〜੒〿ཱ〙〣〤໌〾⿾〜⿴぀ɻ y = q 〜੒ཱ《【〔々ぁ〥ɺq ぇ x < q′ < q ぇຬ〔『〽⿸〟 q′ ∈ Rk 〠 ஔ　׵⿺〛ಉ」ٞ࿦ぇ『぀〈〝〠〽〿౳߸ぇೖぁ〛੒ཱ『぀〈〝⿿֬ ⿾〶〾ぁ぀ɻ3 3౳߸ぇؚ〶぀⿾〞⿸⿾〤ӈ࿈ଓੑ〣ٞ࿦〠⿼⿶〛ຊ࣭త〜〤〟⿶〔〶ɺ౳߸ぇؚ〶』 ٞ࿦「〔ɻ 21 / 43
23. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ 《〾〠ɺF 〣࿈ଓ఺ x 〠⿼⿶〛ɺ೚ҙ〣 ε > 0 〠ର「ɺ⿴぀ q, q′ ∈ Qk ⿿ଘࡏ「〛ɺG(q′) − G(q) < ε ぇຬ〔『ɻ ্ه〣ূ໌ (1/4) x ぇ F 〣೚ҙ〣࿈ଓ఺ɺε > 0 ぇ೚ҙ〠〝〿ݻఆ『぀ɻඇݮগؔ਺ F 〤఺ x ্〜࿈ଓ〕⿾〾ɺ ∃δ > 0 (∈ Rk) s.t. x − δ < ∀y < x + δ ⇒ |F(x) − F(y)| < ε 2 . (7) 22 / 43
24. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ 《〾〠ɺF 〣࿈ଓ఺ x 〠⿼⿶〛ɺ೚ҙ〣 ε > 0 〠ର「ɺ⿴぀ q, q′ ∈ Qk ⿿ଘࡏ「〛ɺG(q′) − G(q) < ε ぇຬ〔『ɻ ্ه〣ূ໌ (2/4) 〈〈〜ɺq0 , q, q′ ∈ Qk ぇɺ x − δ < q0 < q < x < q′ < x + δ ぇຬ〔『〽⿸〠〝぀〝ɺq0 < q < x < q′ 〝 F 〣ඇݮগੑɺ⿼〽〨ࣜ (7)4 ⿾〾ɺ F(x) − ε 2 < F(q0 ) ≤ F(q) ≤ F(x) ≤ F(q′) < F(x) + ε 2 . 4ࣜ (7) 〽〿ɺx − δ < ∀y < x + δ ⇒ F(x) − ε/2 < F(y) < F(x) + ε/2 〜⿴぀⿾ 〾ɺy = p0, q, x, q′ 〝「〔ࣜぇ༻⿶぀ɻ 23 / 43
25. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ 《〾〠ɺF 〣࿈ଓ఺ x 〠⿼⿶〛ɺ೚ҙ〣 ε > 0 〠ର「ɺ⿴぀ q, q′ ∈ Qk ⿿ଘࡏ「〛ɺG(q′) − G(q) < ε ぇຬ〔『ɻ ্ه〣ূ໌ (3/4) 《〾〠ɺF 〣ఆٛ⿾〾ɺ F(q0 ) = inf{G(˜ q) : ˜ q ∈ Qk, ˜ q > q0 } ≤ G(q) (∵ q ∈ Qk, q > q0 ) ≤ F(q). 〈〈〜ɺ࠷ޙ〣ෆ౳ࣜ〠ؔ「〛〤ɺF 〣ఆٛࣜ F(q) = inf{G(˜ q) : ˜ q ∈ Qk, ˜ q > q} (= inf Aq 〝⿼。) ぇৼ〿ฦ぀〝ɺG 〣ඇݮগੑ⿾〾ɺ೚ҙ〣 s ∈ Aq 〠ର「〛 G(q) ≤ s 〜 ⿴぀⿾〾 G(q) 〤 Ax 〣Լք〜⿴〿ɺinf 〣࠷େԼքੑ〽〿 G(q) ≤ inf Aq = F(q) ⿿੒〿ཱ〙〈〝⿾〾ै⿸ɻ 24 / 43
26. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ 《〾〠ɺF 〣࿈ଓ఺ x 〠⿼⿶〛ɺ೚ҙ〣 ε > 0 〠ର「ɺ⿴぀ q, q′ ∈ Qk ⿿ଘࡏ「〛ɺG(q′) − G(q) < ε ぇຬ〔『ɻ ্ه〣ূ໌ (4/4) ಉ༷〠ɺ F(x) = inf{G(˜ q) : ˜ q ∈ Qk, ˜ q > x} ≤ G(q′) (∵ q′ ∈ Qk, q′ > x) ≤ F(q′). ∴ F(x) − ε 2 < G(q) ≤ G(q′) < F(x) + ε 2 . 〽〘〛ɺ G(q′) − G(q) < F(x) + ε 2 − F(x) − ε 2 = ε. 25 / 43
27. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ લらがで〣ٞ࿦⿾〾࣍〣ෆ౳ࣜ⿿੒〿ཱ〙ɿ G(q) ≤ F(x) ≤ G(q′) (8) 〈〈〜ɺྦྷੵ෼෍ؔ਺ Fn j 〣ඇݮগੑ〽〿 Fn j (q) ≤ Fn j (x) ≤ Fn j (q′) ⿿੒〿ཱ〙〈〝⿾〾ɺ֤ล〜্ۃݶɾԼۃݶぇ〝぀〈〝〜 lim inf j→∞ Fn j (q) ≤ lim inf j→∞ Fn j (x) ≤ lim inf j→∞ Fn j (q′), lim sup j→∞ Fn j (q) ≤ lim sup j→∞ Fn j (x) ≤ lim sup j→∞ Fn j (q′). 26 / 43
28. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ 〳〔ɺࣜ (2) 〽〿 G(q) = lim j→∞ Fn j (q), G(q′) = lim j→∞ Fn j (q′) 〜⿴぀⿾〾 G(q) = lim inf j→∞ Fn j (q) = lim sup j→∞ Fn j (q), G(q′) = lim inf j→∞ Fn j (q′) = lim sup j→∞ Fn j (q′) 〜⿴぀〈〝ぇ༻⿶぀〝ɺ ∴ G(q) ≤ lim inf j→∞ Fn j (x) ≤ G(q′), (9) G(q) ≤ lim sup j→∞ Fn j (x) ≤ G(q′). (10) 27 / 43
29. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ Ҏ্〽〿ɺF 〣࿈ଓ఺ x 〠ର「ɺ lim inf j→∞ Fn j (x) − F(x) ≤ G(q′) − G(q) < ε, lim sup j→∞ Fn j (x) − F(x) ≤ G(q′) − G(q) < ε. ⿿੒〿ཱ〙〈〝⿾〾ɺ lim inf j→∞ Fn j (x) = lim sup j→∞ Fn j (x) = F(x). 〽〘〛ɺۃݶ lim j→∞ Fn j (x) ⿿ଘࡏ「ɺ lim j→∞ Fn j (x) = F(x). 28 / 43
30. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ 〈〈〜ɺk ≥ 2 〣৔߹ɺF ⿿ defective 〟෼෍ؔ਺〜⿴぀〔〶〠〤ɺۣܗ 〣࣭ྔ⿿ඇෛ〜⿴぀〈〝ぇࣔ『ඞཁ⿿⿴぀ɻྫ⿺〥ɺk = 2 〣〝　ɺ a ≤ b ぇຬ〔『೚ҙ〣 a = (a1 , a2 ), b = (b1 , b2 ) 〠ର「〛ɺ F(b) + F(a) − F(a1 , b2 ) − F(a2 , b1 ) ≥ 0 (11) ⿿੒〿ཱ〙〝　〠 F ⿿ defective 〟෼෍ؔ਺〠〟぀ɻҎԼɺࣜ (11) ぇ ࣔ『ɻ Fn j 〤෼෍ؔ਺〜⿴぀⿾〾ɺ Fnj (b) + Fnj (a) − Fnj (a1 , b2 ) − Fnj (a2 , b1 ) ≥ 0 (12) ⿿੒〿ཱ〙ɻ 29 / 43
31. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ 〳』ɺ఺ a, b, (a1 , b2 ), (a2 , b1 ) ⿿ F 〣࿈ଓ఺〜⿴぀〝　ɺ lim j→∞ Fnj (a) = F(a), lim j→∞ Fnj (b) = F(b), lim j→∞ Fnj (a1 , b2 ) = F(a1 , b2 ), lim j→∞ Fnj (a2 , b1 ) = F(a2 , b1 ) ⿿੒〿ཱ〙〈〝⿾〾ɺࣜ (12) 〣ۃݶ j → ∞ ぇߟ⿺぀〈〝〜ɺࣜ (11) ⿿੒〿ཱ〙〈〝⿿い⿾぀ɻ Ұൠ〣৔߹ɺ『〟い〖ɺ4 ఺⿿ F 〣࿈ଓ఺〝〤ݶ〾〟⿶〽⿸〟৔߹〤ɺ ӈ࿈ଓੑ⿾〾ࣜ (11) 〣੒ཱ⿿ࣔ《ぁ぀ɻ 30 / 43
32. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ Ұൠ〣৔߹〣ূ໌ํ਑ 1 ఺ a ্〜 F 〤ӈ࿈ଓ〜⿴぀〣〜ɺa 〣〰え〣গ「ӈଆ〣఺ y 〜〤 F(y) − F(a) ⿿ ε ະຬ〠〟぀ɿ ∀ε > 0, ∃δa > 0 s.t. a < ∀y < δa ⇒ F(y) − F(a) < ε. ଞ〣఺ b, c, d 〜〷੒〿ཱ〙ɻ 2 F 〣࿈ଓ఺〤 Rk ্᜚ີ〜⿴぀〈〝ぇ༻⿶〛ɺ֤఺ a, b, c, d 〣〰え 〣গ「ӈଆ〣఺〜ɺa′, b′, (a′ 1 , b′ 2 )(= c′〝⿼。), (a′ 2 , b′ 1 )(= d′〝⿼。) ⿿ F 〣࿈ଓ఺〜⿴぀〽⿸〠 a′, b′ ぇ〝぀ɻ 3 F(a) = F(a′), F(b) = F(b′), F(c) = F(c′), F(d) = F(d′) 〜⿴぀ 〈〝ぇࣔ『ɻ 4 a′, b′, c′, d′ 〤શ〛࿈ଓ఺〕⿾〾ɺલ〣ٞ࿦〽〿 F(a′) + F(b′) − F(c′) − F(d′) ≥ 0 〜⿴぀ɻ〈ぁ〝্ࣜ⿾〾Ұൠ〣 ৔߹⿿ࣔ【぀ɻ 31 / 43
33. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ 4 ఺〣⿸〖ɺ⿶』ぁ⿾〣఺⿿ෆ࿈ଓ఺ぇؚ〵৔߹ぇߟ⿺぀ɻ දهぇ؆ܿ〠『぀〔〶〠 c = (a1 , b2 ), d = (a2 , b1 ) 〝⿼。ɻ5 F 〤֤఺ a, b, c, d ্〜ӈ࿈ଓ〕⿾〾ɺε > 0 ぇ೚ҙ〠〝぀〝ɺ ∃δa > 0 s.t. a < ∀y < a + δa ⇒ F(y) − F(a) < ε, ∃δb > 0 s.t. b < ∀y < b + δb ⇒ F(y) − F(b) < ε, ∃δc > 0 s.t. c < ∀y < c + δc ⇒ F(y) − F(c) < ε, ∃δd > 0 s.t. d < ∀y < d + δd ⇒ F(y) − F(d) < ε ⿿੒〿ཱ〙ɻ 5c 〝 d 〤 a 〹 b 〝ແؔ܎〟〷〣〜〤〟。ɺa 〝 b ⿿ఆ〳ぁ〥ࣗಈత〠ఆ〳぀〷〣〜⿴ ぀〈〝〠஫ҙɻ 32 / 43
34. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ F 〣࿈ଓ఺〤 Rk ্᜚ີ〜⿴぀〈〝⿾〾ɺ6 a < a′ < a + δa , b < b′ < b + δb , c < c′ < c + δc , d < d′ < d + δd ぇຬ〔『〽⿸〠࿈ଓ఺ a′, b′, c′, d′ ぇ〝぀〈〝⿿〜　぀ɻ〔〕「ɺ a′ = (a′ 1 , a′ 2 ), b = (b′ 1 , b′ 2 ) 〝「〔〝　ɺc′ = (a′ 1 , b′ 2 ), d′ = (a′ 2 , b′ 1 ) 〝 『぀ɻ 〈〣〽⿸〟࿈ଓ఺ a′, b′, c′, d′ 〠ର「〛ɺ্ه〣࿈ଓ఺〣ٞ࿦⿾〾ɺ F(a′) + F(b′) − F(c′) − F(d′) ≥ 0. (13) ଓ⿶〛ɺF(a) = F(a′) ぇࣔ『 (ଞ〣఺〷ಉ༷)ɻ 6https://qiita.com/mochimochidog/items/a5091dda350fb00f6086 ぇࢀর「〳 「〔ɻ 33 / 43
35. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ લ〣らがで〜〝〘〔 ε > 0 〠ର「ɺF 〣 x = a 〠⿼々぀ӈ࿈ଓੑ〽〿ɺ F(a′) − F(a) < ε. (14) a′ 〤 F 〣࿈ଓ఺〜⿴぀⿾〾લ〣ٞ࿦〠〽〿 lim j→∞ Fn j (a′) = F(a′). ∴ ∃J ∈ N s.t. j ≥ J ⇒ Fn j (a′) − F(a′) < ε. (15) 〈ぁ〽〿ɺj ≥ J 〣〝　ɺ Fn j (a′) − F(a) = Fn j (a′) − F(a′) + F(a′) − F(a) ≤ Fn j (a′) − F(a′) + |F(a′) − F(a)| < ε + ε = 2ε. ্ࣜ〤 lim j→∞ Fn j (a′) = F(a) ぇҙຯ『぀ɻऩଋઌ〤Ұҙ〕⿾〾 F(a) = F(a′) ⿿⿶⿺〔ɻ 34 / 43
36. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺

ょ゙が〣ิ୊ ょ゙が〣ิ୊〣ূ໌ ଞ〣఺〷ಉ༷〠ߟ⿺ɺF(b) = F(b′), F(c) = F(c′), F(d) = F(d′) 〜⿴ ぀〈〝ぇࣔ【぀ɻ〈〣〈〝〝 F(a′) + F(b′) − F(c′) − F(d′) ≥ 0. ⿿੒〿ཱ〙〈〝⿾〾ɺ F(a) + F(b) − F(c) − F(d) ≥ 0. ⿿੒〿ཱ〙ɻҎ্〽〿ɺҰൠ〣৔߹〠⿼⿶〛〷ۣܗ〣࣭ྔ〤ඇෛ〜⿴぀ 〈〝⿿ݴ⿺〔⿾〾ɺଟ࣍ݩ〣৔߹〜〷 F 〤 defective 〟෼෍ؔ਺〠〟぀ɻ (ょ゙が〣ఆཧ〣ূ໌ऴ) 35 / 43
37. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ໨࣍ 1 〤」〶〠

ࠓճ〣ษڧձ〣಺༰ ه๏〣४උ 2 ょ゙が〣ิ୊ (p.9) ෼෍ؔ਺〝 defective 〟෼෍ؔ਺ ょ゙が〣ิ୊ 3 ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) 36 / 43
38. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ろ゚ぢや〣ෆ౳ࣜ 〳』ɺろ゚ぢや〣ෆ౳ࣜぇ঺հ『぀ɻ Th:

ろ゚ぢや〣ෆ౳ࣜ [3] X ぇ֬཰ۭؒ (Ω, F, P) ্〣 Rk-஋֬཰ม਺〝『぀ɻඇෛؔ਺ h : Rk → R≥0 〠ର「〛 h(X) ⿿Մੵ෼〟〾〥ɺ೚ҙ〣 ε > 0 〠ର「〛ɺ P(h(X) ≥ ε) ≤ E[h(X)] ε . 37 / 43
39. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ྫ 2.6 〣લ൒

(1/3) ྫ 2.6 〣ओு (લ൒) (p.10) ֬཰ม਺ྻ Xn ぇɺ⿴぀ p > 0 ⿿ଘࡏ「〛 E[|Xn |p] = O(1) ぇຬ〔『〷 〣〝『぀ɻ〈〣〝　ɺ{Xn : n ∈ N} 〤Ұ༷ۓີ〜⿴぀ɻ O(1) 〣ղऍ〠ؔ『぀ิ଍ ゘アはげ〣ざがはがه๏〣ఆٛ〠「〔⿿⿺〥ɺ E[|Xn |p] = O(1) ⇔ [∃N ∈ N, ∃C > 0 s.t. n > N ⇒ E[|Xn |p] ≤ C] ぇҙຯ『぀〔〶ɺn > N ぇຬ〔『શ〛〣 n 〜〤 |Xn |p 〤Մੵ෼〜⿴぀ɻ 〝〈あ⿿ɺn ≤ N ぇຬ〔『 n 〜 |Xn |p ⿿Մੵ෼〜⿴぀อূぇ『぀〈〝 〤〜　〟⿶ɻ〷「⿴぀ n(≤ N) 〜 |Xn |p ⿿Մੵ෼〜〟⿶〽⿸〟৔߹ɺ〈 〣ޙ〣ٞ࿦〠ࠩ「ো぀〔〶ɺ〈〈〜〤ɺE[|Xn |p] = O(1) 〣ҙຯぇ ∃C > 0 s.t. ∀n ∈ N ⇒ E[|Xn |p] ≤ C 〝ղऍ「〛ٞ࿦ぇਐ〶぀〈〝〠『぀ɻ 38 / 43
40. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ྫ 2.6 〣લ൒

(2/3) 〟】〟〾ɺろ゚ぢや〣ෆ౳ࣜ〽〿ɺ೚ҙ〣 M > 0 〠ର「〛 P(|Xn | > M) ≤ 1 Mp E[|Xn |p] (16) ⿿੒〿ཱ〙ɻ ࣜ (16) 〣ূ໌ M > 0 ぇ೚ҙ〠〝぀ɻE[|Xn |p] = O(1) 〽〿ɺ೚ҙ〣 n ∈ N 〠ର「 |Xn |p 〤Մੵ෼〜⿴぀⿾〾ɺろ゚ぢや〣ෆ౳ࣜ〽〿ɺ P(|Xn |p ≥ Mp) ≤ E[|Xn |p] Mp . ∴ P(|Xn | ≥ M) ≤ E[|Xn |p] Mp . ∴ P(|Xn | > M) ≤ E[|Xn |p] Mp . 39 / 43
41. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ྫ 2.6 〣લ൒

(3/3) M ぇे෼େ　。〝぀〈〝〠〽〿ɺࣜ (16) 〣ӈล〤⿶。〾〜〷খ《。〟 ぀〔〶ɺ{Xn : n ∈ N} ⿿Ұ༷ۓີ〜⿴぀〈〝⿿い⿾぀ɻ ্〣ओு〣ิ଍ ࣜ (16) 〣྆ล〠ؔ「〛্ݶぇ〝ぁ〥ɺ sup n∈N P(|Xn | > M) ≤ sup n∈N 1 Mp E[|Xn |p] = 1 Mp sup n∈N E[|Xn |p]. 〈〈〜ɺE[|Xn |p] = O(1) 〽〿ɺ⿴぀ఆ਺ C > 0 ⿿ଘࡏ「〛, sup n∈N P(|Xn | > M) ≤ C Mp . 〈ぁ〽〿ɺ lim M→∞ sup n∈N P(|Xn | > M) = 0. 〽〘〛ɺ{Xn : n ∈ N} 〤Ұ༷ۓີ〜⿴぀ɻ 40 / 43
42. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ྫ 2.6 〣ޙ൒

(1/2) ྫ 2.6 〣ޙ൒ (p.10) E[X2 n ] = V[Xn ] + (E[Xn ])2 ⿿੒〿ཱ〙〈〝⿾〾ɺ E[Xn ] = O(1) ⿾〙 V[Xn ] = O(1) 〜⿴぀〈〝〤Ұ༷ۓີੑ〣े෼৚݅ ぇ༩⿺぀ɻ ্〣ओு〣ิ଍ E[Xn ] = O(1) ⿾〙 V[Xn ] = O(1) ⿿੒〿ཱ〙〝　ɺ E[X2 n ] = V[Xn ] + (E[Xn ])2 〽〿 E[X2 n ] = O(1) 〜⿴぀ɻ 〽〘〛ɺp = 2 〜 |Xn |p ⿿Մੵ෼〜⿴぀〈〝⿾〾্ه⿿ै⿸ɻ〔〕「ɺ ٯ〤੒〿ཱ〔〟⿶ɻ 41 / 43
43. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) ྫ 2.6 〣ޙ൒

(2/2) ൓ྫ ֬཰ม਺ྻ Xn ぇɺશ〛〣 n 〜 Xn ぇぢがてが෼෍ 7 〠ै⿸֬཰ม਺〜 ⿴぀〽⿸〟֬཰ม਺ྻ〝『぀ɻ֤ n 〠ର「 Xn 〤ۓີ〕⿾〾ɺε > 0 ぇ ೚ҙ〠〝〘〔〝　ɺ⿴぀ M > 0 ⿿ଘࡏ「〛ɺ P(|Xn | > M) < ε 〜⿴぀ɻ《〾〠ɺશ〛〣 n 〜 Xn ⿿ಉ෼෍〠「〔⿿⿸⿾〾ɺ sup n∈N P(|Xn | > M) < ε ⿿੒〿ཱ〙ɻ〽〘〛ɺ{Xn : n ∈ N} 〤Ұ༷ۓີ〜⿴぀ɻ ҰํɺXn 〤ぢがてが෼෍〠「〔⿿⿸⿾〾ɺظ଴஋〹෼ࢄ〤ଘࡏ「〟⿶ 〔〶ɺE[Xn ] ̸= O(1), V[Xn ] ̸= O(1). 7ぢがてが෼෍〣ಛ௃〝「〛ɺฏۉ〷෼ࢄ〷ଘࡏ【』ɺ྆୺〣֬཰ີ౓〤 0 〠઴ۙ『぀ ࿈ଓత〟֬཰෼෍〜⿴぀〝⿶⿸఺⿿ڍ〆〾ぁ぀ɻશ〛〣 n 〠ର「〛 Xn ぇぢがてが෼෍ 〝「〛「〳⿺〥ɺฏۉ〷෼ࢄ〷ଘࡏ「〟⿶⿾〾 O(1) 〜〟。ɺ֤ Xn 〣ۓີੑ⿿֬཰ม਺ ྻ {Xn} 〣Ұ༷ۓີੑぇ੒〿ཱ〔【〛⿶぀ɻ 42 / 43
44. ### 〤」〶〠 ょ゙が〣ิ୊ (p.9) ろ゚ぢや〣ෆ౳ࣜ (Example 2.6, p.10) Reference I [1]

Aad W Van der Vaart. Asymptotic statistics, Vol. 3. Cambridge university press, 2000. [2] ਿӜޫ෉. جૅ਺ֶ 2 ղੳೖ໳ 1, ୈ 2 ר. ౦ژେֶग़൛ձ. [3] ౷ܭֶ〭〣֬཰࿦, 〒〣ઌ〭: に゜⿾〾〣ଌ౓࿦తཧղ〝઴ۙཧ࿦〭〣Ս々ڮ. ಺ా࿝௽ะ, 2021. [4] स໦௚ٱ. ֬཰࿦. ߨ࠲਺ֶ〣ߟ⿺ํ / ൧ߴໜ [〰⿾] ฤ. ே૔ॻళ, 2004. 43 / 43