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2017 12 12 Makoto Nakashima 2017 12 12 1 / 22

2.1. C, D π- C, D. A 1, A 2 C A 1 A 2 C A 3, A 4 D A 1 A 2 D Makoto Nakashima 2017 12 12 2 / 22

. (,, L p - ). Makoto Nakashima 2017 12 12 3 / 22

. (,, L p - )... Makoto Nakashima 2017 12 12 3 / 22

(Ω, F, P ):. X, {X : 1}: Ω,. X X : X X P ({ }) ω : lim X (ω) = X(ω) = 1 ε > 0 lim P ({ω : X (ω) X(ω) > ε}) = 0 1 p < X, X L p (P ). X X L p lim X X p = 0. X L p (P ). X p = E[ X p ] 1/p Makoto Nakashima 2017 12 12 4 / 22

Figure:. L p -. Makoto Nakashima 2017 12 12 5 / 22. 3.1 X, {X : 1} Ω.. (i) X X, P -a.s. X P X. (ii) X L p X X P X. (iii) X P X {Xk : k 1} X k X, P -a.s..

3.1 ([0, 1], B([0, 1]), dx). [0, 1] X. X (x) = 1 I (x)., I = [ k 1 2 l, k ] 2 l, = 2 l + k, 0 k 2 l 1. x [0, 1] X (x) = 1 X 0, P -a.s., 0 < ε < 1 P ( X > ε) = 2 l ( ) Makoto Nakashima 2017 12 12 6 / 22

3.2 L p -, L p - ([0, 1], B([0, 1]), dx). [0, 1] X. X (x) = 1/p 1 I (x)., I = [ 0, 1 ]. X 0,., E [ X p ] = 1 0 L p -. 0 ( 1/p 1 I ) p dx = 1 Makoto Nakashima 2017 12 12 7 / 22

. 3.1 X:. {X : 1}:. (i) ( ) {X : 1}, X X P -a.s.. lim E[X ] = E[X]. (ii) ( ) {X : 1}. lim E[X ] E[ lim X ]. (iii) ( ) X X, P -a.s.. Y E[Y ] < X Y, P -a.s.. lim E[X ] = E[X] Makoto Nakashima 2017 12 12 8 / 22

Figure:. Makoto Nakashima 2017 12 12 9 / 22

1. S 1 6.? (3.1) Makoto Nakashima 2017 12 12 10 / 22

1. S 1 6.? 3.2 ( ) {X : 1}: R- i.i.d.. m = E[X 1 ], σ 2 = V (X 1 ) <. [ (i) lim E X 1 + + X ] 2 m = 0. X 1 + + X L 2 m. (ii) ε > 0 lim P ( ) X 1 + + X m > ε = 0. (3.1) (iii) R f x = m [ ( )] lim E X1 + + X f = f(m). 3.2 (3.1) L 2 -. Makoto Nakashima 2017 12 12 10 / 22

(i), (ii) (i) 2.1 [ X 1 + + X E ] 2 m = V (X 1) = σ2 L 2 -. (ii) L 2 -. Makoto Nakashima 2017 12 12 11 / 22

(iii) (iii) ε > 0 ( )] [f E X1 + + X f(m) ( ) = [f E X1 + + X f(m)] ( ) ] [f E X1 + + X f(m) : X 1 + + X m > ε ( ) ] + [f E X1 + + X f(m) : X 1 + + X m ε ( ) X 1 + + X sup f(x) f(m) + 2 f P m x m ε > ε σ 2 sup f(x) f(m) + 2 f x m ε ε 2., f x = m. Makoto Nakashima 2017 12 12 12 / 22

. 3.3 ( ) [0, 1] f. f (x) = m=0 ( ) ( m ) x m (1 x) m f m f.. lim sup x [0,1] f (x) f(x) 0 Makoto Nakashima 2017 12 12 13 / 22

3.3 {X : 1} P (X = 1) = p = 1 P (X = 0) (p [0, 1]). E[X ] = p, V (X ) = p(1 p). S = k=1 X k., P (S = m) = [ E f ( ) p m (1 p) m m ( )] S = f (p). Makoto Nakashima 2017 12 12 14 / 22

[0, 1] sup f(x) f(y) ε x,y [0,1], x y <δ δ. δ > 0 3.2(iii) σ 2 [f (p) f(p)] sup f(x) f(p) + 2 f x p δ δ 2 ε + 2 f σ 2 δ 2. p. lim sup p [0,1] f (p) f(p) ε Makoto Nakashima 2017 12 12 15 / 22

3.2 {X : 1}. E[ X ] < 1 X <, P -a.s. 1, E X = E[X ] 1 1. Makoto Nakashima 2017 12 12 16 / 22

E[ X ] = E X 1 1. X <, P -a.s.. ( 1 ). Makoto Nakashima 2017 12 12 17 / 22

. 3.3 X, X ( 1).. (i) ( ) {X : 1}. X P X.. lim E[X ] E[X] (ii) ( ) X P X. Y E[Y ] <.. X Y, P -a.s. lim E[X ] = E[X] Makoto Nakashima 2017 12 12 18 / 22

{X : 1} Ω ( ). X (ω) X(ω) lim X (ω)p (dω) X(ω)P (dω). Ω Ω Makoto Nakashima 2017 12 12 19 / 22

{X : 1} Ω ( ). X (ω) X(ω) lim X (ω)p (dω) X(ω)P (dω)., lim E[X ]. Ω Ω Makoto Nakashima 2017 12 12 19 / 22

3.4 R- {X : 1} R- X. f C(R). f(x ) P f(x) Makoto Nakashima 2017 12 12 20 / 22

3.4 R- {X : 1} R- X. f C(R). f(x ) P f(x) 3.2, 3.3.. Makoto Nakashima 2017 12 12 20 / 22

3.5 d {X : 1}. λ > 0 X Poi(λ ). 1 X 1 λ Makoto Nakashima 2017 12 12 21 / 22

3.5 d {X : 1}. λ > 0 X Poi(λ ). 3.2 1 X 1 λ Makoto Nakashima 2017 12 12 21 / 22

3.6 {X : 1} R- X d Exp(1). lim X log = 1, P -a.s.. Makoto Nakashima 2017 12 12 22 / 22

3.6 {X : 1} R- X d Exp(1). lim X log = 1, P -a.s.. ε > 0 P (X > (1 + ε) log ) = 1 P (X > (1 ε) log ) = 1 ( ). - ( 1.1 2.6.) Makoto Nakashima 2017 12 12 22 / 22

( ).,.(4 ) Makoto Nakashima 2017 12 12 23 / 22