- PDF 無料ダウンロード

n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i

.10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii

1 1 1.1..................................... 1 1............................... 3 1.3......................... 6 1.4...................................... 8 1.1........................... 1.................................. 18.3......................... 5 3 30 3.1 0 1........................... 30 3.................................... 33 3.3.................................. 36 3.4.............................. 40 56 iii

1 0 1.1 1.1 Ω Ω F F σ (i) (ii) (iii) Ω F Λ F Λ { F Λ i F, i = 1,,... Λ i F F 1.1 ( ) F Ω ( ) Λ 1, Λ F, Λ 1 Λ F 1. C Ω C σ C σ σ(c) 1.3 Ω n R n C Ω σ(c) n B n B n n 1.4 C 1 { (, r] ; r }, C { (a, b] ; a < b < } σ(c i ) = B 1, i = 1, 1.5 ξ Ω n. C { } ξ 1 (A) ; A B n C σ ξ σ σ(ξ) 1.6 Ω σ F (Ω, F ), F Ω F 1

1. { n 1.7 (Ω i, F i ), i = 1,..., n C ; Λ i F i, i = 1,..., n Λ i } n Ω i { (ω 1,..., ω n ) ; ω i Ω i, i = 1,..., n } σ σ(c) F i, i = n ( n n ) 1,..., n σ F i Ω i, F i (Ω i, F i ), i = 1,..., n 1.8 (Ω, F ) F P P F (i) Λ F P (Λ) 0 (ii) P (Ω) = 1 (iii) Λ i F, i = 1,,... ( ) P Λ i = P (Λ i ) (σ ) P 1.8 ( ) P Λ F P (Λ) P (Λ) < P P σ Λ i F ( ) 1.9 (Ω, F ) P F (Ω, F, P ) Λ F P (Λ) Λ Λ 1.10 C Ω Q C (i) A C Q(A) 0 (ii) Q(Ω) = 1 (iii) A i C, i = 1,,... ( ) A i C Q A i = Q(A i )

1. 3 Q σ(c) σ(c) P A C P (A) = Q(A) A B (A \ B) (B \ A) A B A B 1.11 (Ω, F, P ) C F C Λ σ(c) A n C, n = 1,, lim P (Λ A n) = 0 σ(c) C n (Ω i, F i, P i ), i = 1,..., n Ω Ω i { n B Λ i ; Λ i F i, i = 1,..., n } C B n n C σ(c) = F i F F i ( n ) Q Q Λ i = A = l j=1 n P i (Λ i ) C n l n Λ (j) i Q(A) = P i (Λ (j) i ) j=1 A Q(A) A Q Q n F P P P 1,..., P n P = P i (Ω, F, P ) (Ω i, F i, P i ), i = 1,..., n 1. (Ω, F ) 1.1 Ω X A B 1 X 1 (A) = { ω Ω ; X(ω) A } F X F F

1. 4 X + + X 1.1 X 1 { } X 1 {} F X F 1.13 X F r X 1 ( (, r ] ) F R n B n n 1.14 R n g n 1.15 X i, i = 1,..., n Ω F g n Y (ω) g(x 1 (ω),..., X n (ω)) F 1.16 X n, n = 1,,... Ω F inf X n, sup X n, lim lim X n Ω Ω F lim X n(ω) R 1 ω F Ω X F Λ i, i = 1,..., n a i R 1, i = 1,..., n X(ω) = a i I Λi (ω) X I ω Λ I Λ (ω) = 1, n n X n, ω / Λ I Λ (ω) = 0 1.17 Ω F X (Ω, F, P ) F X X F A B 1 X 1 (A) F P (X 1 (A)) X A P (X A) 1.18 B 1 µ µ = P X 1, µ(a) = P (X A), A B 1 µ B 1

1. 5 X Ω R 1 X P µ (Ω, F, P ) (R 1, B 1, µ) 1.19 1.18 µ X F (x) µ( (, x] ) = P (X x) R 1 F X µ 1.0 R 1 F F (i) x y F (x) F (y) (ii) x c F (x) F (c) (iii) lim F (x) = 1, x lim F (x) = 0 x ( ) ( ) R 1 F ( ) ( ) X X F F 1.1 δ {a} a R 1 X µ X δ {a} µ(a) = P (X = a) = 1 X a 1 1. Φ(λ, u) λ > 0 u 0 X µ k {0} N X λ u Φ(λ, u) λ λk µ(ku) = P (X = ku) = e k! 1.3 N(m, v) < m < v > 0 X µ A B 1 X m v

1. 6 N(m, v) µ(a) = P (X A) 1 = e (x m) /v dx πv A m = 0 v = 1 X N(0, 1) (Ω, F, P ) X f Λ F P (X Λ) = f(x) dx f P X i, i = 1,..., n (Ω, F, P ) Ω n X(ω) = (X 1 (ω),..., X n (ω)) n 1.4 X n B n µ µ = P X 1 Λ µ(a) = P (X A), A B n µ B n 1.5 1.4 µ X X 1,..., X n F (x 1,..., x n ) µ( (, x 1 ] (, x n ] ) = P (X 1 x 1,..., X n x n ) R n F X µ 1.6 n X X P (X A) = P ( X A), A B n 1.3 1.7 (Ω, F, P ) A i F, i = 1,..., n A 1,..., A n ( n ) P A i = n P (A i )

1. 7 X i, i = 1,..., n (Ω, F, P ) k i 1.8 A i B ki, i = 1,..., n P (X 1 A 1,..., X n A n ) = n P (X i A i ) X 1,..., X n 1.9 n X 1,..., X n X i, i = 1,,... σ 1.30 F C i, i = 1,..., n Λ i C i, i = 1,..., n ( n ) P Λ i = n P (Λ i ) C 1,..., C n n C 1,..., C n F C i, i = 1,,... 1.31 X i, i = 1,..., n σ(x i ), i = 1,..., n ( 1.3 C i, i = 1,,... J 1,..., J p ) σ, l = 1,..., p j J l C j ( k ) ( ) 1.33 C i, i = 1,,... σ C j σ C j j=1 j=k+1 lim X n(ω) R 1 ω 1.16 F ω X(ω) = lim X n (ω), ω / X(ω) = 0 X F 1.34 P ( ) = 1 X n X 1 ω X(ω) = lim X n (ω) X n X X n X a.e.

1. 8 X n (ω), n = 1,,... = p=1 N=1 n,m N { ω Ω ; X n (ω) X m (ω) < 1 p P ( ) = 1 } 1.35 X n ε > 0 lim P ( ω Ω ; N n, m X n(ω) X m (ω) < ε ) = 1 N X n X ε > 0 lim P ( ω Ω ; n m X m(ω) X(ω) < ε ) = 1 1.4 1.36 (Ω, F, P ) X X(ω) P (dω) < Ω X 1.37 (Ω, F, P ) X EX X(ω)P (dω) Ω EX X EX E[X] E(X) Λ F X(ω)P (dω) X(ω)I Λ (ω)p (dω) Λ E(X ; Λ) Ω E(X ; Λ) E[XI Λ ], Λ F

1. 9 1.38 X Y (Ω, F, P ) (i) X Y a, b R 1 E(aX + by ) = aex + bey ( ) (ii) X Y a.e. EX EY (iii) EX E X ( ) (iv) X n, n = 1,, X n X a.e. EX n EX X (Ω, F, P ) n g n 1.15 g(x) Ω Eg(X) X P X µ g(x) µ 1.39 µ X n g Eg(X) Eg(X) Eg(X) = g(x 1,..., x n )µ(dx 1 dx n ) (1.1) (R n, B n, µ) Ẽ (1.1) Ẽg 1.40 X EX k X k E X k X k V (X) E[(X EX) ] X 1.41 δ {a} EX = a, V (X) = 0 1.4 Φ(λ, u) EX = V (X) = uke λ k=0 λk k! = uλ (uk uλ) e λ k=0 λk k! = u λ

1. 10 1.43 N(m, v) EX = V (X) = x 1 πv e (x m) /v dx = m (x m) 1 e (x m) /v dx = v πv m v X 1.46 ( 1.44 1.45),. (Ω i, F i ), i = 1, Ω Ω 1 Ω, F F 1 F Λ Ω Λ ω1 { ω Ω ; (ω 1, ω ) Λ } ω 1 Λ Ω X X ω1 (ω ) X(ω 1, ω ) Ω X ω1 ω 1 X 1.44 Λ F ω 1 Ω 1 Λ ω1 F X F ω 1 Ω 1 X ω1 F 1.45 (Ω i, F i, P i ), i = 1, (Ω, F, P ) X F X ωi (ω j )P j (dω j ), i j + F i Ω j X(ω 1, ω )P (dω 1 dω ) = P 1 (dω 1 ) X ω1 (ω )P (dω ) (1.) Ω Ω 1 Ω = P (dω ) X ω (ω 1 )P 1 (dω 1 ) (1.3) Ω Ω 1 X (Ω, F, P ) X ωi (ω j )P j (dω j ) Ω i (1.) Ω j (1.3) 1.46 X i, i = 1,..., n X i n X i [ n ] n E X i = EX i 1.47 E XY (E X p ) 1/p (E Y q ) 1/q 1 p + 1 q = 1, 1 < p <

1. 11 1.48 lim E( X n X p ) = 0 X n X p p = 1 X n X 1.49 X n, n = 1,,... lim sup X n (ω) P (dω) = 0 a n {ω ; X n (ω) a} a E( X n ; X n (ω) a) n 0 1.50 X n, n = 1,,... X n X a.e. lim E( X n X ) = 0 1.51 Y X n Y a.e., n = 1,,... X n X a.e. lim E( X n X ) = 0

(R n ).1 (Ω, F, P ) Z Z ReZ ImZ Z Z EZ EZ = E[ReZ] + ie[imz].1 X n µ X R n ϕ ϕ(t) Ee i(t,x) = ( exp i t k x k )µ(dx 1 dx n ), t = (t 1,..., t n ) k=1 ϕ X µ (t, X) e i(t,x) e i(t,x) 1 µ f ϕ ϕ(t) = ( exp i t k x k )f(x 1,..., x n ) dx 1 dx n k=1 ϕ f X Y X 1.6 X X. n X ϕ X ϕ X 1

. 13 (i) ϕ X (0) = 1 ϕ X (t) 1 ϕ X ( t) = ϕ X (t) (ii) ϕ X (t + h) ϕ X (t) ϕ X (h) ϕ X (0) (iii) ϕ X R n n 1 n = 1 ( ) ϕ X (t + h) ϕ X (t) = Ee i(t+h)x Ee itx = E[e itx (e ihx 1)] (E e ihx 1 ) (E e ihx 1 ) E[ e ihx 1 ] = E[(cos(hX) 1) + sin (hx)] = E(1 cos(hx)) = Re (ϕ X (0) ϕ X (h)) ϕ X (0) ϕ X (h) ϕ X (t + h) ϕ X (t) ϕ X (h) ϕ X (0) ( ) ϕ X (h) ϕ X (0) = ϕ X (0) ϕ X (h) = E(1 e ihx ) E 1 e ihx 1 e ihx lim 1 e ihx = 0 h 0 lim E 1 h 0 eihx = 0 ( ) ε > 0 δ > 0 h < δ ϕ X (t + h) ϕ X (t) < ε ϕ X R 1 n n 1 n = 1.3 X i, i = 1,..., n n X X i X n ϕ X (t) = ϕ Xi (t), t = (t 1,..., t n ) (.1)

. 14 n n = ϕ X (t) = E[e it(x 1+X ) ] = E[cos(tX 1 + tx ) + i sin(tx 1 + tx )] = E[cos tx 1 cos tx sin tx 1 sin tx ] + ie[sin tx 1 cos tx + cos tx 1 sin tx ] = E[cos tx 1 cos tx ] E[sin tx 1 sin tx ] + ie[sin tx 1 cos tx ] + ie[cos tx 1 sin tx ] (.) (.) 1.46 ϕ X (t) = E[cos tx 1 ]E[cos tx ] E[sin tx 1 ]E[sin tx ] = E[cos tx 1 ](E[cos tx ] + ie[sin tx ]) = E[e itx 1 ]E[e itx ] + ie[sin tx 1 ]E[cos tx ] + ie[cos tx 1 ]E[sin tx ] + ie[sin tx 1 ](E[cos tx ] + ie[sin tx ]) n = n n + 1 X 1 + + X n X n+1 ϕ X (t) = Ee it(x 1+ +X n +X n+1 ) = E[e it(x 1+ +X n ) e itx n+1 ] = E[e it(x 1+ +X n ) ]E[e itx n+1 ] ( n ) = ϕ Xi (t) ϕ Xn+1 (t) = n+1 ϕ Xi (t) X 1,..., X n X (.1).4 δ {a} ϕ(t) = e ita.5 Φ(λ, u) ϕ(t) = k=0 e ituk e λ λ k k! = exp{λ(e itu 1)}

. 15.6 N(m, v) ϕ(t) = e itx 1 πv e (x m) /v dx = e itm vt / f ˆf ˆf f.7 n X µ ϕ n A = (a i, b i ) µ( A) = 0 µ(a) = ( 1 ) n lim π T T T T T n e it ka k e it k b k k=1 it k ϕ(t 1,..., t n ) dt 1 dt n (.3) n n A A A = [a i, b i ] (a i, b i ) 0 sin t t [0, T ] T T 0 sin t t dt = = T dt = π ( 0 0 T J e tu sin t dt 0 J = [ e tu cos t ] T 0 0 T u 0 0 ) e tu du sin t dt (.4) e tu sin t dt du (.5) e tu cos t dt [ = 1 e T u cos T + ue tu sin t J = 1 u + 1 e T u u + 1 T sin t dt = 0 t cos T + u sin T u + 1 ] T = 1 e T u cos T ue T u sin T u J 0 0 T u e tu sin t dt 0 (cos T + u sin T ) (.5) 1 u + 1 du u + 1 0 e T u (cos T + u sin T ) du u + 1 u + 1 0 e T u du = T

. 16 0 sin t t dt = T lim T 0 sin t t dt = 0 1 [ u + 1 du = tan 1 u ] 0 = π (.4) 0 sin xt t dt = π, x > 0 0, x = 0 π (.6), x < 0 (.3) T T e ita e itb it ϕ(t) dt = T T e ita e itb it ( ) e itx µ(dx) dt (.7) e ita e itb it e itx e ita e itb it e it(a b) e itb e itb T it (e it(a b) 1) e itb it e it(a b) 1 it a b e ita e itb e itx eit(x a) e it(x b) dt µ(dx) < it T it (.7) T T e ita e itb it ϕ(t) dt = T T e it(x a) e it(x b) it dt µ(dx) (.8) T J(T, x, a, b) T e it(x a) e it(x b) it dt T J(T, x, a, b) = 0 sin (x a)t t T dt 0 sin (x b)t t dt (.9) (.6) lim J(T, x, a, b) = T 0, x < a b < x π, π, x = a x = b a < x < b

. 17 (.9) (.6) sup J(T, x, a, b) < x,t lim J(T, x, a, b) µ(dx) = T lim J(T, x, a, b)i A(x) µ(dx) T + lim J(T, x, a, b)i A(x) µ(dx) T µ( A) = 0 0 lim T J(T, x, a, b) µ(dx) = (π) µ(a) (.8) (.3).8 n X Y µ X µ Y ϕ X ϕ Y ϕ X = ϕ Y µ X = µ Y C µ X ( A) + µ Y ( A) = 0 A = (, a] µ X ({a}) + µ Y ({a}) > 0 a { a ; µ X ({a}) > 0 } = n N { a ; µ X({a}) > 1/n } { a ; µ X ({a}) > 1/n } a n { a ; µ X ({a}) > 0 } a µ X ({a}) + µ Y ({a}) > 0 a B = (, b] A m C, m = 1,,... µ X (B) = lim m µ X(A m ), µ Y (B) = lim m µ Y (A m ) ϕ X = ϕ Y µ X (A m ) = µ Y (A m ), m = 1,, µ X (B) = µ Y (B) B X Y µ X = µ Y.

. 18. µ, µ k, k = 1,,... R n.9 g k g(x 1,..., x n )µ k (dx 1 dx n ) g(x 1,..., x n )µ(dx 1 dx n ) (.10) µ k µ µ k µ k.10 (i) µ k µ (ii) (iii) (iv) (v) Γ R n lim µ k(γ) µ(γ) k G R n lim µ k (G) µ(g) k µ( A) = 0 A B n lim µ k(a) = µ(a) k F F k µ µ k (vi) F x lim F k(x) = F (x) k f lim k f(x 1,..., x n )µ k (dx 1 dx n ) = f(x 1,..., x n )µ(dx 1 dx n ) ( ) ) : Γ R 1 ρ(x, Γ) inf{ x y ; y Γ } ( 1 ) j h j (x) hj (x) I Γ (x) 1 + ρ(x, Γ) h j (x) I Γ (x), x R 1 h j, j = 1,,... lim k h j (x)µ k (dx) µ k (Γ) h j (x)µ k (dx) lim k µ k (Γ)

. 19 lim b j k h j (x)µ k (dx) {b j } j lim j b j lim k µ k (Γ) ( ) lim b j = lim j = j µ(γ) lim µ k (Γ) k h j (x)µ(dx) I Γ (x)µ(dx) = µ(γ) ( ) ( ) ( )( ) ( ) : A B 1 A i A Ā A A i A Ā Ai Ā ( )( ) µ(a i ) lim µ k (A i ) lim µ k (A) lim µ k (A) lim µ k (Ā) µ(ā) (.11) Ā = A i ( A) µ( A) = 0 µ(ā) µ(ai ) + µ( A) = µ(a i ) (.1) (.11) (.1) µ(ā) = µ(ai ) (.11) lim µ k(a) = µ(a) k ( ) ( ) : x F A (, x] A i = (, x) µ( A) = µ(a) µ(a i ) = F (x) lim y x F (y) = 0 A ( ) F (x) = µ(a) = lim k µ k (A) = lim k F k (x) ( ) ( ) ( ) : f ε > 0 f f sup x R 1 f(x) f(x) < ε (.13) f f K M > 0 K [ M, M] f [ M, M]

. 0 ε > 0 δ > 0 x, y [ M, M] x y < δ f(x) f(y) < ε (.14) l j a j b j < δ/ Λ j = (a j, b j ] Λ j Λ j = [ M, M] α j f(b j ) f j=1 f(x) l α j I Λj (x) j=1 (.14) f(x) f(x) = < ε l f(x) f(b j ) I Λj (x) j=1 l I Λj (x) j=1 = ε I [ M,M] (x) ε (.13) f F R 1 Λ j F F x A(x) (, x] A(x) B B { x ; µ( A(x)) > 0 } B = { x ; µ( A(x)) > 1/n } n=1 { x ; µ( A(x)) > 1/n } x n B B { R 1 F R 1 Λ j F F R 1 Λ j F Λ j F µ(λ j ) = µ((, b j ]) µ((, a j ]) = F (b j ) F (a j ) = lim k {F k (b j ) F k (a j )} = lim k µ k (Λ j )

. 1 f(x)µ(dx) = l α j µ(λ j ) j=1 = lim k l j=1 α j µ k (Λ j ) = lim k f(x)µ k (dx) (.15) (.13) (.15) ε > 0 k 0 k k 0 f(x)µ(dx) f(x)µ k (dx) f(x)µ(dx) f(x)µ(dx) + f(x)µ(dx) f(x)µ k (dx) + f(x)µ k (dx) f(x)µ k (dx) < 3ε ( ) ( ) ( ) : a > 0 J a [ a, a] J a 1 J { a+1 0 [0, 1] f a µ ε > 0 a, ( ) µ(j a ) I Ja (x)µ(dx) 1 µ(j a ) < ε (.16) = lim f a (x)µ(dx) k lim k lim k f a (x)µ k (dx) I Ja+1 (x)µ k (dx) µ k (J a+1 ) = lim inf k n µ k(j a+1 ) n 0 k n 0 1 µ k (J a+1 ) < ε (.17) ε > 0 a = a(ε) g gf a g(x) = g(x)f a (x) + g(x)(1 f a (x)) g(x)f a (x) + g (1 f a (x)) g(x)µ(dx) g(x)µ k (dx) g(x)f a (x)µ(dx) g(x)f a (x)µ k (dx) { } + g (1 f a (x))µ(dx) + (1 f a (x))µ k (dx)

. (.16) (.17) k n 0 (1 f a (x))µ(dx) + (1 f a (x))µ k (dx) 1 µ(j a ) + 1 µ k (J a+1 ) < ε ( ) ε > 0 k 0 k k 0 g(x)f a (x)µ(dx) g(x)f a (x)µ k (dx) < ε j max(n 0, k 0 ) k j g(x)µ(dx) g(x)µ k (dx) < ε + ε g µ k µ.11 M R n M ε > 0 R n Γ µ M µ(γ) > 1 ε (.18) ( ) : M ε > 0 Γ µ(γ) 1 ε µ µ Γ M Γ l [ l, l] µ l µ Γl µ l, l = 1,,... µ lj, j = 1,,... G k ( k, k) l j k j G k Γ lj l j k µ lj (G k ) µ lj (Γ lj ) 1 ε µ lj µ µ M.10 ( ) k µ(g k ) lim µ lj (G k ) 1 ε j µ(r 1 ) = lim k µ(g k ) 1 ε µ ( ) : (.18) µ k M, k = 1,,... Helly F k µ k { a j ; j = 1,,... } R 1

. 3 F 0 F k (a j ) 1, k = 1,,... F k a j { F k ; k = 1,,... } { F k(j, l) ; l = 1,,... } ρ F k(1, l) (a 1 ) ρ(a 1 ) { k(, l) ; l = 1,,... } { k(1, l) ; l = 1,,... } F k(, l) (a ) ρ(a ) { k(3, l) ; l = 1,,... } { k(, l) ; l = 1,,... } { k(l, l) ; l = 1,,... } m k(1, 1),, k(m 1, m 1) { k(m, l) ; l = 1,,... } F k(m, l) (a m ) ρ(a m ) F k(l, l) (a m ) ρ(a m ), m = 1,,... J x { y R 1 ; y > x } F F (x) inf ρ(a m ) a m J x a mj x a mj J x ρ(a mj ), j = 1,,... ρ(a mj ), j = 1,,... F (x) = lim j ρ(a mj ) (.19) a p J x l a p a ml ρ(a p ) ρ(a ml ) lim j ρ(a mj ) inf ρ(a m ) = F (x) a m J x a p J x F (x) = inf a p J x ρ(a p ) lim j ρ(a mj ) F (x) (.19) F h 0 a mj x a pj x + h a pj a mj a mj J x a pj J x+h, j = 1,,... F 1.0 a pj a mj 0 ρ(a pj ) ρ(a mj ) 0 lim ( ρ(a pj ) ρ(a mj ) ) j = F (x + h) F (x)

. 4 F F k x k c F (x k ), k = 1,,... lim F (x k) F (c) k a p J c a p J xk k ρ(a p ) inf ρ(a m ) = F (x k ) lim F (x k ) F (c) a m J xk k a p J c F (c) = inf a p J c ρ(a p ) lim k F (x k ) F (c) F (c) = lim F (x k ) F (x k ) F k (.18) ε > 0 A > 0 sup k µ k ([ A, A] c ) < ε (.0) a m < A sup F k(l, l) (a m ) ε l ρ(a m ) ε a m J x F (x) = inf ρ(a m ) ε a m J x x < A F (x) ε lim F (x) = 0 (.0) a m > A x inf F k(l, l) (a m ) > 1 ε l ρ(a m ) > 1 ε a m > x > A x a m J x F (x) = inf ρ(a m ) > 1 ε a m J x x > A F (x) > 1 ε lim F (x) = 1 F 1.0 F x F k(l, l) (x) F (x) x F ε > 0 F (x) > F (a p ) ε x a p J x a j J ap a p (.19) F (x) > ρ(a j ) ε = lim F k(l, l) (a j ) ε l lim F k(l, l) (x) ε l (.1)

. 5 a m a m < x F (x) < F (a m ) + ε a m < a j < x a j F (x) < ρ(a j ) + ε = lim F k(l, l) (a j ) + ε l F k(l, l) (x) + ε (.) lim l (.1) (.) x F (x) = lim F k(l, l) (x) l µ F.10 µ k(l, l) µ µ k, k = 1,,....1 µ R n M {µ k, k = 1,,... }.11 µ k µ µ k µ µ k µ g k j g(x)µ kj (dx) lim g(x)µ kj (dx) g(x)µ(dx) (.3) j.11 { µ kj ; j = 1,,... } µ kjl, l = 1,,... µ kjl µ lim l g(x)µ kjl (dx) = g(x)µ(dx) (.3) µ k µ.3.13 X k, X n µ k, µ µ k µ X k X µ k µ (.10) X k X g Eg(X k ) Eg(X)

. 6 X k X.14 ϕ ϕ k µ µ k µ k µ ϕ k ϕ ε > 0 δ > 0 h < δ ϕ k (t + h) ϕ k (t) < 3ε (.4) k M { µ, µ k, k = 1,,... } M µ.11 ε > 0 µ(γ) > 1 ε inf k µ k(γ) > 1 ε (.5) Γ R 1 e ihx { h ; h 1 } Γ sup e ihx 1 h ε > 0 x Γ h < δ sup e ihx 1 < ε (.6) x Γ δ > 0 (.5) (.6) k h < δ h ϕ k (t + h) ϕ k (t) = (e R i(t+h)x e itx )µ k (dx) 1 e R ihx 1 µ k (dx) 1 = e ihx 1 µ k (dx) + Γ Γ{ eihx 1 µ k (dx) < ε + µ k (Γ { ) < 3ε (.4) k ϕ ϕ(t + h) ϕ(t) < 3ε K R 1 { t 1,..., t l } K δ t K t t j < δ t j, j = 1,..., l µ k µ t R 1 ϕ k (t) ϕ(t)

. 7 k 0 k k 0 ϕ k (t j ) ϕ(t j ) < ε, j = 1,..., l k k 0 sup ϕ k (t) ϕ(t) t K l { ϕ k (t) ϕ k (t j ) I { t tj <δ} + ϕ k (t j ) ϕ(t j ) j=1 } + ϕ(t j ) ϕ(t) I { t tj <δ} < 7lε ε ϕ k ϕ K.15 ϕ k µ k ϕ k ϕ ϕ µ k µ ϕ µ ϕ k ϕ.16 ν R n ψ ν A > 0 ν([ 4A, 4A] n ) A n A 1 (n 1)/n A 1 (n 1)/n A 1 (n 1)/n A 1 (n 1)/n ψ(t 1,..., t n ) dt 1 dt n 1 (.7) n = 1 1 T T T ψ(t) dt = 1 T = = T T T T sin T x T x e itx ν(dx) dt 1 T eitx dt ν(dx) ν(dx) (.8) (.8) sin T x/t x 1 x > A sin T x 1 T x T x < 1 AT (.8) ν([ A, A]) + 1 AT T = A 1 (.8) ( 1 ν([ A, A]) ) A A 1 A 1 ψ(t) dt ν([ A, A]) + 1 ( 1 ν([ A, A]) ) 1 ν([ A, A]) + 1 1 ( ν([ 4A, 4A]) + 1 )

. 8 (.7) (.15) δ δ > 0 ϕ k (t) dt δ > 0 1 δ δ δ δ ϕ k (t) dt 1 δ δ δ δ δ ϕ(t) dt 1 δ ϕ(t) dt δ δ ϕ k (t) ϕ(t) dt (.9) ε > 0 k 0 k k 0 1 δ δ δ ϕ k (t) dt 1 ε (.30) ϕ(0) = lim ϕ k (0) = 1 ϕ t = 0 k ε > 0 δ > 0 t < δ ϕ(t) 1 < ε ϕ k ϕ ϕ k (t) 1 δ k 0 k k 0 1 δ δ δ ϕ k (t) ϕ(t) dt < ε (.31) δ δ δ ϕ(t) dt = δ δ δ δ ε(δ) (1 ϕ(t)) dt 1 ϕ(t) dt δ δ ϕ(t) dt δ(1 ε) (.3) (.31) (.3) (.9) (.30) ε > 0 inf k µ k ([ Ã, Ã]) > 1 4ε (.33) Ã > 0.16 (.7) n = 1 ν = µ k ψ = ϕ k A = δ 1 µ k ([ 4A, 4A]) 1 δ δ δ ϕ k (t) dt 1

. 9 (.30) k k 0 µ k ([ 4A, 4A]) (1 ε) 1 = 1 4ε µ j µ j ([ A j, A j ]) > 1 4ε, j = 1,,..., k 0 1 A j Ã max(a 1,..., A k0 1, 4A) (.33).11 µ k, k = 1,,... µ kj, j = 1,,... µ µ kj µ ϕ kj µ ϕ kj ϕ µ ϕ µ kl µ k ν µ kl ν ϕ kl ϕ ν ϕ ν = µ M {µ k, k = 1,,... }.11 µ k µ.1 µ k µ.14 ϕ k ϕ ϕ k ϕ ϕ

3 0 1 3.1 0 1 ( ) B j, j = 1,,... F σ σ k σ B j σ k k σ σ k {B j, j = 1,,... } k=1 σ k k=1 ξ j, j = 1,,... B j = σ(ξ j ), j = 1,,... {ξ j, j = 1,,... } j=k 3.1 Λ j B j, j = 1,,... lim Λ k lim Λ k ξ j 0 ω Λ { ω ; lim ξ j (ω) = 0 } j k Λ ξ 1,..., ξ k Λ σ 3. 0 1 B j F, j = 1,... σ {B j, j = 1,,... } 0 1 ( k F k σ ( C = σ j=1 j=1 ) B j F k C F k ) B j C σ(c) = σ k=1 ( ) B j j=1 30

3. 31 Λ {B j, j = 1,,... } ( ) k Λ σ B j Λ σ(c) j=k 1.11 ε > 0 Λ ε C ( l l Λ ε F l = σ P ( (Λ\Λ ε ) (Λ ε \Λ) ) < ε (3.1) j=1 ( Λ σ l+1 = σ ) B j Λ j=l+1 B j ) 1.33 Λ ε Λ (3.1) P (Λ) = P (Λ Λ) P (Λ Λ ε ) + P ( (Λ\Λ ε ) (Λ ε \Λ) ) P (Λ)P (Λ ε ) + ε P (Λ)( P (Λ) + P ( (Λ\Λ ε ) (Λ ε \Λ) ) ) + ε P (Λ) + P (Λ)P ( (Λ\Λ ε ) (Λ ε \Λ) ) + ε P (Λ) + ε ε P (Λ) P (Λ) 0 P (Λ)(1 P (Λ)) = 0 P (Λ) 0 1 3.3 σ X P (X = c) = 1 c X F F (x ) lim F (y) F (x ) x y x F σ(x) X 1 ( (, x] ) σ(x) X 1 ( (, x] ) 0 1 F (x) = P (X 1 ( (, x] ) ) 0 1 F c sup{ x R 1 ; F (x) = 0 } c < F (c ) = 0 F (c) = 1 P (X = c) = 1

3. 3 3.4 ξ i, i = 1,,... ω ξ i (ω), i = 1,,... 0 < lim b n b n Y n = 0 1 1 lim lim Y n ξ i b n b n = 1 k n k X (k) n = n Y n X (k) n = (ξ 1 + + ξ k 1 )/b n (ξ 1 + + ξ k 1 )/ lim n Λ k = i=k ξ i b n b n X (k) n (ω) ω Λ k { ω Ω ; X (k) n (ω) X (k) m (ω) < 1 } p p=1 N=k n,m=n ( Λ k σ j=k ) σ(ξ j ) lim Λ k Y n (ω) ω Λ P (Λ k Λ) = 0 (Y n X (k) n )(ω) (ξ 1 (ω) + + ξ k 1 (ω))/ lim b n ω Γ k Λ k Λ P (Λ k Λ) = P (Λ k Λ { ) + P (Λ k { Λ) ω Λ k Λ { ω Γ { k P (Λk Λ { ) = 0 P (Λ { k Λ) = 0 P (Λk Λ) = 0 P ((lim Λ k ) Λ) = 0 ( ω Λ j ) Λ ω (Λ k Λ) k=1 j k k=1 ( ) P ((lim Λ k ) Λ) P (Λ k Λ) lim Λ k P (lim Λ k ) 0 1 P (Λ) 0 1 k=1 = 0 lim b n = Y n lim Y n = lim X (k) n lim k σ k a R 1 { lim Y n a } = { lim X (k) n a } σ k X n (k) k lim Y n 3.3 lim Y n

3. 33 3. ξ i, i = 1,,... (ξ 1 + + ξ n )/n 3.4 1 3.5 ( S i ξ 1 + +ξ i ξ i, i = 1,..., n Eξ i = 0, i = 1,..., n P ( max S i a ) ES n, a > 0 i n a ES n < Eξ i v i <, i = 1,..., n Λ { ω Ω ; max i n S i(ω) a } i = 1,..., n Λ i Λ i { ω Ω ; S 1 (ω) < a,..., S i 1 (ω) < a, S i (ω) a } n Λ 1,..., Λ n Λ i = Λ ES n E(S n ; Λ) = E(S n ; Λ i ) = = E((S i + ξ i+1 + + ξ n ) ; Λ i ) { E(S i ; Λ i ) + E(S i (ξ i+1 + + ξ n ) ; Λ i ) j=i + E((ξ i+1 + + ξ n ) ; Λ i ) } (3.) S i I Λi (ξ 1,..., ξ i ) ξ i+1 + + ξ n E(S i (ξ i+1 + + ξ n ) ; Λ i ) = E[S i I Λi (ξ i+1 + + ξ n )] = E[S i I Λi ]E[(ξ i+1 + + ξ n )] = 0 (3.) Λ i S i a ES n E(S i ; Λ i ) a P (Λ i ) = a P (Λ)

3. 34 3.6 Λ n, n = 1,,... P (Λ k ) < k=1 P (lim Λ k ) = 0 3.7 ξ i, i = 1,,... Eξ i = 0, i = 1,,... V (ξ i ) < S n ξ 1 + + ξ n Λ k P (Λ k ) < Λ k { ω Ω ; k=1 sup S n S pn 1 } p n n k v i V (ξ i ), i = 1,,... Eξ i = 0 u < t E(S t S u ) = E(ξ u+1 + + ξ t ) = V (ξ u+1 + + ξ t ) t t t = V (ξ i ) = i=u+1 p n i=u+1 v i i=u v i i=p n v i < ( 1 k ) 3 v i < p n ( P (Λ k ) = lim P max S n S pn 1 ) l p n n l k = lim l P ( max S n S pn 1 ) n p n l p n k lim k E(S l S pn ) l l lim k = k l i=p n v i P (Λ k ) < k=1 i=p n v i < 1 k

3. 35 ( ) P Λ j = 0 ( P Λ { ) j = 1 k=1 j=k K = K(ω) k=1 j=k k K ω Λ { k n, m p n K(ω) S n (ω) S m (ω) S n (ω) S pn (ω) + S pn (ω) S m (ω) < 1 k + 1 k = k {S n (ω)}, n = 1,,... lim S n(ω) = ξ i (ω) S n 3.8 u i ( > 0) x i lim x i = 0 lim u i y n x i u i u n 1, v i u i u i 1, v 0 0 x i = = = i u i (y i y i 1 ) = v j (y i y i 1 ) j=1 v j (y i y i 1 ) = i=j p v j (y n y j 1 ) + j=1 v j (y n y j 1 ) j=1 j=1 j=p+1 v j (y n y j 1 ) (3.3) y n y n K, n = 1,,... (3.3) Ku p max y n y i p i n j=p+1 v j max y n y i (u n u p ) p i n max y n y i u n p i n

3. 36 y n ε > 0 sup y n y m < ε p<n<m p > 0 Ku p p < ε n 0 u i u n0 n n 0 1 Ku p u n x i < u n + max p i n y n y i < ε lim x i = 0 u n 1 3.9 ξ i, i = 1,,... Eξ i = 0, i = 1,,... ξ i 0 n Eξ i i < η i ξ i η i 3.7 i ξ i i ξ i 3.8 (ω) i ω n 1 ξ i (ω) 0 n 1 ξ i 0 3.9 3.10 ξ i, i = 1,,... i ξ i Eξ i a (< ), V (ξ i ) < a n η i ξ i a η i 3.9 n 1 η i 0 n 1 ξ i a 3.3 (ξ 1 + + ξ n )/n

3. 37 3.11 n ξ n,i, i = 1,..., n Eξ n,i = 0 V (ξ n,i ) = 1 (L) S n ξ n,i N(0, 1) (L) τ > 0 lim L n(τ) lim E(ξ n,i ; ξ n,i > τ) = 0 ϕ n,i, ϕ Sn ξ n,i, S n ξ n,i, i = 1,..., n n.3 ϕ Sn (t) = ϕ n,i (t) N(0, 1) ϕ.6 ϕ(t) = e t S n N(0, 1).15 t R 1 n ϕ n,i (t) e t (3.4) a R 1 e ia 1 a (3.5) e ia 1 ia a e ia 1 ia + a n t R 1 a 3 6 (3.6) (3.7) max i n 1 ϕ n,i(t) 0 (3.8) n Eξ n,i = 0 (3.6) ϕ n,i (t) 1 = E(e itξ n,i 1) = E(e itξ n,i 1 itξ n,i ) t Eξ n,i = t V (ξ n,i) τ > 0 i = 1,..., n V (ξ n,i ) = Eξ n,i = E(ξ n,i ; ξ n,i > τ) + E(ξ n,i ; ξ n,i τ) L n (τ) + τ i = 1,..., n ϕ n,i (t) 1 t (L n(τ) + τ ) max i n ϕ n,i(t) 1 t (L n(τ) + τ ) ε > 0 τ ε τ (L) n 0 n n 0 L n (τ) < ε

3. 38 n n 0 max i n ϕ n,i(t) 1 t ε n t R 1 (3.8) t R 1 lim ϕ Sn (t) exp e (ϕn,i(t) 1) 1 ϕ n,i (t) 1 { } n ϕ Sn (t) exp (ϕ n,i (t) 1) = ϕ n,i (t) { } (ϕ n,i (t) 1) = 0 (3.9) = n e (ϕ n,i(t) 1) ϕ n,i (t) e (ϕn,i(t) 1) (ϕ n,i (t) 1) + 1 e (ϕn,i(t) 1) C > 0 z 1 z C e z 1 1 (z 1) C z 1 { } ϕ Sn (t) exp (ϕ n,i (t) 1) C ϕ n,i (t) 1 (3.6) Eξ n,i = ϕ Sn (t) exp { C max i n ϕ n,i(t) 1 = C max i n ϕ n,i(t) 1 V (ξ n,i ) = 1 ϕ n,i (t) 1 E(e itξ n,i 1 itξ n,i ) (ϕ n,i (t) 1) } C max i n ϕ n,i(t) 1 t = t C max i n ϕ n,i(t) 1 Eξ n,i n (3.8) t R 1 (3.9) t R 1 { lim exp (ϕ n,i (t) 1) } = e t (3.10)

3. 39 ρ n (t) (ϕ n,i (t) 1) + t Eξ n,i = 0 V (ξ n,i ) = 1 ρ n (t) = E(e itξ n,i 1 itξ n,i + t ξ n,i ) ρ n (t) E(e itξ n,i 1 itξ n,i + t ξ n,i ; ξ n,i τ) + E(e itξ n,i 1 itξ n,i + t ξ n,i ; ξ n,i > τ) (3.7) (3.6) E( tξ n,i 3 6 ; ξ n,i τ) t 3 6 τ Eξ n,i = E(t ξ n,i ; ξ n,i > τ) = t L n (τ) t 3 6 τ ε > 0 τ = 3ε/ t 3 ε t R 1 n (L) t L n (τ) < ε lim ρ n(t) = 0 { e ρn(t) = exp (ϕ n,i (t) 1) (3.10) lim eρn(t) = 1 } e t t R 1 (3.9) (3.10) t R 1 (3.4) S n N(0, 1) ξ n,i ξ n,i (L) S n 3.1 ξ i, i = 1,,... V (ξ 1 ) v E ξ 1 ξ i Eξ i < N(0, 1) nv 3.1 3.11 3.1 ξ n,i ξ i Eξ i, S n ξ n,i ξ i, i = 1,,... nv n ξ n,i, i = 1,..., n Eξ n,i = 0

3. 40 V (ξ n,i ) = 1 (L) 3.1 3.11 3.4 n ξ n,i, i = 1,, n S n ξ n,i n ξ i, i = 1,, ξ n,i ξ i n, S n ξ n,i S n n δ {Eξ1 } S n 0 (infinitely divisible distribution) 3.14 3.18 3.19 3.13 µ R 1 ϕ µ n = 1,,... ϕ(t) = ( n ϕ(t) ) n, t R 1 (3.11) nϕ ϕ µ µ Y 1,..., Y n Y i µ

3. 41 3.14 n ξ n,i, i = 1,..., n S n ξ n,i n ϕ α R 1 ν ϕ { ϕ(t) = exp itα + ( e itx 1 itx ) 1 + x 1 + x x } ν(dx) α Eξ 1 ν ν = 0 α = 0 ν δ {0} 3.15 f R 1 f(0) = 1 [ T, T ] f 0 t [ T, T ] λ(0) = 0 f(t) = e λ(t) (3.1) λ(t) f (, ) 0 (3.1) (, ) z = 1 0 L { z ; z 1 1/ } L(z) j=1 ( 1) j 1 (z 1) j (3.13) j L L(1) = 0 e L(z) = z (3.14) L L L(1) = 0 1 z c c 0 1/z c dz z L(z) z dζ ζ c 1

3. 4 c L(z) = log z + iargz 1 ζ = ( (ζ 1)) k k=0 L(z) = = = z ( 1) k (ζ 1) k dζ k=0 j=1 1 k (z 1)k+1 ( 1) k + 1 k=0 j 1 (z 1)j ( 1) = L(z) j e L(z) = e L(z) e log z +iargz = e = z L (3.14), δ > 0 [ δ, δ] (3.1) λ f [ T, T ] ρ inf f(t) > 0 t, s [ T, T ] t T ( ρ t s δ f(t) f(s) min, 1 δ > 0 ) λ(t) L(f(t)), t < δ f(t) f(0) 1/ λ [ δ, δ] (3.1), [δ, δ] (3.1) λ g(s) f(s + δ)/f(δ) g(0) = 1 s δ g(s) g(0) = f(s + δ)/f(δ) 1 f(s + δ) f(δ) = f(δ) ( ρ min, 1 ) f(δ) ( ρ min, 1 ρ ) 1 λ(s) L(g(s)) g(s) = e λ(s) λ(s + δ) λ(δ) + λ(s), 0 s δ λ e λ(s+δ) = e λ(δ)+ λ(s) = e λ(δ) e λ(s) = f(δ)g(s) = f(s + δ) [δ, δ] (3.1)

3. 43 [ δ, kδ] λ 0 s δ [ δ, T ] λ g(s) f(s + kδ)/f(kδ) λ(s) L(g(s)) λ(s + kδ) λ(kδ) + λ(s) δ δ λ [ T, T ] (3.1) λ (3.1) λ ˆλ (3.1) exp( ˆλ(t) λ(t) ) = 1 m(t) ˆλ(t) λ(t) = πi m(t) ˆλ λ m(t) m(0) = ˆλ(0) λ(0) = 0 m(t) = 0 ˆλ = λ (3.1) λ f (, ) 0 (3.1) λ (, ) f (, ) 0 [ T, T ] (3.1) λ T T < T λ T (t) = λe T (t) t T (, ) λ λ(t) = λ T (t), t T 3.15 λ log f 3.16 R 1 f f k, k = 1,,... f(0) = f k (0) = 1 [ T, T ] 0 f k f [ T, T ] log f k log f [ T, T ] log f k f = log f k log f (3.15)

3. 44 [ T, T ] f k /f 0 (f k /f)(0) = 1 3.15 f k f = elog fk e log f = e log f k log f (3.15) log(f k /f) 0 [ T, T ] f k /f 1 [ T, T ] ε > 0 k k 0 sup (f k /f)(t) 1 < ε 1 t T k 0 k k 0 (3.13) sup L((f k /f)(t)) 0 sup t T t T < j=1 j=1 j=1 1 j 1 j (f k/f)(t) 1 j sup (f k /f)(t) 1 j t T 1 j εj ε j=1 ε j 1 = ε 1 ε ε log(f k /f) 0 [ T, T ] (3.15) log f k log f [ T, T ] Φ 3.17 (i) (ii) (iii) (iv) ϕ Φ ϕ Φ ϕ i Φ i = 1, ϕ 1 ϕ Φ ϕ Φ ϕ 0 ϕ k Φ ϕ ϕ ϕ Φ ( ) ϕ Φ ϕ = ( n ϕ) n ϕ = ( n ϕ) n nϕ ϕ ϕ Φ ( ) ϕ i Φ i = 1, ϕ i = ( n ϕ i ) n i = 1, ϕ 1 ϕ = ( n ϕ 1 n ϕ ) n nϕ 1 n ϕ.3 ϕ 1 ϕ Φ ( ) ϕ Φ ϕ = ( n ϕ) n nϕ f ϕ, n f n ϕ ϕ = ϕ ϕ, n ϕ = n ϕ n ϕ 3.17 ( )( ) f, n f

3. 45 f, n f f = ϕ = ( n ϕ) n = ( n f) n nf(t) = (f(t)) 1/n 0 f(t) 1 h(t) lim nf(t) = lim (f(t)) 1/n = { 1, f(t) 0 0, f(t) = 0 h f(0) = ϕ(0) = 1 f t = 0 0 h t = 0 1 nf h h t = 0.15 h h R 1 t R 1 h(t) = 1 ϕ(t) = f(t) 0 ϕ 0 ( ) ϕ Φ (3.11) ϕ n ϕ 0 ϕ(0) = n ϕ(0) = 1 3.15 n log n ϕ(t) = log ϕ(t) (3.11) nϕ ϕ 1/n ( 3.17 ( )) ϕ 0 f ϕ, f k ϕ k f, f k f k 1/n = ( ϕ k ) 1/n = ϕ k 1/n ϕ k 1/n f 1/n k k f 1/n k f 1/n f.15 f 1/n f = ( f 1/n ) n Φ 3.17 ( ) f 0 ϕ 0 ϕ log ϕ ϕ k Φ log ϕ k ϕ k ϕ ϕ.15 ϕ k ϕ 3.16 log ϕ k log ϕ 1 ) ( 1 ) ϕ 1/n k = exp( n log ϕ k exp n log ϕ 1 ) ( 1 ) exp( n log ϕ k exp n log ϕ ( 1 ).15 exp n log ϕ ( ( 1 )) n ϕ = e log ϕ = exp n log ϕ ϕ Φ

3. 46 3.18 3.14 (3.16) 3.18 ϕ α R 1 ν ϕ { ϕ(t) = exp itα + ( e itx 1 itx ) 1 + x 1 + x x ( ) : ψ(t) ψ(t) itα + } ν(dx) (3.16) ( e itx 1 itx ) 1 + x ν(dx) (3.17) 1 + x x e ψ(t) Φ e itx ( e itx 1 itx ) 1 + x = it 1 + x x x + (eitx 1) 1 + x x = it x + = k= k=1 (it) k k! (it) k (1 + x ) k! x k 1 + x x x k x 1 t e itx 1 itx 1 + x e itx 1 1 + x x ( x < 1 e itx 1 itx tx / e itx 1 itx e itx 1 itx 1 + x tx + itx t itx 1 + x e itx 1 + tx 1 + x ) ( + t ) + x 1 + x itx itx 1 + x e itx 1 itx 1 + x t (1 + x ) + tx t + t 1 + x x

3. 47 t ψ(t) itα ( e itx 1 itx ) 1 + x ν ε > 0 1 + x x M > 0 ( e itx 1 itx ) 1 + x ν(dx) < ε (3.18) 1 + x x { x >M} ( e itx 1 itx ) 1 + x 1 + x x { x M} ν(dx) l k = (x k, x k+1 ] = [ M, M] k=1 k ( f(x) e itx 1 itx ) 1 + x 1 + x x h l (x) x [ M, M] l 0 l k=1 f(x k )I k (x) ε > 0 l l 0 h l (x) f(x) < ε f l l 0 h l (x) ν(dx) f(x) ν(dx) < ε [ M,M] [ M,M] ε l 0 l l 0 l ( e itx k 1 itx ) k 1 + xk ν( 1 + x k x k ) k k=1 ( e itx 1 itx ) 1 + x ν(dx) < ε (3.19) 1 + x x { x M} (3.18) (3.19) l l 0 ( e itx 1 { x >M} itx ) 1 + x 1 + x x ( + ν(dx) l ( e itx k 1 itx k 1 + x k ν(dx) k=1 e itx 1 itx ) 1 + x 1 + x x ( e itx 1 { x M} l ( e itx k 1 k=1 itx ) 1 + x 1 + x x itx k 1 + x k ν(dx) ) 1 + xk ) 1 + xk x k ν( k ) x k ν( k ) < ε

3. 48 l t R 1 ψ(t) itα k=1 ( e itx k 1 itx k 1 + x k ) 1 + xk x k ν( k ) l ( e itx k 1 itx k 1 + x k k=1 ( l = it ) 1 + xk k=1 { ( l e ψ(t) itα exp it k=1 x k ν( k ) ν( k ) x k ) + ν( k ) x k ) + l {( 1 + xk ) } ν( x k ) (e itx k 1) k k=1 l {( 1 + xk ) }} ν( x k ) (e itx k 1) k t R 1 { ( l exp it = k=1 l ( { exp k=1 ν( k ) x k ) + k=1 l {( 1 + xk ) }} ν( x k ) (e itx k 1) k k=1 it ν( k) x k } exp {( 1 + xk ) }) ν( x k ) (e itx k 1) (3.0) k { exp it ν( } k).4 δ ν( x { k ) } k x k {( 1 + xk ) }.5 exp ν( x k ) (e itx k 1) Φ( 1+x k x k ν( k ), x k ) k 3.17 ( ) (3.0) Φ (3.17) lim ψ(t) = 0 t 0 e ψ(t) itα l ( { exp it ν( } {( k) 1 + xk ) } ) exp ν( x k x k ) (e itx k 1) Φ k k=1 e ψ(t) itα e ψ(t) itα.15 e ψ(t) itα 3.17 ( ) e ψ(t) itα Φ e itα e ψ(t) Φ ( ) : ϕ α R 1 ν log ϕ(t) = itα + ( e itx 1 itx ) 1 + x ν(dx) (3.1) 1 + x x ϕ n ϕ 1/n λ log ϕ t R 1 t < T T T > 0 lim sup n(ϕ n (t) 1) λ(t) = 0 (3.) t T

3. 49 n(ϕ n (t) 1) = n(e λ(t) n 1) C > 0 sup t T e λ(t) λ(t) n 1 n sup t T n C λ(t) n sup n(e λ(t) n 1) λ(t) sup t T t T C λ(t) n n lim sup n(e λ(t) n 1) λ(t) = 0 t T (3.) sup n sup n ϕ n (t) 1 < (3.3) t T µ n ϕ n ε > 0 A > 0 sup n nµ n ([ A, A] { ) < ε (3.4)..16 Â > 0 1/ Â µ n ([ 4Â, 4Â]{ ) Â ϕ n (t) dt 1/Â = Â = Â = Â = Â = Â 1/ Â 1/ Â 0 1/ Â 1/Â 1/ Â 1/Â 1/ Â 0 dt Â n nµ n ([ 4Â, 4Â]{ ) Â 0 e itx µ n (dx) dt (cos tx + i sin tx) dt µ n (dx) cos tx dt µ n (dx) 1/ Â Re (1 ϕ n (t)) dt 1/ Â 0 0 cos tx µ n (dx) dt Re {n(1 ϕ n (t))} dt

3. 50 n (3.) Re {n(1 ϕ n (t))} Re λ(t) [0, 1/Â] lim nµ n([ 4Â, 4Â]{ ) lim Â 1/ Â = Â 0 1/ Â 0 1/ Â Re {n(1 ϕ n (t))} dt Re λ(t) dt Â Â Re λ(t) dt Re λ(0) lim Â 0 lim nµ n([ 4Â, 4Â]{ ) Re λ(0) = 0 ε > 0 Â(ε ) > 0 Â Â(ε ) n 0 lim nµ n([ 4Â, 4Â]{ ) < ε n n 0 nµ n ([ 4Â, 4Â]{ ) < ε n < n 0 µ n Ã nµ n ([ Ã, Ã]{ ) < ε n = 1,,..., n 0 1 4Â Ã A ε > 0 A > 0 (3.4) ν n (dx) x /(1 + x ) nµ n (dx) sup ν n (R 1 ) < (3.5) n ( sin tx ) 1 + x L(t) t inf 1 x R 1 tx x sin tx x 1 1 1 + x tx x < 1 sin tx tx tx 3 4. x sin tx 1 3! tx sin tx 1 tx 1 + x t (1 + x ) x 6 t x 6 < t 3 t 0 0 < L(t) < t t n(1 Re ϕ n (s)) ds = (1 cos sx) nµ n (dx) ds 0 = t 0 ( 1 L(t)ν n (R 1 ) sin tx tx ) 1 + x x ν n (dx)

3. 51 (3.3) n (3.5) g lim j g(x) ν nj (dx) = g(x) ν(dx) (3.6) n n j ν (3.5) a n ν n (R 1 ) a {n} {n } ν n (3.4) n ν n ([ A, A] { ) = [ A,A]{ x 1 + x nµ n(dx) nµ n ([ A, A] { ) < ε ν n ([ A, A] { ) < ε ν n ν n (B) ν n (B) ν n (R 1 ), B B 1 ν n ([ A, A] { ) < ε ν n (R 1 ) ε inf n ν n (R 1 ) (i) inf ν n (R 1 ) 0 n ε > 0 (3.4) ε ε < ε inf n ν n (R 1 ) ν n ([ A, A] { ) inf n ε ν n (R 1 ) < ε.11 ν n ν n ν n ν ν n (R 1 ) a g g(x) ν n (dx) = = g(x)ν n (R 1 ) ν n (dx) g(x)(ν n (R 1 ) a) ν n (dx) + a g(x) ν n (dx) lim n g(x) ν n (dx) = a g(x) ν(dx) ν = a ν ν n = ν nj g (3.6) n j ν

3. 5 (ii) inf ν n (R 1 ) = 0 n ν n (R 1 ) 0 {n } {n } ν 0 n j ν x lim j R 1 + x n jµ 1 nj (dx) α t R 1 ( lim n j(ϕ nj (t) 1) = itα + e itx 1 itx ) 1 + x ν(dx) (3.7) j 1 + x x ϕ n (t) t R 1 lim n(ϕ n(t) 1) = lim (e itx 1) 1 + x ν x nj (dx) = = j = lim = lim (e itx 1) nµ n (dx) (e itx 1) 1 + x x ν n (dx) (e itx 1) 1 + x ν x nj (dx) (3.8) {( e itx 1 ( e itx 1 itx ) 1 + x } + itx 1 + x x x itx ) 1 + x ν 1 + x x nj (dx) + it ν n j (dx) x 1 + x n jµ nj (dx) ( e itx 1 itx ) 1 + x 1 + x x (3.6) (3.8) x it lim j 1 + x n jµ nj (dx) = lim j (e itx 1) 1 + x lim j = lim n(ϕ n (t) 1) ( x ν nj (dx) ( e itx 1 e itx 1 itx ) 1 + x 1 + x x itx ) 1 + x 1 + x x ν nj (dx) ν(dx) (3.) ν x α lim j 1 + x n jµ nj (dx) t R 1 (3.7) lim n(ϕ n(t) 1) = lim n j (ϕ nj (t) 1) (3.7) j t R 1 (3.) t R 1 (3.1)

3. 53 3.14.13.15 3.14 3.19 ϕ ϕ n n (ϕ n ) n ϕ ϕ δ > 0 ϕ(t) ϕ n (t) [ δ, δ] 0 ϕ n.3 (ϕ n ) n (ϕ n (t)) n ϕ n n (t) ϕ n n ϕ ϕ.15 ϕ n n ϕ δ > 0 n n 0 sup ϕ n n (t) ϕ(t) < 1 4 t δ (3.9) n 0 ϕ R 1 ϕ(0) = 1 δ (3.9) t δ ϕ(t) 1 < 1 4 (3.30) 1 ϕ(t) 1 ϕ(t) t δ ϕ(t) 3/4 inf ϕ(t) > 0 (3.31) t δ ϕ n n (t) 1 ϕ n n (t) ϕ(t) + ϕ(t) 1 t δ (3.9) (3.30) n n 0 sup ϕ n n (t) 1 1 4 t δ 1 ϕ n n (t) 1 ϕ n n (t) t δ ϕ n n (t) 1/ n n 0 inf ϕ n n (t) 1/ t δ inf ϕ n n (t) = ( inf ϕ n(t) ) n t δ t δ n n 0 inf ϕ n (t) t δ ( 1 ) 1 n inf ϕ n(t) > 0 (3.3) t δ (3.31) (3.3) ϕ(t) ϕ n (t) [ δ, δ] 0

3. 54 ϕ(0) = 1 ϕ n n (0) = 1 3.15 log ϕ(t) log ϕ n n (t) ϕ n n (t) ϕ(t) [ δ, δ] 3.16 log ϕ n n (t) log ϕ(t) [ δ, δ] log ϕ n n (t) = n log ϕ n (t) lim sup n log ϕ n (t) log ϕ(t) = 0 (3.33) t δ lim sup n(ϕ n (t) 1) n log ϕ n (t) = 0 (3.34) t δ C > 0 sup e log ϕn(t) 1 log ϕ n (t) sup C log ϕ n (t) t δ t δ n sup n(e log ϕn(t) 1) n log ϕ n (t) sup C n log ϕ n (t) t δ t δ = sup t δ C n log ϕ n (t) n (3.33) lim sup t δ n(e log ϕn(t) C n log ϕ n (t) 1) n log ϕ n (t) lim sup t δ n (3.34) (3.33) (3.34) n C log ϕ(t) = lim sup t δ n = 0 lim sup n(ϕ n (t) 1) log ϕ(t) = 0 (3.35) t δ µ n ϕ n ν n (dx) x /(1 + x ) nµ n (dx) t < δ t 3.18 ( ) g (3.6) n n j ν (3.35) 3.18 ( ) x lim j R 1 + x n jµ 1 nj (dx) α t R 1 ( lim n j(ϕ nj (t) 1) = itα + e itx 1 itx ) 1 + x ν(dx) (3.36) j 1 + x x

3. 55 ϕ R 1 0 ( c(t) itα + e itx 1 itx ) 1 + x ν(dx) + 1 1 + x x R 1 (3.36) ϕ nj (t) 1 < c(t) n j ϕ nj (t) > n j 1 c(t) n j ϕ(t) = lim j ϕ nj (t) n j ( lim 1 c(t) ) nj = e c(t) j n j ϕ R 1 0 (3.35) δ (3.36) t R 1 log ϕ(t) = lim n(ϕ n (t) 1) = lim n j (ϕ nj (t) 1) j ( = itα + e itx 1 itx ) 1 + x 1 + x x 3.18 ( ) ϕ Φ ν(dx) ( 3.14) 3.18 ξ n,i, i = 1,..., n n n ξ n,i, i = 1,..., n ϕ n.3 S n (ϕ n ) n ϕ ϕ (ϕ n ) n ϕ.15 (ϕ n ) n ϕ S n n 3.19 ϕ ϕ 3.14 ξ n,i

[1],,, 1976 [],,, 1963 [3],,, 1997 [4],,, 1973 [5],,, 1980 [6],,, 1978 56