2011 ( ) ( ) ( ),,.,,.,, ,.. (. ), 1. ( ). ( ) ( ). : obata/,.,. ( )

2011 () () (),,.,,.,,. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.,.. (. ), 1. ( ). ()(). : www.math.is.tohoku.ac.jp/ obata/,.,. () obata@math.is.tohoku.ac.jp http://www.dais.is.tohoku.ac.jp/ amf/, (! 22 10.6; 23 10.20; ) 10.13 () 11.10 ()

,, 1. : (), 1982... 2. W. Feller: An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, 1957.., (Vol. 2!).. W. () : (). 4. 3. B. V. Gnedenko: The Theory of Probability and the Elements of Statistics, AMS Chelsea Publishing Co., 6th ed. 1989. 4. R. Durrett: Probability: Theory and Examples, Duxbury Press, 1996. 2,.,. 5. (), 1994..,., (, ). 6.,, 2004. 7.,, 1978. 2. 8. (), 2000.,. 9. P. (, ) (),, 2004...,. 10. (), 2002. 11. (), 2007. 2. [1,2]. 2

1 1.1 (), (, ),.,.,., 1,.,.,.,,, 1/2. 1,,,,..,,,., 98%, 2%.,,..,...,.,. (,.),,..,,.,,.,,. 1.1: 1, (K. Pearson). 1

1.2: 1.3: 1.2 1.2.1 () A,B 2. A, B, 1, 1. 0, A + B. A,B.,? 1.2.2 () (1 ).,?. 1.2.3 () A,B 2. 1, 1. A,B 0.,, (), ()?. 2

1.2.4 (),.,?,. 1.2.5 (),. 1,? 1.3 ()., x, y, z, t,...., 0 x 1, x 0 1.,,.,,.., X, Y, Z, T,...., X, X {1, 2, 3, 4, 5, 6},. P (X = 1) = P (X = 2) = = P (X = 6) = 1 6,., 1 x. X,., X, X., (), (),, S, S-. S.,,. 1.4, t.,, t {X(t)},. 2. 1 {X n ; n = 1, 2,... }..,,.,, {X(t) ; 0 t T }., t,,,,,.. 1.4.1 (), n Z n., P (Z n = 1) = p, P (Z n = 0) = q. 3

, 0 p, q 1 p + q = 1.,, {Z 1, Z 2,... }., n P (Z 1 = ξ 1, Z 2 = ξ 2,..., Z n = ξ n ) = P (Z k = ξ k ) ξ 1, ξ 2,..., ξ n {0, 1}. (= ) p. 1.4.2 () ( )., t = 0 0,, (H) 1, (T) 1. 1.4,. 30 20 10 0 200 600 800 1000-10 -20-30 1.4: (p = q = 1/2) t = n, n. X n. X n, {Z n }., Z n ±1, P (Z n = 1) = p, P (Z n = 1) = q = 1 p,., n X n = Z k (1.1). {X n } 1. X n, P (X n = 2k n) =., X n. ( ) n p k q n k, k = 0, 1, 2,..., n, (1.2) k 4

2 2.1 ()., x, y, z, t,...., 0 x 1, x 0 1.,,.,,.., X, Y, Z, T,...., X, X {1, 2, 3, 4, 5, 6},. P (X = 1) = P (X = 2) = = P (X = 6) = 1 6,., 1 x. X,., X, X., (), (),, S, S-. S.,,. 2.2 2.2.1 X 1,. X {a 1, a 2,... }, X a i P (X = a i ) = p i X., p i = 1, p i 0 (2.1) i. p i = 0, X a i., p i = 0,. 1. ( N ),. 5

X, a 1, a 2,... p 1, p 2,.... X., a 1, a 2,... p 1, p 2,... (2.1), 1., µ = i p i δ ai, µ(dx) = i p i δ(x a i )dx,, R. 2.2.2, X., X, a P (X = a) 0.,,, P (a X b). P (a X b) = b a f(x)dx, f(x), f(x) X ()., + f(x)dx = 1, f(x) 0, (2.2)., X 0, P (a < X < b)., µ(dx) = f(x)dx (2.3) X., f(x) (2.2), X.,,. 2.2.1 R 1 A, O X = OA [0, R]. O X a b 2.1: 1 X. 0 < a < R P (X = a). X = a, A O, a C(a).,,., P (X = a) = C(a) πr 2 = 0 πr 2 = 0 6

., C(a), 1, C(a) 0., P (X = a), X., P (a X b). 0 a < b 1,., P (a X b) = πb2 πa 2 πr 2 = b2 a 2 R 2 P (a X b) = 1 R.,, b a 2xdx, 0 < a < b < R, 0, x 0, f(x) = 2x, 0 x 1, 0, x > 1,., x = 1 (). 2.2.3 2,.,. 2.2.2 X, F (x) = F X (x) = P (X x), x R, X () ((probability) distribution function, pdf). ( R.) 2.2.3 1, X. X, 0 x < 1 1/6 1 x < 2 F (x) =.. 5/6 5 x < 6 1 6 x 0 1 2 3 4 5 6 x 7

,, F (x) x. 2.2.1, 0, x < 0, F X (x) = x 2, 0 x 1, 1, x > 1., x < 0 x 1, 0 x 1,. F X (x) = P (X x) = πx2 π = x2 x 2.2: 2.2.1 2,, ( ). 2.2.4 F (x). (1) () x 1 x 2 F (x 1 ) F (x 2 ). (2) lim F (x) = 0, lim x F (x) = 1. x + (3) () lim ϵ 0 F (x + ϵ) = F (x). X F X (x) 2.2.4., (i) (iii)., 2.2.4 F (x) 1. 2.2.5. 2.2.6 F (x) (),., X. 2.2.7 F (x), F (x) (). f(x) = F (x) 8

. f(x) 0, + f(x)dx = 1., F (x) = x, X [a, b] f(t)dt P (a X b) = F (b) F (a) = b a f(t)dt, a < b. F (x)., P (X = a) = F (a) F (a 0),. 1/6 x x 2.3: ) 2.3, X µ(dx). X, µ X = n p n δ xn µ X (dx) = n p n δ(x x n )dx. x 1, x 2,... p 1, p 2,.... p p p p i x x x x i 9

X f(x),. µ(dx) = f(x)dx 2.3.1 X (mean value, expectation) E[X] = + xµ(dx). ( X,.) X, E[X] = i x i p i., X f(x), E[X] =., φ(x), E[φ(X)] = + + xf(x)dx φ(x)µ(dx)., m. E[X m ] = + x m µ(dx) 2.3.2 X (variance).,. V[X] = + V[X] = E[(X E[X]) 2 ] = E[X 2 ] E[X] 2 (x E[X]) 2 µ(dx) = + ( + ) 2 x 2 µ(dx) xµ(dx) 2.4 2.4.1 () p, 1, 0, (1 p)δ 0 + pδ 1. p (Bernoulli distribution). p, p(1 p). 10

2.4.2 () 0 p 1, n 1, n ( ) n p k (1 p) n k δ k k (binomial distribution), B(n, p). k=0 B(100, 0.4) p n, X, X B(n, p). B(n, p) np, np(1 p). 2.4.3 () 0 p 1, p(1 p) k 1 δ k p (geometric distribution). 2 p, (). p 1 p, 1 p 2. 2.4.4 () λ > 0, k=0 λ λk e k! δ k λ (Poisson distribution). λ λ, λ. 2.5 2.5.1 () a < b. 1 f(x) = b a, a x b, 0, 2,, k {0, 1, 2,... }, X p(1 p) k δ k k=0., p,.,. 11

[a, b] (uniform distribution). [a, b] a + b (b a)2,. 2 12 1 b a a b x 2.5.2 () λ > 0. λe λx, x 0, f(x) = 0, λ. λ λ 1, λ 2. λ x 2.5.3 () σ > 0, m R, { } 1 exp (x m)2 2πσ 2 2σ 2 (2.4), m, σ 2 (normal distribution) (Gaussian distribution), N(m, σ 2 )., N(0, 1). 12

,, (2.4).,, m = σ 2 = 1 2πσ 2 1 2πσ 2 + + (x m)2 x exp { 2σ 2 (x m) 2 exp {, m σ 2. } dx, } (x m)2 2σ 2 dx 1 1 1, X. (1) X,,,. (2) S = πx 2,. 2 L, 2., X, X,,,. 3 a, b, 0 p 1, q = 1 p, 2 pδ a + qδ b. 0, 1 2. 4 () {0, 1, 2,... } µ = µ. G(z) = (1) µ m σ 2. (2). p k z k k=0 m = G (1), σ 2 = G (1) + G (1) {G (1)} 2 (3). 5 (1) t, +., n = 0, 1, 2,.... 0 e tx2 dx = + 0 π 2 t, t > 0 x 2n e x2 dx p k δ k, (2) f(x) = ax 2 e x2 a,. k=0 13

3 3.1. 2, 1,0. n Z n, P (Z n = 1) = p, P (Z n = 0) = q = 1 p, 0 < p < 1, (3.1)., p., p n., Z n p. 2, P (Z 1 = 1, Z 2 = 1) = p 2 = P (Z 1 = 1)P (Z 2 = 1), P (Z 1 = 1, Z 2 = 0) = pq = P (Z 1 = 1)P (Z 2 = 0), P (Z 1 = 0, Z 2 = 1) = qp = P (Z 1 = 0)P (Z 2 = 1), P (Z 1 = 0, Z 2 = 0) = q 2 = P (Z 1 = 0)P (Z 2 = 0),., n P (Z 1 = ξ 1, Z 2 = ξ 2,..., Z n = ξ n ) = P (Z k = ξ k ) (3.2)., p Z 1, Z 2,..., (3.2).,.,, Z 1, Z 2,....,, Z n, (3.1). p. 2,,. 3.2,. 3.2.1 () A 1, A 2,... (pairwise independent), 2 A i1, A i2 (i 1 i 2 ) P (A i1 A i2 ) = P (A i1 )P (A i2 ). 14

3.2.2 () A 1, A 2,... (independent), A i1,..., A in (i 1 < i 2 < < i n ) P (A i1 A i2 A in ) = P (A i1 )P (A i2 ) P (A in ). 3.2.3 52 1, A B. 3.2.4 6 1, 2, 3, 4, 5, 6,., 1 2 A, 3 4 B, 5 6 C. 3.2.5 112, 121, 211, 222 4. 1, 100 1 A 1, 10 1 A 2, 1 1 A 3., A 1, A 2, A 3. 3.2.6 A 2 A, A,, P (A A) = P (A)P (A)., P (A) = 0 P (A) = 1.,, Ω. (P (A) = 0 A =, P (A) = 1 A = Ω.) 3.2.7 A, A # A A c.. (1) 2 A B, A # B #. (2) A 1, A 2,..., A # 1, A# 2,.... (1). (2). 3.3 3.3.1 () X 1, X 2,..., X i1,..., X in (i 1 < i 2 < < i n ) a 1,..., a n P (X i1 a 1, X i2 a 2,..., X in a n ) = P (X i1 a 1 )P (X i2 a 2 ) P (X in a n ) (3.3)., X 1, X 2,..., a 1, a 2,... A 1 = {X 1 a 1 }, A 2 = {X 2 a 2 },..., A n = {X n a n },... ( 3.2.2). 15

X 1, X 2,..., (3.3), P (X i1 = a 1, X i2 = a 2,..., X in = a n ) = P (X i1 = a 1 )P (X i2 = a 2 ) P (X in = a n ). 3.3.2 () X 1, X 2,... 3.3.1. 3.3.3 Ω = {(x, y) ; a x b, c y d} 1. x X, y Y, X, Y. 3.4 n X 1, X 2,..., X n X = (X 1, X 2,..., X n ) n ()., F X (x 1, x 2,..., x n ) = P (X 1 x 1, X 2 x 2,..., X n x n ) n F X X = (X 1,..., X n )., X i F Xi (x) = P (X i x) X i. 1,., lim x 2 + x n + F X (x 1, x 2,..., x n ) = F X1 (x 1 ). X i,, X. n X = (X 1, X 2,..., X n ), X 1,..., X n, F X,.,, F X (x 1,..., x n ) = P (X 1 = x 1, X 2 = x 2,..., X n = x n ) x1 xn f X (s 1,..., s n )ds 1 ds n, n f X X = (X 1,..., X n )., 1 R,.,. + + f X (x 1, x 2,..., x n )dx 2... dx n = f X1 (x 1 ) 3.4.1 n (X 1,..., X n ), F (x 1,..., x n ), X i F i (x i )., X 1,..., X n,. F (x 1,..., x n ) = n F i (x i ) i=1 16

3.4.2 n (X 1,..., X n ) f(x 1,..., x n ), X i f i (x i )., X 1,..., X n, f(x 1,..., x n ) =. n f i (x i ), n (X 1,..., X n )., X 1,..., X n, (X 1,..., X n ) ( 3.4.1), X 1,..., X n. i=1 3.5 X. m X = E(X) = + xµ X (dx) 3.5.1 () X, Y a, b, E(aX + by ) = ae(x) + be(y ). 3.5.2 () X 1, X 2,..., X n, E[X 1 X 2 X n ] = E[X 1 ] E[X n ]. (3.4), X k = 1 Ak (). X 1,..., X n A 1,..., A n., E[X 1 X n ] = E[1 A1 A n ] = P (A 1 A n ) = P (A 1 ) P (A n ) = E[X 1 ] E[X n ]., (3.4).,, X k ( ). X k,,. X. X µ(dx),. V(X) = σ 2 X = V(X) = E[(X m X ) 2 ] = E[X 2 ] E[X] 2 + (x m X ) 2 µ(dx) = + ( + ) 2 x 2 µ(dx) xµ(dx) 3.5.3 X, Y. σ XY = E[(X E(X))(Y E(Y ))] = E[XY ] E[X]E[Y ] X, Y. (, σ XX = σx 2 X.) σ XY = 0 X Y., σ XY > 0, σ XY < 0, X Y. 17

3.5.4 X, Y,. X, Y, E[XY ] = E[X]E[Y ], σ XY = E[XY ] E[X]E[Y ] = 0. 3.5.5 X P (X = 1) = P (X = 1) = 1 4, P (X = 0) = 1 2, Y = X 2., X, Y, σ XY 3.5.4. = 0., 3.5.6 () X 1, X 2,..., X n, 2, [ n ] n V X k = V[X k ]. 3.5.7 X, Y σ X > 0, σ Y > 0, X, Y. ρ XY = σ XY σ X σ Y 3.5.8 X, Y, 1 ρ XY 1., σ X > 0, σ Y > 0. X, Y, X = X E[X], Ȳ = Y E[Y ],. E[ X 2 ] = V[X] = σ 2 X, E[Ȳ 2 ] = V[Y ] = σ 2 Y, (3.5)., t R (t X + Ȳ )2 0, 0 E[(t X + Ȳ )2 ] = t 2 E[ X 2 ] + 2tE[ XȲ ] + E[Ȳ 2 ]., 2 E[ XȲ ]2 E[ X 2 ]E[Ȳ 2 ] 0. (3.5), E[ XȲ ]2 E[ X 2 ]E[Ȳ 2 ] = σxσ 2 Y 2., ρ 2 XY = σ2 XY σ 2 X σ2 Y = E[(X E(X))(Y E(Y ))]2 σ 2 X σ2 Y = E[ XȲ ]2 σ 2 X σ2 Y 1. 18

, 1 ρ XY 1. 2 X, Y 1 1., Y = ax + b, a > 0, a < 0, ρ XY = 1 ρ XY = 1. 3.5.9 X E[X] V[X]. n X = (X 1, X 2,..., X n ),, m X = (E[X 1 ], E[X 2 ],..., E[X n ]) σ X1 X 1 σ X1 X 2 σ X1 X n σ X2 X Σ X = 1 σ X2 X 2 σ X2 X n...... σ Xn X 1 σ Xn X 2 σ Xn X n. σ XiX i = σx 2 i = V[X i ] X i., X 1, X 2,..., X n. 6 ρ 1. (r, θ) (0 r ρ, 0 θ < 2π)., R, Θ, R, Θ. 7 2 L, S. (, L = S.), (1) σ LS r LS. (2) L, S. 8 a > 0, b > 0. Ω = {(x, y) ; 0 bx + ay ab, x 0, y 0},. x X, y Y, (1) (X, Y ) X, Y, X, Y. (2) X, Y. 19

4 4.1. n X n, 1, X n = 0, (4.1)., S n =, n., S n n = 1 n n n, n..,.,, n S n /n 1/2.. X k X k 4.1:,,, S n lim n n = 1 (4.2) 2., S n /n, 1, 0,, {0, 1/n, 2/n,..., 1}. n 20

, 1/2., ω = (ω 1, ω 2,... ) S 1 (ω), S 2(ω), S 3(ω),..., S n(ω) 2 3 n,...., ω = (1, 1, 1,... ), S n /n 1, ω = (0, 0, 0,... ), S n /n 0., 0 t 1, S n /n t ω,, S n /n ω., ω (4.2).,, (4.2). 4.2 4.2.1 () X 1, X 2,..., m, σ 2 (, )., X 1, X 2,..., ϵ > 0,., 1 n lim P n ( 1 n ) n X k m ϵ = 0 n X k m. 4.2.2, X 1, X 2,...,. 4.2.3 () X m, σ 2, ϵ > 0,., P ( X m ϵ) σ2 ϵ 2 m = E[X] = X(ω) P (dω), Ω σ 2 = E[(X m) 2 ] = (X(ω) m) 2 P (dω)., 2. σ 2 = (X(ω) m) 2 P (dω) Ω = (X(ω) m) 2 P (dω) +,. X m ϵ X m ϵ X m ϵ (X(ω) m) 2 P (dω) ϵ 2 P (dω) = ϵ 2 P ( X m ϵ). Ω X m <ϵ (X(ω) m) 2 P (dω) 21

(),., Y = Y n = 1 n E[Y ] = 1 n n X k n E[X k ] = m.. E[X k X l ] = E[X k ]E[X l ] (k l),, E[Y 2 ] = 1 n 2,,. n k,l=1 E[X k X l ] = 1 n n 2 E[X 2 k] + E[X k X l ] k l = 1 n ( V[Xk n 2 ] + E[X k ] 2) + E[X k ]E[X l ] = 1 n 2 { nσ 2 + nm 2 + (n 2 n)m 2} = σ2 n + m2. k l V[Y ] = E[Y 2 ] E[Y ] 2 = σ2 n. P ( Y m ϵ) V[Y ] ϵ 2 = σ2 nϵ 2. lim P ( Y n m ϵ) = 0. n 4.2.4. 1/2., 1, n, X n = 0, n,. X 1, X 2,...,, m = 1 2, σ2 = 1 4., ϵ > 0, ( ) lim P 1 n X k 1 n n 2 ϵ = 0., 1 n n X k n,. 1/2 ϵ n 0., 1/2. 22

4.3, ϵ > 0, ϵ 0.. 4.3.1 () X 1, X 2,..., m (, )., X 1, X 2,..., ( ) 1 n P lim X k = m = 1 n n.,. 1 lim n n n X k = m 4.3.2,.,,., N. Etemadi (1981).,., Durrett. a.s. 4.4,. B(n, p) m = np, σ 2 = np(1 p), B(n, p) N(np, np(1 p)) (4.3)., n. 4.2: B(100, 0.4) (4.3).,., B(n, p) S, P (S x) 1 2πσ 2 x e (t m)2 /2σ 2 dt, m = np, σ 2 = np(1 p) 23

.,. P (S x) 1 (x m)/σ e t2 /2 dt 2π ( S m P (S x) = P σ, (x m)/σ x, ( ) S np P x np(1 p) x m ) σ 1 2π x e t2 /2 dt (4.4). N(0, 1), n. (4.3) n, n (4.4)., (). 4.4.1 ( ) 0 < p < 1. B(n, p) S n,. lim P n ( ) S n np x np(1 p) = 1 2π x e t2 /2 dt (4.5), B(n, p) n, N(np, np(1 p)). 4.5, 4.4.1. p Z 1, Z 2,..., n S n = B(n, p)., Z k Z k = Z k Z k p p(1 p), Z 1, Z 2,... 0, 1., S n np = 1 np(1 p) n., (4.5), ( lim P n 1 n n n Z k x Z k p = 1 p(1 p) n ) = 1 2π x n Z k e t2 /2 dt.,. 4.5.1 () X 1, X 2,..., 0, 1., x R, ( n ) lim P 1 X k x = 1 x n n 2π., 1 n n e t2 /2 dt X k n N(0, 1). 24

,. 4.5.2 X, φ(z) = E[e izx ], z R φ(z). X µ(dx), φ(z) = + e izx µ(dx), z R. µ(dx)..,.,. 4.5.3 () 1 µ 1, µ 2,..., µ φ 1, φ 2,..., φ. z C lim n φ n (z) = φ(z), µ n µ. µ 1, µ 2,..., µ, F x,. lim F n(x) = F (x) n,., F 1, F 2,..., F 4.5.4 x R, ) { } eix (1 + ix + (ix)2 x 3 min 2! 6, x 2. () 1 n n k=0 X k φ n (z) = E [ exp { iz n n k=0 X k } ] (4.6)., N(0, 1) e z2 /2 (??).,, z R, lim φ n(z) = e z2 /2 n (4.7). X 1, X 2,... φ(z) = E[e izx 1 ].,, (4.6) φ n (z) = n [ { }] iz ( z ) n E exp X k = φ n (4.8) n 25

., e i n z X 1 z = 1 + i X 1 z2 n 2n X2 1 + R n (z), E[X 1 ] = 0, V[X 1 ] = 1,., (4.8),., ( z ) φ n = E [ e i n z X 1 ] z 2 = 1 2n + E[R n(z)] φ n (z) =, ( 34),, (4.7). ) n (1 z2 2n + E[R n(z)] (4.9) lim ne[r n(z)] = 0 (4.10) n ) n lim φ n(z) = lim (1 z2 n n 2n + E[R n(z)] = e z2 /2, (4.10). 4.5.4, { 1 R n (z) min z 3 } 6 X 1, z 2 n X 1. n n,, ne[r n (z)] E[n R n (z) ] z 2 E { } z min 6 n X 1 3, X 1 2 X 1 2 [ { }] z min 6 n X 1 3, X 1 2. (4.11), E[ X 1 2 ] <,, [ { }] [ { }] z z lim E min n 6 n X 1 3, X 1 2 = E lim min n 6 n X 1 3, X 1 2, (4.11), lim ne[r n(z)] = 0. n, (4.10)., E[ X 1 3 ] <. E[ X 1 3 ] <,, (4.10). = 0. 9 p {Z k }, { Z k }., S n np = 1 np(1 p) n., n, - ( ). n Z k 26

10 (1) x R, n e ix (ix) k k! k=0 = in+1 n! x 0 (x t) n e it dt. (2) x R, eix. n (ix) k k! x n+1 (n + 1)! k=0 (3) (1), x R, n (ix) k eix k! 2 x n n!. k=0 ( 4.5.4 ().) 11 f(x) [0, 1]. x 1, x 2,... [0, 1], 1 n 1 0 n f(x k ) f(x)dx ().,. 27

5.,.,, 1.,.. 5.1 1 ()., 1, p, q (p > 0, q > 0, p + q = 1)., p, q. q p 3 2 1 0 1 2 3 n X n. 0., X 0 = 0. n X n { n, n + 2,..., n 2, n}.,. {Z n } (iid) P (Z n = 1) = p, P (Z n = 1) = q. ( {0, 1} )., n, X n = Z 1 + Z 2 + + Z n, X 0 = 0,. {X n } 1. 5.1.1 X n { n, n + 2,..., n 2, n},, ( ) n P (X n = n 2k) = p n k q k, k = 0, 1, 2,..., n. k k = 0, 1, 2,..., n. X n = Z 1 + Z 2 + + Z n = n 2k = (n k) k 28

, Z i = 1 i k, Z i = 1 i n k. Z 1, Z 2,..., Z n p n k q k., ( ) n P (X n = n 2k) = p n k q k k. 5.1.2 E[X n ] = (p q)n, V[X n ] = 4pqn., E[Z k ] = p q, V[Z k ] = 4pq ().,, n E[X n ] = E[Z k ] = (p q)n., {Z n },, n V[X n ] = V[Z k ] = 4pqn. n X n. (p q)n, n, X n,., n, X n ( ). x (p q)n 0 n 5.2.,.,., p, q, n {X n }.., 2n () p 2n, ( ) 2n p 2n = P (X 2n = 0) = p n q n = (2n)! n n!n! pn q n, n = 1, 2,..., (5.1) 29

., p 0 = 1. R ( ) R = P {X 2n = 0} n=1.,, R p 2n., 2n 0 q 2n = P (X 2 0, X 4 0,..., X 2n 2 0, X 2n = 0) n = 1, 2,...,. p 2n., T = inf{n 1 ; X n = 0} (5.2), q 2n = P (T = 2n) (5.3). 5.2.1 (5.2) inf{n 1 ; X n = 0},, X n = 0 n., inf min. {n 1 ; X n = 0},, inf{n 1 ; X n = 0} = +., T (), +., R = q 2n (5.4) n=1.., q 2n, R ()., {p 2n } {q 2n } n p 2n = q 2k p 2n 2k, n = 1, 2,..., (5.5)., 2n,. 2 {p 2n } {q 2n } g(z) = p 2n z 2n, h(z) = q 2n z 2n (5.6) n=0 n=1. z 1. (5.5) z 2n, n, n g(z) 1 = q 2k z 2k p 2n 2k z 2n 2k = n=1 n=0 = h(z)g(z). q 2k z 2k p 2n z 2n 30

, n=0 h(z) = 1 1 g(z). (5.7), g(z) ( 12)., ( ) 2n g(z) = p 2n z 2n = p n q n z 2n 1 = n 1 4pqz 2, (5.7), z 1 0, R = h(1) = n=0 h(z) = 1 1 g(z) = 1 1 4pqz 2. (5.8) q 2n = 1 1 4pq = 1 p q n=1 ( 13)., 5.2.2 1 R,. R = 1 p q,., R = 1, R < 1,.,,. 5.2.3 1, p = q = 1/2 ( ). p q.,. 5.2.4 1. T. E(T ), E(T ) =, p = q = 1/2, (5.6) (5.8), h(z) = q 2n z 2n = 1 1 z 2.,, z 1 0, E(T ) = h (z) = 2nq 2n = n=1,. n=1 n=1 2nq 2n. (5.9) n=1 2nq 2n z 2n 1 z =. 1 z 2 lim z 1 0 h (z) = lim z 1 0 z 1 z 2 = +. 31

5.2.5,. 12 α. (1 + x) α = n=0 ( ) α x n n,. n=0 ( ) 2n z n 1 =, z < 1 n 1 4z 4, 13 a n 0, f(x) = a n x n. 1,, =. n=0 lim f(x) = x 1 0 n=0 a n 5.3,. ( 19.),.,, (?).,,.,. n 1, (ϵ 1, ϵ 2,..., ϵ n ) ±1 n., { 1, 1} n., ϵ 1 0 ϵ 1 + ϵ 2 0 ϵ 1 + ϵ 2 + + ϵ n 1 0 ϵ 1 + ϵ 2 + + ϵ n 1 + ϵ n = 0, n.,. 5.3.1 2n C n., C 0 = 1., C 0 = C 1 = 1, C 2 = 2, C 3 = 5, C 4 = 14, C 5 = 42, C 6 = 132,.... 32

,. n n, (0, 0) xy-., ±1 (ϵ 1, ϵ 2,..., ϵ n ), ϵ k = +1 u k = (1, 0) ϵ k = 1 u k = (0, 1), (0, 0), u 1, u 1 + u 2,..., u 1 + u 2 + + u n 1, u 1 + u 2 + + u n 1 + u n., ϵ 1 + ϵ 2 + + ϵ n 1 + ϵ n = 0, u 1 + u 2 + + u n 1 + u n = (n, n),, (0, 0) (n, n) 1. 5.3.2 (0, 0) (n, n) y = x 1 1. 5.3.3 () C n = (2n)!, n = 0, 1, 2,..., (n + 1)!n! n = 0, 0! = 1. n 1,, ( ) ( ) 2n 2n C n = = (2n)! n n + 1 n!(n + 1)!. 5.3.4 C n, f(z) = n=0 C n z n = 1 1 4z 2z (5.10). 33

14.,. xy-. (0, 0), (1, ϵ 1 ), (2, ϵ 1 + ϵ 2 ),..., (n 1, ϵ 1 + ϵ 2 + + ϵ n 1 ), (n, ϵ 1 + ϵ 2 + + ϵ n 1 + ϵ n ), 0 0 2n 0, y 0., 5.3.5 n 1. 0 0 2n 0, y 0 C n. 5.3.6 p, q 1 {X n }, q 2n = P (X 2 0, X 4 0,..., X 2n 2 0, X 2n = 0) = 2C n 1 (pq) n, n = 1, 2,....., 5.2.2. 5.3.7 1 R, R = 1 p q. 5.3.6, R = q 2n = 2C n 1 (pq) n n=1 n=1,, R = 2pqf(pq) = 2pq 1 1 4pq 2pq = 1 1 4pq = 1 p q,. 14 C n. n (1) C n = C k 1 C n k. (2) (1), f(z) = (3) f(z). (4) (3) C n. C n z n f(z) 1 = z{f(z)} 2. n=0 15 m m + n, (0, 0) (m, m), (0, 0) (m + n, m),. (m + n)! (m + 1)!n! 34

16 {X n } p, q = 1 p, X 0 = 0., n = 1, 2,..., P (X 1 0, X 2 0,..., X 2n 1 0) n 1 = P (X 1 0, X 2 0,..., X 2n 0) = 1 q C k (pq) k., C k.,,. k=0 1 q P ( n 1 X n 0 ) = p, p > q, 0, p q,, 1. : (), 1982.. 2. W. Feller: An Introduction to Probability Theory and Its Applications, Vol. 1, Vol. 2, Wiley, 1957..,,,.. W. () : (). 4. 3. B. V. Gnedenko: The Theory of Probability and the Elements of Statistics, AMS Chelsea Publishing Co., 6th ed. 1989... B. V. (): I, II,, 1971, 1972. 4. R. Durrett: Probability: Theory and Examples, Duxbury Press, 1996.. 5. (), 2000.. 6. R. B. ):,, 2001.. 7. Z. Brzeźniak and T. Zastawniak: Basic Stochastic Processes, Springer, 1999.. 8. F. Spitzer: Pronciples of Random Walk, Springer, 2nd Ed., 1976.. 9. K. L. Chung: Markov Chains, Springer, 1960.. 35

5.4 {X n }., {Z n } P (Z n = +1) = P (Z n = 1) = 1 2, n X 0 = 0, X n = Z k.,, X 0, X 1, X 2,..., X 10000., 2., ±1 5000. (X 10000, ±1 5000 ± 50.),,? X 0, X 1,..., X 2n. X i 0 X i+1 0, [i, i + 1], X i 0 X i+1 0, [i, i + 1]. [0, 2n], 2k W (2k 2n), k = 0, 1,..., n,. 5.4.1 n = 1, 2,..., p 2n P (X 2n = 0) = ( 2n n )( 1 2 ) 2n.. 5.4.2 n = 1, 2,..., f 2n P (X 1 > 0, X 2 > 0,..., X 2n 1 > 0, X 2n = 0) = C n 1 ( 1 2 ) 2n = 1 4n p 2n 2.. 5.4.3 n = 1, 2,..., P (X 1 0, X 2 0,..., X 2n 1 0, X 2n 0) = p 2n. 36

., p 2n 2 p 2n = 2f 2n = 1 2n p 2n 2 5.4.4 n = 1, 2,..., P (X 1 0, X 2 0,..., X 2n 1 0, X 2n 0) = p 2n.., p 2n 2 p 2n = 2f 2n = 1 2n p 2n 2 5.4.5 n = 1, 2,...,, W (2k 2n) = p 2k p 2n 2k = W (2k 2n) = k r=1 ( 2k k )( 2n 2k n k )( 1 2 ) 2n, k = 0, 1,..., n. n k f 2r W (2k 2r 2n 2r) + f 2r W (2k 2n 2r)., k, n ( 17). n. 0 < a < b < 1, P (a < < b) = = = bn k=an r=1 W (2k 2n) n χ [an,bn] (k)w (2k 2n) k=0 n ( k χ [a,b] n k=0 )( 2k k )( 2n 2k n k χ I (x) I, x I 1, 0.,, n! ( ) n n 2πn n, e ( 2k k )( 1 2 ) 2k 1 πk P (a < < b) = k=0 k=0 1 )( 1 2 n ( ) k 1 χ [a,b] n π k(n k) n ( ) k 1 χ [a,b] n k π n (1 k n ) 0 dx χ [a,b] (x) π x(1 x). 1 n ) 2n. 37

5.4.6 dx π x(1 x). 0 < x < 1, (arcsine law). F (x) = x 0 dt π t(1 t) = 2 π arcsin x = 1 2 + 1 π arcsin(2x 1)., F (0.9) = 2 π arcsin 0.9 0.795,, 90% 1 F (0.9) 0.205! 17 5.4.5. 18,.,,. ( 2n n )( 1 2 ) 2n 1 πn 5.5 n = 0 0 Z., A B (A 1, B 1),, (, ).. 0 < p < 1 Z 1, Z 2,..., X 0, X 1, X 2,.... X n 1 + Z n, A < X n 1 < B, X 0 = 0, X n = A, X n 1 = A, (5.11) B, X n 1 = B. {X n }. 38

B 0 A., ( ) R = P ( n X n = A) = P {X n = A}, n=1 ( ) S = P ( n X n = B) = P {X n = B}.. R, S. X 0 = 0., A k B k, X 0 = k X n (k). X n = X n (0). X n (k) A, B R k, S k. R = R 0, S = S 0. n=1 5.5.1 {R k ;, A k B}. R k = pr k+1 + qr k 1, R A = 1, R B = 0. (5.12), {S k ;, A k B}. S k = ps k+1 + qs k 1, S A = 0, S B = 1. (5.13) 5.5.2 A 1, B 1. {X n } A B (5.11)., (q/p) A (q/p) A+B P ( n X n = A) = 1 (q/p) A+B, p q, B A + B, p = q = 1 2, 1 (q/p) A, p q, P ( n X n = B) = 1 (q/p) A+B A A + B, p = q = 1 2,., 1., (the gambler s ruin problem). A,B 2. A, B, 1., 1 39

, 1., 0, A + B. A,B. n A A + X n, X n, {X n }. 5.5.2, A,B P (A), P (B), P (A) = A A + B, P (B) = B A + B (5.14),., A = 1, B = 100 P (A) = 1/101, P (B) = 100/101, A., 5.2.2., 0.,,. (5.14)..,. 5.5.3 {X n } 5.5.2.. A q p A + B 1 (q/p) A, p q, q p 1 (q/p) A+B AB, p = q = 1 2., 0 k ( A k B), Y k. Y k = min{j 0 ; X (k) j E(Y 0 ).,. A < k < B, = A X (k) j = B }. E(Y A ) = E(Y B ) = 0 (5.15) E(Y k ) = 2. 5.5.2, (5.16), jp (Y k = j) (5.16) j=1 P (Y k = j) = pp (Y k+1 = j 1) + qp (Y k 1 = j 1) E(Y k ) = p jp (Y k+1 = j 1) + q jp (Y k 1 = j 1) j=1 j=1 = pe(y k+1 ) + qe(y k 1 ) + 1. (5.17), E(Y k ) (5.17) (5.15)., A + k E(Y k ) = q p A + B 1 (q/p) A+k, p q, q p 1 (q/p) A+B (A + k)(b k), p = q = 1 2. 40

. k = 0,., p = q = 1/2, A = 1, B = 100 AB = 100. A (, A 1/101 ), 100,.,,. 5.5.4., A 1, B 1, A B.,,.. 0 < p < 1 Z 1, Z 2,..., X 0, X 1, X 2,.... X n 1 + Z n, A < X n 1 < B, X 0 = 0, X n = A + 1, X n 1 = A, (5.18) B 1, X n 1 = B. 19 t = 0 0, {0, 1, 2,... }., 0. t = 2n P n. 41

6,,.. (),.,., ().,,. 1. (),.. (?) 6.1 2 A, B, P (B) > 0 P (A B) = P (A B) P (B), B A., B A. 6.1.1 2 A, B 5. (i) A B. (ii) P (B A) = P (B). (ii ) P (A B) = P (A). (iii) P (B A) = P (B A c ). (iii ) P (A B) = P (A B c ).,,. (). A B, (i) (ii) (iii). (i) (ii) (ii) = (iii)., P (B A c ) = P (Ac B) P (A c ) P (A B) = P (A)P (B), = P (B) P (A)P (B) 1 P (A) = = P (B) P (A B) 1 P (A) P (B)(1 P (A)) 1 P (A). = P (B). 42

, P (B A c ) = P (B) = P (B A). (iii) = (ii)., P (B A) =, P (A B) P (A), P (B A c ) = P (Ac B) P (A c ) P (A B) P (A) = = P (B) P (A B) 1 P (A), P (A B) = P (A)P (B), A, B. P (B) P (A B) 1 P (A) P (A B) = P (A), B A.,. 6.1.2 () A 1, A 2,..., A n, P (A 1 A 2 A n ) = P (A 1 )P (A 2 A 1 )P (A 3 A 1 A 2 )... P (A n A 1 A 2 A n 1 ). (6.1). (6.1)., 2 A, B., P (A B).. P (A) P (A c ) A A c P (B A) P (B c A) P (B A c ) A B A B c A c B P (B c A c ) A c B c 6.1:. ( A B), A B., ( 1, 2, ). 20 () A 1,..., A n Ω, n Ω = A k, A j A k = (j k)., B.. n P (B) = P (A k )P (B A k ) 43

6.2 () S {X n ; n = 0, 1, 2,... }. S R, S = {1, 2,..., N} S = {0, 1, 2,... }. S., S. 6.2.1 S {X n ; n = 0, 1, 2,... }, 0 i 1 < i 2 < < i k < n a 1, a 2,..., a k, a S, P (X n = a X i1 = a 1, X i2 = a 2,..., X ik = a k ) = P (X n = a X ik = a k ), {X n } S.. 6.2.2 S {X n }, P (X n+1 = j X n = i) n (i j )., n ()., p ij = p(i, j) = P (X n+1 = j X n = i),., P = [p ij ]. (., S.), i S p(i, j) = P (X n+1 = j X n = i) = 1. j S j S,. 6.2.3 S P = [p ij ], 2. (i) p ij 0. (ii) j S p ij = 1. 6.2.4 ( 6.2.3 ).,.., p(i, j) > 0 i j. 6.2.5 (2 ) 2 {0, 1}, p(0, 1) = p, p(0, 0) = 1 p, p(1, 0) = q, p(1, 1) = 1 q 44

., [ 1 p p q 1 q.. p = p ] p = 1p p = 1q p = q 6.2.6 (Z 1 ), p, j = i + 1, p(i, j) = q = 1 p, j = i 1, 0,., :............... q 0 p 0 0 q 0 p 0 0 q 0 p 0. 0 q 0 p.............. 6.2.7 () A > 0, B > 0. A B S = { A, A + 1,..., B 1, B}.. A < i < B, p, j = i + 1, p(i, j) = q = 1 p, j = i 1, 0,. i = A i = B, 1, j = A, 1, j = B, p( A, j) = p(b, j) = 0,, 0,., 1 0 0 0 0 0 q 0 p 0 0 0 0 q 0 p 0 0............. 0 0 q 0 p 0 0 0 0 q 0 p 0 0 0 0 0 1 45

6.2.8 () A > 0, B > 0. A B S = { A, A + 1,..., B 1, B}.. A < i < B, p, j = i + 1, p(i, j) = q = 1 p, j = i 1, 0,. i = A i = B, 1, j = A + 1, 1, j = B 1, p( A, j) = p(b, j) = 0,, 0,., 0 1 0 0 0 0 q 0 p 0 0 0 0 q 0 p 0 0............. 0 0 q 0 p 0 0 0 0 q 0 p 0 0 0 0 1 0 6.2.9 n, p n (i, j) = P (X m+n = j X m = i), i, j S,. (, m.), 0 r n, p n (i, j) = p r (i, k)p n r (k, j) (6.2) k S. -. p n (i, j) = P (X m+n = j X m = i) = k S P (X m+n = j, X m+r = k X m = i)., P (X m+n = j, X m+r = k X m = i) = P (X m+n = j, X m+r = k, X m = i) P (X m+r = k, X m = i),, P (X m+r = k, X m = i) P (X m = i) = P (X m+n = j X m+r = k, X m = i)p (X m+r = k X m = i). P (X m+n = j X m+r = k, X m = i) = P (X m+n = j X m+r = k) P (X m+n = j, X m+r = k X m = i) = P (X m+n = j X m+r = k)p (X m+r = k X m = i).,, P (X m+n = j, X m+r = k X m = i) = p n r (k, j)p r (i, k). 46

, (6.2). (6.2), 1 p 1 (i, j) = p(i, j), p n (i, j) = p(i, k 1 )p(k 1, k 2 ) p(k n 1, j). (6.3) k 1,...,k n 1 S, n p n (i, j) n (i, j). 6.2.10 P (X m+n = j X m = i) = p n (i, j) = (P n ) ij 6.2.11, -, P n = P r P n r. (, P 0 = E ().) {X n }, n i S π i (n) = P (X n = i). π(n) = [ π i (n) ], X n., X 0 π(0). 6.2.12, π(n) = π(0)p n π i (n) = π j1 (0)p(j 1, j 2 )p(j 2, j 3 )... p(j n 1, j n )p(j n, i). j 1,j 2,...,j n, π i (n) = P (X n = i) = k S P (X n = i X 0 = k)p (X 0 = k) = k S π k (0)p n (k, i). p n (k, i) = (P n ) ki,., π i (n) = (π(0)p n ) i., π(n) = π(0)p n. 21 A B,., A 30% B, B 20% A., A 80%, B 20%., B?,,. 47

6.3, S () {X n }. 6.3.1 2 i, j S, p n (i, j) > 0 n 0, i j., i. 2 i, j S, i j j i,, p n (i, j) > 0 p m (j, i) > 0 m 0 n 0, i, j. 6.3.2 2 i, j S,, i j., S., (i) i i; (ii) i j j i; (iii) i j j k i k.. ). (i), (ii). (iii) ( 6.3.3 i, 1, j = i p(i, j) = 0,,.,. 6.3.4. 6.3.5,. 6.3.6 i S, i (, first hitting time) T i = inf{n 1 ; X n = i} 48

., X n = i n 1 T i =. P (T i < X 0 = i) = 1, i., P (T i = X 0 = i) > 0,., i, i, i 1.,. 6.3.7 i S, p n (i, i) =. n=0 (1.), p n (i, j) = P (X n = j X 0 = i), n = 0, 1, 2,..., f n (i, j) = P (T j = n X 0 = i) = P (X 1 j,..., X n 1 j, X n = j X 0 = i), n = 1, 2,...,. p n (i, j) n., f n (i, j) i n j., i n j, j, n p n (i, j) = f r (i, j)p n r (j, j), i, j S, n = 1, 2,.... (6.4) r=1.,. G ij (z) = p n (i, j)z n, n=0 F ij (z) = f n (i, j)z n, n=1. (6.4),, G ij (z) = p 0 (i, j) + F ij (z)g jj (z) (6.5). (6.5) i = j, G ii (z) = 1 + F ii (z)g ii (z).,, n=0 G ii (z) = 1 1 F ii (z). G ii (1) = p n (i, i), F ii (1) = f n (i, i) = P (T i < X 0 = i), F ii (1) = 1 G ii (1) =.,. n=1 49

6.3.8 i, p n (i, i) < n=0,. 1 p n (i, i) = 1 P (T i < X 0 = i) n=0 6.3.9 (Z )., p 2n (0, 0),. 2n,,., p 2n (0, 0) = (2n)! n!n! pn q n, p + q = 1. (, 6 (5.1.1).) n! ( n ) n 2πn (6.6) e,., p q, p = q = 1 2, p 2n (0, 0) 1 πn (4pq) n p 2n (0, 0) <. n=0 n=0 p 2n (0, 0) =., 1, p q, p = q = 1 2. a n 22 (1) a n, b n. lim = 1, a n b n ( n b n )., c 1 > 0, c 2 > 0 c 1 a n b n c 2 a n. (2) (1), a n b n. n=1 n=1 (3) Z, 6.3.9. 6.3.10 (Z 2 ) ()., p 2n (0, 0),. 2 50

, x y 2, 2n x 2i y 2j., p 2n (0, 0) = ( ) 2n (2n)! 1 i!i!j!j! 4 i+j=n = (2n)! ( ) 2n 1 n!n! n!n! 4 i!i!j!j! i+j=n ( ) ( ) 2n 2n 1 n ( ) 2 n = n 4 i., (),., (), n i=0 p 2n (0, 0) = ( ) 2 n = i ( 2n n i=0 ( ) 2n n ) 2 ( 1 4 p 2n (0, 0) 1 πn p 2n (0, 0) = n=1 ) 2n ()., 2. 6.3.11 (Z 3 ).. 3, x, y, z 3, p 2n (0, 0) = i+j+k=n = (2n)! n!n! ( 2n = n ( 1 6 ) ( 1 6 (2n)! i!i!j!j!k!k! ) 2n i+j+k=n ) 2n i+j+k=n ( ) 2n 1 6 n!n! i!i!j!j!k!k! ( n! ) 2 i!j!k!., 2., (), i+j+k=n (6.7) n! i!j!k! = 3n. (6.8), M n = max i+j+k=n n! i!j!k!, n 3 1 i, j, k n + 1. 3 M n 3 3 2πn 3n. 51

,., p 2n (0, 0) ( 2n n ) ( 1 6 ) 2n 3 n M n 3 3 2π π n 3/2 p 2n (0, 0) < n=1, 3., ( (1911 2004) ). i P (T i < X 0 = i) = 1., E(T i X 0 = i). 1,.. E(T i X 0 = i) <, i. 6.3.12,,,.,,,,. 6.3.13 S. 23 2 {X n }. p = p p = 1p p = 1q, P (T 0 = 1 X 0 = 0), P (T 0 = 2 X 0 = 0), P (T 0 = 3 X 0 = 0), P (T 0 = 4 X 0 = 0)., P (T 0 = n X 0 = 0), p = q P (T 0 = n X 0 = 0), n=1 np (T 0 = n X 0 = 0) n=1. 24, O T 6. t = 0 O, 1/2., T, T. (1). (2)? (3) t = 12. (4) t. 52

6.4 6.4.1 S P. π = [ π i ] π i 0, S. S π π i = 1 i S π = πp (6.9), ().,. 6.4.2 (2 ) [ ] 1 p p P = q 1 q. π = [π 0 π 1 ], (6.9) [ ] 1 p p [π 0 π 1 ] = [(1 p)π 0 + qπ 1 pπ 0 + (1 q)π 1 ] = [π 0 π 1 ] q 1 q.,. pπ 0 qπ 1 = 0 π 0 + π 1 = 1. p + q > 0, π 0 = q p + q, π 1 = p p + q,. p = q = 0, π π 0 + π 1 = 1.. (, ). 6.4.3, 2 : (i). 53

(ii)., π,. π i = 1 E(T i X 0 = i), i S, S ( 6.3.13), 6.4.4 S 1 π., i S π i > 0. 6.4.5 (2 ) [ ] 1 p p P = q 1 q, P n., p + q > 0., 1, 1 p q. p + q > 0, 2., [ ] 1 p T = 1 q,.,, n, P T = T P n = T lim P n = T n [ 1 0 0 1 p q [ 1 0 0 (1 p q) n [ ] 1 0 0 0 ] ] T 1 [ T 1 = 1 q p + q q ] p. p, π(0) = [π 0 (0) π 1 (0)] n π(n), 6.2.12, π(n) = π(0)p n. n, [ lim π(n) = 1 n p + q [π q 0(0) π 1 (0)] q ] p p = 1 [ q [q p] = p + q p + q ] p. p + q, π., 2 p + q > 0,,, X n π.,,., (, 6.4.4 ),. 54

6.4.6 2 [ ] 0 1 P = 1 0., ()., π(0), lim π(n). n 6.4.7 S π ( 1 )., {X n }, j S. lim n P (X n = j) = π j,, 6.4.6.,,. 25.,. 26.,. 1. 1 15 2 16 2 4 (1 25 ). 0. 2. 3. 2 6 2 10 55

7 T R, {X(t) ; t T }. T = [0, 1] T = [0, ). t T, X(t), ω Ω., X(t, ω) X t (ω). ω Ω, X(t, ω), t X(t, ω) X(t).,., ()., (),. 7.1. t {X t }., {X t } {0, 1, 2,... }, (counting process).,., E., : (i) (ii) (iii) (iv) (v) (vi) (vii) t = 0, t E X t., E t 1, t 2,...,. 0 t t t tn 2. 56

(i) t E X t., {X t, t 0}. (ii) n 1 n T n (, T 1 E )., {T n ; n = 1, 2,... }, [0, ) ( ). X t T n T T 0 t 0 t t t tn (i), ()., t > 0, [0, t] n. t = t n,. t 1 2 n 0 t E : (1) E, λ > 0, P (E 1 ) = λ t + o( t), P (E ) = 1 λ t + o( t), P (E 2 ) = o( t),. (2) E. ( 1, ). (1) P (E 2 ) = o( t), t, E 2., E., P (E 1 ) = λ t + o( t), t, E. (2), E 57

,.,., i E Z i, Z 1, Z 2,..., Z n, P (Z i = 0) = 1 λ t + o( t), P (Z i = 1) = λ t + o( t), P (Z i 2) = o( t),. [0, t] E, n i=1. t, t 0 n, X t = lim Z i t 0 i=1 n Z i (7.1) X t, {X t }., t. λ. (7.1) Z i t ( n ). t, n,, Z i 0,., ( n ) P Z i = k = i=1 ( ) n (λ t) k (1 λ t) n k + o( t), k, t = t n.,, (λt) k ( n(n 1)... (n k + 1) P (X t = k) = lim t 0 k! n k 1 λt ) n k = (λt)k e λt. n k!, X t λt. 7.1.1, B(n, p), n p, np ( 10.0.6). 7.1.2 {X t ; t 0}. (1) () X t {0, 1, 2,... }. (2) X 0 = 0. (3) () 0 s t X s X t (4) () 0 t 1 < t 2 < < t k, X t2 X t1, X t3 X t2,..., X tk X tk 1,. (5) () 0 s < t, h 0, X t+h X s+h X t X s. 58

(6) λ > 0, P (X h = 1) = λh + o(h), P (X h 2) = o(h). (1) X t λt.,. (2). (3) s = m t, t = n t, m < n,. X s = lim t 0 i=1 m Z i lim (4) t 1 = n 1 t,..., t k = n k t, n 1 < < n k,., X t2 X t1 X t2 X t1 = lim n 2 t 0 i=1 Z i lim t 0 i=1 n 1 t 0 i=1 n Z i = X t Z i = lim n 2 Z i t 0 i=n 1 +1 [t 2, t 1 ) Z i., X t2 X t1,..., X tk X tk 1 Z i. {Z i }, X t2 X t1,..., X tk X tk 1. (5) X t+h X s+h X t X s, Z i, Z i,. (6) X h λh., P (X h = 0) = e λh = 1 λh + = 1 λh + o(h), P (X h = 1) = λhe λh = λh(1 λh +... ) = λh + o(h)., P (X h 2) = 1 P (X h = 0) P (X h = 1) = o(h). 7.1.3 1 2... (1), 2. (2), 2. (3) 2, 1 2. t X t. λ = 2. (1) P (X 2 = 0). X 2 2λ = 4, P (X 2 = 0) = 40 0! e 4 0.018 (2) t 0,, P (X t0 +2 X t0 = 0).,, (1). P (X t0 +2 X t0 = 0) = P (X 2 X 0 = 0) = P (X 2 = 0) 59

(3) P (X 2 = 0, X 3 X 2 = 2). X 2 X 3 X 2, P (X 2 = 0, X 3 X 2 = 2) = P (X 2 = 0)P (X 3 X 2 = 2), = P (X 2 = 0)P (X 1 = 2) = 40 0! e 4 22 2! e 2 0.00496 27 1 20.,. (1) 2 1. (2) 2 1, 3 2. (3) 10. 28 {X t }. 0 < s < t, ( ) n (s ) k ( P (X s = k X t = n) = 1 s ) n k, k = 0, 1,..., n, k t t.. 7.2 λ {X t ; t 0}., X 0 = 0, 1., t., T 1 = inf{t 0 ; X t 1} (7.2)., E. T 2 E 2., T 2 = inf{t 0 ; X t 2} T 1., T n = inf{t 0 ; X t n} T n 1, n = 2, 3,..., (7.3), T n n 1 E E. 7.2.1 λ {X t }, T n (7.2) (7.3)., {T n ; n = 1, 2,... } λ. t = n t, t. Z i. P (T 1 > t) = lim t 0 P (Z 1 = = Z n = 0) = lim (1 λ t)n t 0 ( = lim 1 λt ) n t 0 n = e λt 60

, P (T 1 t) = 1 e λt., T 1 λ. T 2, E 1,, T 1., T 3,.... 7.2.2 λ {X t } E(X 1 ) = λ.,., 1/λ. 7.2.1, λ, 1/λ.,. 29 () λ, E n S n = T 1 + T 2 + + T n. T n 7.2.1. P (S 2 t), S 2. [, S n.] 7.3 (7.1),.,, {X t }., (Ω, F, P )? {T n }., λ > 0 {T n ; n = 1, 2,... } (, (Ω, F, P ) )., S 0 = 0, S n = T 1 + + T n, n = 1, 2,...,, t 0, X t = max{n 0 ; S n t}., t 0 X t ({T n } )., {X t ; t 0}. λ. 7.1.2. 7.4 M/M/1/. A/B/c/K/m/Z. A:, B:, c:, K:, m:, Z: (,, ). M/M/1/,,,.,. 61

. 1,,.,.,,,.,,.. t ( ) X(t). X(t),,. 2., λ > 0, µ > 0. 1/λ, 1/µ., λ, µ., t t + t. t 1.,,,., λ t, µ t, 1 λ t µ t. λ t n n n µ t n t t + t, P (X(t) = n) : P (X(t + t) = n) = P (X(t + t) = n X(t) = n 1)P (X(t) = n 1) + P (X(t + t) = n X(t) = n)p (X(t) = n) + P (X(t + t) = n X(t) = n + 1)P (X(t) = n + 1) = λ tp (X(t) = n 1) + (1 λ t µ t)p (X(t) = n) + µ tp (X(t) = n + 1), P (X(t + t) = 0) = (1 λ t)p (X(t) = 0) + µ tp (X(t) = 1). p n (t) = P (X(t) = n), p n(t) = λp n 1 (t) (λ + µ)p n (t) + µp n+1 (t), n = 1, 2,..., p 0(t) = λp 0 (t) + µp 1 (t).,. t () p n. t p n (t) p n 62

., 0,, λ µ, λp n 1 (λ + µ)p n + µp n+1 = 0 n = 1, 2,..., λp 0 + µp 1 = 0. p n = A ( ) n λ (A ) µ., λ = µ, p n = A (). p n, p n = 1, λ < µ, n=0 p n = ( 1 λ ) ( ) n λ, n = 0, 1, 2,... µ µ.,,.,. np n = λ/µ 1 λ/µ = λ µ λ n=0, λ µ ρ = λ µ. ρ 1. ρ < 1, (1 ρ)ρ n, n = 0, 1, 2,...,., 1 ρ, ρ. ρ. ρ/(1 ρ). 7.4.1 ATM 3, 5., λ = 1 5, µ = 1 3, ρ = 3 5., ATM, 1 ρ = 2 5., ATM,, 2 5 3 5 = 6 25., ATM,, 3 1. 1 2 5 6 25 = 9 25 = 0.36. 7.4.2 X(t),. 63

8,,., (Robert Brown) (1827)., 1905 () (Albert Einstein),,.,, (Marian Smoluchowski, 1872 1917). 1906,, ().,. (Norbert Wiener, 1894 1964) (Paul Lévy, 1886 1971).. 1940, (Kiyoshi Itô, 1915 2008) ().,,.,. 8.1., t = 0 0, t, (H) ϵ, (T) ϵ. 1/2 {Z n }, t = n t X n, X n =. n ϵz k x t ε t 64

, P (Z k = 1) = P (Z k = 1) = 1 2 Z 1, Z 2,..., X n () : n E(X n ) = ϵ E(Z k ) = 0, (8.1) 8.1 (8.1) (8.3). V(X n ) = ϵ 2 n, (8.2) m n Cov (X m, X n ) = ϵ 2 Cov (Z j, Z k ) = min {ϵ 2 m, ϵ 2 n}. (8.3) j=1 8.2 n, t nt., t n t = 1, t, t., t X t (t)., nt X t (t) = ϵz k. (8.4) t 0 1 t, n t = 1, t 0, n, (8.4)., X t (t) V(X t (t)) = ϵ 2 nt = ϵ2 t t, ϵ 2 α () (8.5) t., t, 1 ϵ. (8.5)., (8.5) α = 1. t X t (t). ( t 0).,, nt B(t) = lim X t (t) = lim ϵz k (8.6) n t = 1, ϵ 2 t = 1 65

, n, t 0, ϵ 0.,, (8.1) (8.3) :, (8.6) E(B(t)) = 0, (8.7) V(B(t)) = E(B(t) 2 ) = t, (8.8) Cov (B(s), B(t)) = min{s, t}. (8.9) B(t) = lim ϵ 2 nt 1 nt Z k = t lim 1 nt nt nt., - (), B(t) 0, t N(0, t). {B(t)}. 8.2 (8.7) (8.9). 30 Z k 20 10 0 200 600 800 1000-10 -20-30 8.2.1 {B t ; t 0} : (1) () {B t },, a 1 B t1 + + a n B tn, a i R, t i 0,. (2) () E[B t ] = 0 (3) () Cov(B s, B t ) = E[B s B t ] = min{s, t} (4) () B 0 = 0 (5) () t B t 1. (6) () 0 t 1 < t 2 < < t n,. B t1, B t2 B t1, B t3 B t2,..., B tn B tn 1 66

(2) (6). t B t N(0, t). (6) (1) ( ).,,,. (Ω, F, P )?,,, (1 3) {B t ; t 0}., (4) (6)., (1) (5) (). (5), (1) (3). 8.2.2.,. 8.3 x = x(t) dx dt = f(t, x), x(0) = x 0 () (8.10). x = x(t)., α, dx dt = αx, x(0) = x 0 (8.11), t x x(t)., (8.11)., (8.10) dx(t) = f(t, x)dt, x(0) = x 0 (8.12). t t + t, x(t) x(t + t) x(t) = f(t, x(t)) t + o( t) (8.13). (8.12), (8.10) (8.12). x(t) = x 0 + t 0 f(s, x(s))ds (8.14), 2 f(t, x) (), (8.10).,, t = 0, x = x(t). x(t) x(t + t) 67

,., (8.13). x(t + t) x(t) = f(t, x(t)) t + (), x(t) ().,. x(t) x(t + t) {B t }, x(t + t) x(t) = f(t, x(t)) t + (B(t + t) B(t)),.,,, x(t + t) x(t) = f(t, x(t)) t + g(t, x(t))(b(t + t) B(t)). {B(t)} {x(t)}.,, x(t) X(t). t 0, dx(t) = f(t, X(t))dt + g(t, X(t))dB t, X(0) = x 0 () (8.15)., X(t) = x 0 +. t f(s, X(s))ds + t 0 0 g(s, X(s))dB s (8.16) (8.15) {X(t)}..,,.,, db t. db t, (1915 2008).. 8.4, (8.15) (8.16), (8.16). 2, 3 db t. (, )., {B t } (, ) {G(t)}, : I(t) = t 0 G(s)dB s. 68

{I(t)} {B t }., (8.15) {B t } {X(t)} (8.16)., {X(t)} X(t) = x 0 + t F (s)ds + t 0 0 G(s)dB s, (Itô process). dx = F ds + GdB. X(t) f(t, x) Y (t) = f(t, X(t))., Y (t) = y 0 + t F 1 (s)ds + t 0 0 G 1 (s)db s dy = F 1 dt + G 1 db. F 1 (s) G 1 (s) : 8.4.1 () 2 f(t, x) t 1, x 2,,, dy = f f (t, X(t))dt + t x (t, X(t))dX + 1 2 dx = F dt + GdB, (dx) 2 = G 2 dt. 2 f (t, X(t))(dX)2 x2 F 1 = f f (t, X(t)) + t x (t, X(t)) F (t) + 1 2 f 2 x 2 (t, X(t)) G(t)2, F 2 = f (t, X(t)) G(t) x.,, : (dt)(dt) = (db)(dt) = (dt)(db) = 0, (db)(db) = dt. 8.3 F 1, F 2. 8.5 8.5.1 γ > 0, σ > 0. dx = γxdt + σdb, X 0 = x 0 (8.17) (Langevin equation). dx = γxdt, x(0) = x 0, 69

γ ( x 0 ).,, ()., (8.17).,. (f(t, x) = e γt x.),, dy = γe γt Xdt + e γt dx Y (t) = e γt X(t) = γe γt Xdt + e γt ( γxdt + σdb) = e γt σdb e γt X(t) = x 0 + X(t) = x 0 e γt + σ. (σ = 0), t 0 t 0 x(t) = x 0 e γt e γs σdb s e γ(t s) db s., 0.,, t., lim E[X t] = 0, t, X(t) N lim E[Xt 2 ] = σ2 t 2γ ) (0, σ2. 2γ 8.5.2 α R, σ > 0. dx = αxdt + σxdb, X 0 = x 0 (8.18). (, σ = 0), dx = αxdt, x(0) = x 0., x(t) α, α > 0, α < 0. α (), x(t) ()., (8.18), dx = (αdt + σdb)x,, α,., (8.18)., X(t) = Y (t)e σb t 70

., (8.18),, dx = dy e σb t + Y d(e σb t ) = dy e σb t + Y dy e σbt + Y, ) (σe σb t db + σ2 2 eσb t dt. ) (σe σbt db + σ2 dt = αy e 2 eσbt σbt dt + σy e σbt db. Y = K exp dy = ) (α σ2 Y dt 2 ) } {(α σ2 t, K:, 2., (8.18), ) } X(t) = x 0 exp {(α σ2 t + σb t 2 (8.19)., a, b, exp(at + bb t ). (8.19) (8.18). {X(t)} (X(t), α, σ)., (Black-Scholes model). (1997 ),. 8.1: ()., 1. B.,, 1999. 2.,, 2005. 3. Z.Brzeźniak and T. Zastawniak: Basic Stochastic Processes, Springer, 1999. 71

9,. (Galton-Watson branching process),.,, ( ) ().,,., F. Galton 1873, Galton Watson,., Irénée-Jules Bienaymé (1796 1878), 1845,, Bienaymé BGW. 9.1 n = 0, 1, 2,..., n X n..,.,,,., Y, P (Y = k) = p k, k = 0, 1, 2,.... Y, Y 1, Y 2,...., n i 1, n + 1 j () ( i ) p(i, j) = P (X n+1 = j X n = i) = P Y k = j, i 1, j 0, 72

. 0,, 0, j 1, p(0, j) = 1, j = 0., 0. p(i, j) {0, 1, 2,... } {X n }., {X n } Y {p k ; k = 0, 1, 2,... }., {X n }., X 0 = 1 ( 1 ). 9.1 p 0 + p 1 = 1, (X 0 = 1 )?, q 1 = P (X 1 = 0), q 2 = P (X 1 0, X 2 = 0),..., q n = P (X 1 0,..., X n 1 0, X n = 0),..., ( ) P = {X n = 0} = P ( n 1 X n = 0 ). n=1 9.1.1 {X n },.,. p 0 + p 1 < 1, p 2 < 1,..., p k < 1,... (9.1) 9.2 {p k ; k = 0, 1, 2,... } {X n }., X 0 = 1. f(s) = p k s k (9.2). s 1., f 0 (s) = s, f 1 (s) = f(s), f n (s) = f(f n 1 (s)),. k=0 9.2.1 p(i, j) p(i, j)s j = [f(s)] i, i = 1, 2,.... (9.3) j=0, p(i, j) = P (Y 1 + + Y i = j) = k 1 + +k i =j k 1 0,...,k i 0 P (Y 1 = k 1,..., Y i = k i ) 73

, Y 1,..., Y i, p(i, j) = P (Y 1 = k 1 ) P (Y i = k i ) = k 1 + +k i =j k 1 0,...,k i 0 k 1 + +k i =j k 1 0,...,k i 0 p k1 p ki., p(i, j)s j = p k1 p ki s j j=0 = j=0 k 1+ +k i=j k 1 0,...,k i 0 p k1 s k1 p ki s ki k 1 =0 k i =0 = [f(s)] i. 9.2.2 n p n (i, j) p n (i, j)s j = [f n (s)] i, i = 1, 2,.... (9.4) j=0, p 1 (i, j) = p(i, j), f 1 (s) = f(s), n = 1, p(i, j)s j = [f(s)] i, i = 1, 2,.... (9.5) j=0., 9.2.1. n 1, (9.4) n., p n+1 (i, j)s j = p(i, k)p n (k, j)s j.,, j=0 j=0 k=0 p n (k, j)s j = [f n (s)] k j=0 p n+1 (i, j)s j = p(i, k)[f n (s)] k j=0 k=0, (9.5) s f n (s), p n+1 (i, j)s j = [f(f n (s))] i = [f n+1 (s)] i j=0.. X 0 = 1, P (X n = j) = P (X n = j X 0 = 1) = p n (1, j),, P (X 1 = j) = P (X 1 = j X 0 = 1) = p 1 (1, j) = p(1, j) = p j. 74

9.2.3, m = kp k < k=0 E[X n ] = m n., (9.2), f (s) = kp k s k 1, s < 1, (9.6). s 1 0, m.,, (9.4) i = 1, k=0 lim f (s) = m. s 1 0 p n (1, j)s j = f n (s) = f n 1 (f(s)). (9.7) j=0, s 1 0, f n(s) = jp n (1, j)s j 1 = f n 1(f(s))f (s). (9.8) j=0 lim f n(s) = s 1 0 jp n (1, j) = lim f n 1(f(s)) lim f (s) = m lim f n 1(s). s 1 0 s 1 0 s 1 0 j=0,., j=0 lim f n(s) = m n s 1 0 E(X n ) = jp (X n = j) = jp n (1, j) = m n j=0., n E(X n ) n, m < 1 0, m > 1, m = 1., m = 1. 30 V[Y ] = σ 2. 9.2.3, σ 2 m n 1 (m n 1), m 1, V[X n ] = m 1 nσ 2, m = 1,. 75

9.3 {X n } n, {X n = 0} n., q, ( ) q = P {X n = 0} n=1.,., {X 1 = 0} {X 2 = 0} {X n = 0}... q = lim n P (X n = 0) (9.9). q = 1, (), q < 1, 1 q > 0., q = 1. 9.3.1 f(s), n f n (s) = f(f n 1 (s)), q = lim n f n(0)., q, q = f(q) (9.10)., 9.2.2, f n (s) = p n (1, j)s j j=0 f n (0) = p n (1, 0) = P (X n = 0 X 0 = 1) = P (X n = 0)., X 0 = 1. (9.9),., f(s) [0, 1]. 9.3.2 f(t). (, (9.1).) (1) f(s)., 0 s 1 s 2 1, f(s 1 ) f(s 2 ). (2) f(s),, 0 s 1 < s 2 1, 0 < θ < 1, f(θs 1 + (1 θ)s 2 ) < θf(s 1 ) + (1 θ)f(s 2 ). (1) f(s) 0. (2) f (s) > 0. 9.3.3 (1) m 1, 0 s < 1 f(s) > s. (2) m > 1, 0 s < 1 f(s) = s s 1. 76

9.3.4 f 1 (0) f 2 (0) q. 9.3.5 q, s = f(s), 0 s 1, s., m 1 q = 1, m > 1 q < 1., m < 1, m = 1, m > 1, (subcritical), (critical), (supercritical). m, m = 1.. 31 p 0 < p < 1., p 0 = 1 p, p 1 = 0, p 2 = p, p 3 = p 4 = = 0., P (X 1 0,..., X n 1 0, X n = 0), P (X n = 0), lim n P (X n = 0),,. 32 b, p b > 0, 0 < p < 1, b + p < 1. p k = bp k 1, k = 1, 2,..., p 0 = 1, f(s)., m = 1, f n (s). p k 33 100. q = 9/10,?., 1. R. B. ):,, 2001. 2. K. B. Athreya and P. E. Ney: Branching Processes, Dover 2004 (original version, Springer 1972) 77