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()63,,, POI SSO N MARKOV BROWN, ( CIP ) /, :, 7 ISBN 7-3 - 3673 - - 6 CIP ()397 : : : (54, 8 ) : : : : : 787 9 : 3 5 : : : : I SBN 7-3 - 3673 - / O74 : 5 8,,

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( ) 3 3 4 6 5 Lplce Lplce - Stieltjes 8 6 7 7 8 Erlg 9 3 5 7 7 8 3 9 4 3 33 34 34 39 3 4 4 43 5 48 6 5 54 3 Poisso 56 Poisso 56 Poisso 59 3 Poisso 65 3 68 4 69 69 N t 7

3 7 4 75 4 79 5 Mrkov 8 Mrkov 8 Chpm - Kolmogorov 83 3 Mrkov 85 4 9 5 9 6 94 7 8 4 5 7 6 Mrkov Mrkov Pi j ( t) 3 3 7 6 7 Brow Brow 8 3 3 4 3 5 Brow 34 6 37 7 43 8 46 46 48 3 49 8 54 9 57 Fourier 57 64 3 73 4 75 9 83 86 86 88

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,,,, ( rdom vrible),, ( ), ( ), F, F ( ), (, F ) F P()F, : ( ): P( A ), AF ; ( ): P() = ; (3 ): A, A,, A, F, Ai Aj =, ij, i j =, P( = A ) = = P( A ) P( A )A, (,F, P) E,, (, F, P) X (), X (), x, {: X()x} F, X () (: X()x) F, X(), XX () R = ( -, ) x, ( ), (,F, P ), X (), X, ( : X () x) ( Xx ), ( X x )F, P( Xx ),, F F, F ( ) F ; ( ) AF, A = - AF ; ( 3 ) A, A,, A, F, U A F =

: X, { xi }, ij xi xj, X 3 P( X = xi ) = pi > pi = i F( X) = P( Xx ) = pi x R ( ) x x i X, R f( x ), x R, x F( x ) = f ( y) d y ( ) - f ( x) d x = -, - < b < R P( x = ) = lim P( - P( < x b) = f ( x) d x b < x ) = lim - f ( x ) d x =,, f( x ) X () x EX, ( ): EX = xd F( x ) = i - - xi f ( xi ) = xipi, i x f ( x) d x X, X Vr X = E ( X - EX) = EX - ( EX ) X Vr X EX X, X,, f ( x ) p i i ( x ), x, F ( x ) = x xi - f ( y) d y, xi ( x ) = x = x i ; xi ( x ) =, xx i,

Vr X X EX Vr X,,, Vr X,,, Vr X =, X (EX ), X k v k = EX k X k, ( X - EX) k k = E ( X - EX ) k X k, k, r, r, kr, k, X, Vr X X X,, EX = kp( X = k) = P( X = ) k = EX = P( Xk) ( 3) k = + P( X = ) + P( X = ) + P( X = 3 ) + P( X = 3 ) + P( X = 3) + X, EX,, ( 4) + P( X = k) + P( X = k) + + P( X = k) + ( k ) EX = ( - F( x ) ) d x ( 4) lim - F( x) ) + x x( xd F( x) EX, EX <, lim x x ( - F( x) ) = lim x x lim x x x d F( y) yd F( y) = EX =, lim x x( - F( x ) ), ( 4 ) x, EX, EX = ( - F( x) ) d x - F( x ) d x - 3,, ( X, Y ),, ( X, X, X ) ( > ), 3

X Y, X Y ( ) ( ) F( x, y) = P( Xx, Y y), - < x, y < X Y FX ( x ), F Y ( y) : FX ( x) P( Xx ) = y lim P( Xx, Yy)F( x, ) FY ( y) P( Yy) = x lim P( Xx, Yy)F(, y) ( X, Y ), ( ) XY ( ) f ( x, y) = P( X = x, Y = y) f X ( x) = f( x, y) y f Y ( y) = f ( x, y) x ( X, Y ), RR f ( x, y), x, yr F( x, y) = x - y - f ( u, v) d ud v f ( x, y)( X, Y ), X Y ( X, Y ): f X ( x) = f ( x, y) d y - f Y ( y) = f ( x, y) d x - x x FX ( x) = f X ( u ) d u = - - - y y FY ( y) = f Y ( v) d v = - - - XY, xy F( x, y) = F X ( x ) F Y ( y) f ( u, y) d ud y f( x, v) d xd v, ( X, Y ), f( x, y) = f X ( x ) f Y ( y), x, yr (( X, Y ), x, y(lebesgue ) ) X, Y Cov( X, Y ) XY X Y Cov( X, Y ) = E{ ( X - EX ) ( Y - E Y ) } = E( X Y ) - EXE Y X Y = Cov( X, Y ) Vr XVr Y XY, EX Y = EXE Y, Cov( X, Y ) =, Cov( X, Y ) = XY, E : XY E( X + by ) = EX + be Y b, Vr, 4

Vr( X + Y ) = Vr X + Vr Y + Cor( X, Y ) Vr( X ) = Vr X ( X, X,, X ) F( X,, X ) P( X x,, X x ) X x,, X x,, : ( ) F( x,, xi -,, xi +,, x ) = i =,,, ; ( ) F(,,, ) = ; (3 )F( x, x ) f ( x,, x ) = F( x,, x ) x x, f( x,, x ), F( x,, x ) = x x - - G R, G f ( x,, x ) d x d x P( ( x,, x )G ) = G f ( x,, x ) d X d x, Y = X + b X Fx ( x),, Y F Y ( y) = P( X + by) = P( Xy - b) = P( X y - b ) = F X ( y - b ) > P( X y - b ) = - FX ( y - b - ) < FX ( x) f X ( x), Y f Y ( y), f Y ( y) = f X ( y - b ) 3 X = ( X,, X ) f X ( x,, x ), yi = ui ( x,, x ), i =,, : ( ) xi = vi ( y,, y ), i =,, ( ) ui ( x,, x ) vi ( y,, y ), J yi = ui ( x,, x ): J = x y x y x y x y Y i = ui ( X,, X ), i =,, Y = ( Y,, Y ) 5

f Y ( y,, y ) = J f X ( x ( y,, y ),, x ( y,, y ) ) ( 5), Y = x + b, = X = Y - b, J = x = y f y - b y - b X, ( 5) f Y ( y) = f X 4 X ( t) = Ee t X = e t x d F( x ) - t = X, t = ( t) = E ( Xe t X ) ( t) = E( X e t X ) ( ) ( t) = E( X e t X ) ( ) ( ) = EX,, 4 X Y,,, X + Y ( t) = Ee t ( X + Y ) = Ee t X Ee t Y ( ) = X ( t) Y ( t) = exp{ ( + ) t + ( + ) t / }, X + Y +, +, X + Y p( x) ( t ), p, P > x p x ( - p ) - x x =,,, e - x x! x =,,, [ pe t + ( - p) ] p p ( - p) e xp { ( e t - ) } p p( - p ) x - x =,, p e t - ( - p ) e t p - p p r, p x - r - p r ( - p) x - r x = r, r +, pe t - ( - p) e t r r p r ( - p ) p 6

f ( x ) ( t ) (, b) b -, < x < b e tb - e t t ( b - ) + b ( b - ) > e - x, x - t (, )> e - x ( x ) - ( - )! x - t (, ) e - ( x - )/ - < x < exp{t + t } B -, b, >, b > cx - ( - x ) b - < x < ( + b) c = ( )( b) + b b ( + b) / ( + b + ), X P( X = k) = pk k =,,,, X G( s) = pks k k = pk, pk = X, k G( s)x G( s) = Es X, { pk } G( s) ;, G( s) { pk }, pk G( s) pk = k! d k G( s) d s k s = G( s), s < G X ( s) = G k = ( s) = k = k =,,, kpks k - k( k - ) pks k - EX = kpk = G ( ) k = E ( X ( X - ) ) = k( k - ) pk = G ( ) k = 7

EX Vr X = EX, X k : = E( X( X - ) ) + EX = G ( ) + G ( ) - ( EX ) = G ( ) + G ( ) - ( G ( ) ) E[ X( X - ) ( X - k + ) ] = d k G( s) d s k s = { k} { bk}, ck = bk + bk - { ck } { k }{ bk }, { ck } = { k } { bk } + + kb X Y, { pi } { qj } P( X = i, Y = j) = pi qj X = X + Y k rk P( z = k) = p( X = i, Y = k - ) i = k = piqk - i = { rk }{ pk }{ qk } 4 X, X,, X ZX + X + + X Gz ( s) X, X,, X G ( s),, G ( s) : : Gz ( s) = G ( s) G ( s) G ( s) Gz ( s) = Es z = E( s X + + X ) = E ( s X s X ) = Es X Es X = G ( s)g ( s), X,, X, G( s), Gz ( s) = ( G( s) ) 4 3 X, P( s), m, m, Y = mx + Q ( s) = s P( s m ) Y Q( s) = ES Y = ES m X + = s ES m X = s P( s m ) i 5 Lpce Lplce - Stieltjes Lplce, 8 f ( t) R + =, ), f( s) = e - st f ( t) d t ( 6)

f ( t)f ( t)lplce (L ), f ( s)l{ f ( t)} f ( s), s, L{ f ( t) } f ( t) L, ( 6 ) s, Lplce, s, s L ( ) L ci fi ( t) i = = i = ci f i ( s), ci, L{cosht} = L et + e - ( ) L{ t} = s L{e - t t} = L{ g( t) } = e - (3 ) s f ( s) (4 )L (5 )L (6 ) L{e - t f( t)} = f ( s + ) ( s + ) g( t) = L{si t} = f ( t - ) t >, t, L{ f ( t)} = f s s + - L{ f ( t)} = s f ( s) - f ( ) = L{ t f( x ) d x } = s f ( s) lim t f( t) = lim s f ( s) s lim t f( t) = lim s s f ( s) t s + = s s - X, F( x ) X f ( x), L, L L{ f ( x) } = e - sx f ( x) d x, X, L L - S { F( x) }Ee - sx = e - sx d F( x) ( 7) Stieltjes, s F( x ) ( X )Lplce - Stieltjes (L - S ) 9

X, G( s) = pksk k =, X L - S 7 ( s) = pke - sk = G(e - s ) k =, X L - S, X G( s) e - s s, F( x)l - S, L - S, L - S 6 X Y, P ( Y = y ) > y, Y = y, X P( X = x Y = y) = Y = y, X Y = y, X P( X = x, Y = y) P( Y = y) F( x y) = P{ Xx Y = y} E[ X Y = y] = xd F( x y) = xp{ X = x Y = y} x X Y f ( x, y), f Y ( y) > y, Y = y, X Y = y, X, X f ( x y) = f ( x, y) f Y ( y) x F( x y) = P{ Xx Y = y} = f( x y) d x - E [ X Y = y] =xd F( x y) =x f( x y) d x Y = y, E[ X Y] Y, Y = y, E [ X Y = y] X Y,, E[ X] = E[ E [ X Y ] ] =E[ X Y = y] d F Y ( y) ( 8) Y, ( 8 ) E [ X] = E[ X Y = y] P{ Y = y} y Y, f ( y), 8 + E[ X] = E[ X Y = y] f ( y) d y - X Y ( 8 ) ( 8 ) X Y

E [ X] = y E[ X Y = y] P{ Y = y} E[ X Y = y] P{ Y = y} y = y xp{ X = x Y = y} P{ Y = y} x = y xp{ X = x, Y = y} x = x x P{ X = x, Y = y} y = xp{ X = x } x = E[ X ] ( 8 ), E[ X ]Y = y, X, E [ X Y = y] X, X,, ; N, X, X, N Y N = X i i = E exp = E exp = E exp = ( X ( t) ) X ( t) = E[ e t X ]X E exp Y ( t) = E exp N t N = i = N txi i = N txi i = N = N txi N = ( X ( t) ) N i = N txi N Y = X i, Y ( t) : i = t =, i = ( ) = E[ ( X ( t) ) N ] Y ( t) = E[ N ( X ( t) ) N - X ( t) ] Y ( t) = E[ N ( N - ) ( X ( t) ) N - ( X ( t) ) + N ( X ( t) ) N - X ( t) ] E( Y ) = E [ NE ( X) ] = E[ N] E[ X ] E[ Y ] = E[ N ( N - ) E [ X ] + N E[ X ]] = E[ N] Vr( X) + E[ N ] E [ X ]

Vr Y = E[ Y ] - E [ Y ] = E[ N] Vr( X ) + E [ X] Vr( N ),,,, X Y 3 Ee t X = E( E( e t X Y ) ) = E [e t X Y = i] P[ Y = i] i = Y =, X =, = 3 { E[e t X Y = ] + E[ e t X Y = ] + E [e t X Y = 3 ]} ( 9) E[e t X Y = ] = e t Y =, X = 3 + X, X,, XX ( 9 ) E[ e t X Y = ] = E [e t ( 3 + X ) ] = e 3 t E [e t X ] E[e t X Y = 3 ] = e 5 t E[ e t X ] E[e t X ] = 3 { e t + e 3 t E[e t X ] + e 5 t E[ e t X ] } E[e t X ] = e t 3 - e 3 t - e 5 t :,,, E X X X = E, E E[ X ] = P( E) E[ X Y = y] = P( E Y = y), Y, 8 P( E) = P( E Y = y) d F Y ( y) 3 X Y, F G, X + Y, F G, F G, ( F G) ( ) = P{ X + Y} + = P{ X + Y Y = y}d G( y) -

F F F, F F - + = P{ X + Y Y = y}d G( y) - + = F( - y}d G( y) - F = F F, F, 4, A B m, > m,, :, A ( - m)/ ( + m) P, m, P, m = P{ A A } + m + P{ A B } + m, A, A A ( - ) B m B, P, m = + m P -, m +, + m, P, m = - m + m m m + P, m - ( ) + m =, P, = + m = k, + m = k +, P, m = + m = - m + m - - m - + m + m m + - m + + m -,, p, P{= } = P{= } p ( - p ), (), m = - P{= } = P, - = p ( - p ) - p ( - p ) 3

5,, k E P = P( E), M M c, P( E M ) =, P = P( E) = P( E M ) P( M ) + P( E M c ) P( M c ) P = P( E M c ) - ( ) P( E M c ) - -, - ( ) ( ), P -, - P -, P( E M c ) = P - + - P -, ( ), P = - P - + P - P - P - = - ( P - - P - ) ( ), ( ), P3 - P = - P - P 3 P4 - P3 = - P3 - P 4 P =, P = = - 3! P3 =! - 3! = - P4 = 4!! - 3! + 4! P =! - 3! + ( - ) - + 4!! k, k, P - - - ( k - ) P - k = ( - k)!! k - k k P - k k, k k ( - k)! P - k = k! e - / k! 4! - 3! + + ( - ) - k ( - k)!

,,,, p,, p Xi = i, Xi, i =,, P{ Xi = } = / P{ Xi = Y j = } = / ( - ), ji, 6,, ( i, i + ) / ( - ), i =,,, -,,, = 8, (, 3 ), ( 7, 8 ), ( 3, 4 ) ( 4, 5 ), ( ( 3, 4) ), 3 4 5 6 7 8 Ii, = i, Ii,, i = E[ ] = E[ Ii, ] i = = Pi, i = Pi,, i P P, = P, P ( ) Pi,,,,,, i i, i +,, i, Pi, = PiP - i + ( 3), Pi, P, ( ( i, i + ) ),,,, i - i +,,, ( i, i + ),, ( - i - ) 5

- - P = i = P - i - - = P + + P - - ( - ) P = P + + P - ( - ) P - = P + + P - 3 P = P =,, ( 3 ) ( - ) P - ( - ) P - = P - P - P - = - P3 - P = - P - P ( P - - P - ) - = P 3 =! P4 - P3 = - 3 ( P 3 - P ) = - 3! P 4 =! - 3! P =! - 3! + + ( - ) - Pi, = - ( - ) j j = j! - ( - )! = j = ( - ) j, j! i =,, i =, -, i - ( - ) j j = j! - i ( - ) j j = j! < i < - i - i, Pi, e -, Pi, ( ) i = Pi, ( + ) e -, i = 7,, () G, x,, x N P{ N > k}, Ei, i =,,,, P{ N > k} = P( E i ) i = i k, = P( Ei ) - P( EiEl ) + + ( - ) + P( E E E ) i = i < l, pj, j, 6

p = P(j x) d G( x ) = ( - x ) j d G( x ) N : P( Ei ) = p k P( EiE ) = p k,, P( E E E ) = p k P( N > k) = p k -, N p k + 3 E ( N ) = P( N > k) k = = k = i = = i = = i = i i p k 3 + + ( - ) + p k i ( - ) i + p k i ( - ) i + p k i ( - ) i + - pi E ( N ) = P( N > k) k = k = 7 X X (> ) X, f( x ) = e - x x, x < Ee t X = e t x e - x d x = - t ( 4) L - S ( s) = EX =, Vr X = ( s) = e - - ( s) = (+ s) e - x d x = + s sx (+ s) 3,, ( s) ( ) ( s) = ( - )! / (+ s) + 7

X, EX = ( ) = EX = ( - ) ( ) ( ) =! 7, : X, s, t P( x > s + t xt) = P( x > s) ( 5), X F( x), 7 X X, (> ), Z X + X (, ), X + X, X (X )L S ( s) = (, )L S, Z = X + X L S, s + s +, k k (, k) 7 3 X, X,, (> ), N, Xi N GN ( s ) N = p s, Z = = = X L - S Z ( s) = Ee s Z = G N ( X ( s) ) = GN = p = s + s +, N p ( < p < ), p = p ( - p ) -, GN ( s) = ps, Z L - S - qs Z ( s) = GN s + = p = s + p Z p p / ( s + ) - ( q / ( s + ) ), X, X, P( X > s + t X > t) = P( X > s), s, t ( 6) t, s + t s, t, 8 7 4 6 F( s + t) = F( s) F( t)

F() = - F( ), F, F ( s + t ) = e - s e t ( ) 7 5,,,,?, :,,,, :, ( ) X, F( x) = P( X > x ), F( x ) F( s + t) = F( s) F( t) g( s + t) = g( s) g( t) g() : ( ) g( t) ; ( ) g( t); (3 ) g( t) g( t) ;, g g g( t) = e - t, ( ), g, g ( s + t ) = g ( s ) g ( t) g + = g, g m, g = [ g( ) ] m m = g m g( ) = g + + + = = t g( t) = [ g( ) ] t g, t g ( ) = g, g( t), g( ) >, = - l g( ), g( t) = e - t, F(), (> )F( x) = e - x 7 6 ( ) F ( x ) X, f ( x), X( F( x ) ) ( x) f ( x), F( x ) > x F( x ) F( x ) = - F( x) = P( X > x), : F( x ) ( x ), ( x ), F ( x ) ( x ), F( ) = 9

F() =, ( x ) x f ( x ) = F( x ) = - x f( t)d t F ( x) = - f ( x) ( x ) d x = - d F( x)/ F( x) x ( t) d t = - l [ F( t) ] = - l F( x) F( x) = exp{ - x ( t) d t} x F( x ) = - exp{ - ( t) d t} x <, x ( x)exp{ - ( t) d t} x, ( ) ( x ) :X, ( x) d x X ( x, x + d x ),, P( X( x, x + d x ) X > x) = P( X( x, x + d x ) ) P( X > x ) = f ( x) d x = ( x) d x F( x),( x ) x, ( ), ( x )x, X, f ( x) = x <, e - x x,, ( x ), F( x ) = - e - x, F( x ) = e - x, L - S ( s) = - (+ s), ( s) = ( s) = e - (+ s) 3, e - x d x = + s sx ( ) ( s) = ( - )! (+ s) + EX = - ( ) = EX = ( ) =

EX = ( - ) ( ) ( ) =! / Vr X = EX - ( EX) = 8 Erlg X f, k ( x ) = k x k - e - x / ( k) x, x < k, (, k), >, k > ( x) e - t t x - d t, x = k,( k) = ( k - )! L - S ( s) = e - s x f, k ( x ) d x = ( s) = ( s) = - k k (+ s) k + k( k + ) k (+ s) k + EX = - ( ) = k EX = ( ) = k( k + ) k + s Vr X = - ( ( ) ) = k L - S k, L - S : k + s + s, k, k, ; k = 8 : Xi ( i =, ) ki = X + X (, k + k ), Z, Xi L - S ( + s ) k k + k i, Z L - S, (, k + s + k )L - S, Z k + k k, f k, k ( x ) = (k) k x - k x / ( k) x, x < Erlg, E k () (k, k)e k ()L - S

( s) = k s + k k k 9 X, X,, X, F ( x ), f ( x ) X( ), X ( ),, X( ) X,, X, X( i ) Xi, X ( ) X,, X, X ( ),, X( ) X( ) X ( ) X( ) E :X ( k) x, x + x ), x, F ( x ), X ( ),, X( k - ) ( k - ) x, - k X ( k + ),, X( ) x + x (), x x +x X ( ) X ( k - ) X ( k ) X ( k + ) X ( ) P( E) = P( x X ( k) < x + x)! = ( k - )! ( - k)! ( F( x) ) k - f( x )x - F( x + x ) - k x x X( k) f X ( k) ( x ) = k k =, X ( ) k = X ( ) k ( F( x) ) k - ( - F( x ) ) - k f ( x) f X ( ) ( x) = ( - F( x ) ) - f( x ) f X ( ) ( x) = ( F( x) ) - f ( x) X ( ) X ( ) FX ( ) ( x) = - P( X > x,, X > x) = - ( - F( x) ) FX ( ) ( x ) = P( X x,, X x) = ( F( x) ) r X ( ),, X( r ), : x < x < < xr = P( x X ( ) < x + x,, x rx( r ) < xr + xr )! - r f( x ) x f ( x r )x r ( - F( xr + xr ) ) ( - r )! x x r xi ( j =,, r) f( X ( ),, X ( r ) ) ( x,, xr ) =! ( - r)! ( - F( x r ) ) - r r f( xj ) j =

r =, X ( ),, X( ) f X,, X ( ) ( ) ( x,, x ) =! f ( x j ) j =, X,, X, T, f X ( ),, X ( ) ( x,, x ) =! T x x x T,, f ( ) N = {,,, }, f ( ) f ( ) f ( ) = f ( + ) - f ( ) f ( ) f ( ) f ( ) = ( f ( ) ) = f( + ) - f( ) = f ( + ) - f ( + ) + f ( ), k, f( )k + k + E, E f ( ) f ( + ), f( ) = k ( f( ) ) = k f( + ) - k f ( ) E k f ( ) = f ( + k) f( ) = E f ( ) - f ( ) = ( E - I) f ( ) E - I, EI +, ( ) f ( + ) - f ( + ) - f ( ) = ; ( ) (+ ) p = + p + + p -, ; (3 ) p - qp - - p( - p - ) =, p q ; (4 ) U x + U x + + + k U x + k = g( x), i ( ), +, ( + ) - =,,, E, ( ) E f( ) - E f ( ) - f ( ) = D = d, d x E, f ( ) = c, f( + ) - f ( ) =,,?, b, E f ( + ) + f ( + ) + bf ( ) = ( 7) 3

r ( E) E r( E) f( ) = + E + b, f ( ) = ( ) ( 7 ) + pm + q = ( 7), : ( ), ( 7 ),, c c c + c ( 7) c c ( ), = = = - c ( 7) f ( ) = c + c p c ( 7 ), ( + ) + + p ( + ) + + q = + + p + + ( + + p + + q ) = + ( + p ) = f ( ) = ( c + c ) (3 ) = re i = re - i + = rcos= - p = r = q, r = qcos= - p q = r e i = r (cos + isi ) = r e - i = r (cos - isi )( 7 ), r cos r si ( 7), f( ) = r ( Acos + Bsi ) ( 7), AB k, k f ( ) + f( + ) + + k f( + k) = ( 8) r ( m) + m + + k m k = k m,, m k, ( 8) f ( ) = c m + c m + + ck m k m, m = m,, f ( ) = ( c + c ) m + c3 m 3 + + ck m k c,, c, r ( E ) f( ) = g( ) ( 9) f ( ) = u + v u ( 9) r ( E) f ( ) =, v ( 9 ) v g( E) 4

( Fibocci ) f = f - + f - f = f = m - m - =, m = + 5 m = ( - 5 ) f = c m + c m c = - c =, 5 f ( ) = ( + 5 ) - ( - 5 ) / 5 l U = ( U + + U - ) l - U = U = r ( m) = m - m + =, m = m =, U = c + c c = c = - - l (l) l, U = N,, X F, F = - F X Y E[ X Y = y] f ( x, y) = E [ N] = P( Nk) = k = k = E[ X] = F( x) d x E[ X ] = X - F( x) d x P( N > k) 6 x y( - x - y) < x <, < y <, 3 X Y Poisso, X + Y =, X = k( k ) 4 X f ( x) = e - x x >, x 5

( ) Y = X; ( ) Y = e - X 5 Poisso P( s) = exp{ ( s - ) }, 6X, X,, X,, : Xi (, ), X i i = f ( t) = e - t ( t) - / ( - )!, t 7XY, /, /, Z = mi( X, Y ), Z = X, Z? 8 X, X, i ( t)xi ( i =, ) ( t) P( X < X mi( X, X ) = t) = ( t) + ( t) 9 ( s) = t s - e - t d t( s > )( s), s ( s) = ( s - )( s - ) ( ) = ( ) f ( + ) - f ( ) = ( ) U + + U + + U = i = 6

,, X, X,, { X, =,, }, ; ;,,, (),,, : ( )?,?,,,? :????, 97, Mrkov, ; 93, N Wieer, ( Wieer),, 3 93, Kolmogorov ;, Khitchie,, P Levy, 953, J L Doob ; ( Mrtigle), 95,, ;, ;, 6,,, :, ;,, : 7

; ;,,,,,,,,,, t,, t t, t ( t),, t[, 4],,,,, t ( t), t =,, 3,,,, t,,,,,, =,,, 4,, O, A,,, t t,, t, t, + ) 5,,, ( t) t,, t, t, + ) 6,,, t() Y t,, Yt = Ct + It, Ct It t, t Y t Ct It, t =,, 7 t,,, ( x, y, z, t), ( x, y, z ), t 8,,,

, 3 ( ) 3 (, F, P) T, tt (, F, P), E ( t, ),, t {( t, ) : tt } { ( t) }{ t t, T, T : T = N + = {,,, } ; T = N = {,,, } ; T =, b; T = R + =, ) ; T = R = ( -, ) T E, N +, N, {,, }, R +, R, b,,, { ( t, ), tt} t, t T, ( t, ) (, F, P) ;, ( t, ) T E (, ),, { t }M t M t, M t, E t = x t, M x, : t T, t,,, T = N N +,, T = R R +,, E t T,, { t }, tt, t Ft ( x )P( t x ) ; t, t T, t t Ft, t ( x, x ) P( t x, t x ) ;,, t,, t T, t,, t Ft,, t ( x,, x )P( t x,, t x ) ( ) 3 ( ), { Ft,, t ( x,, x ) :, t,, t T } 9

{ t }, : ( )xi ( i =,, ), Ft,, t ( x,, x ) ; ( ) lim x i - F t,, t ( x,, x ) = lim F t,, t x +,, x + ( x,, x ) = ; (3 )xi < yi ( i =,, ), Ft,, t ( y,, y ) - Ft,, t ( y,, yi -, xi, yi +,, y ) i = + Ft,, t ( y,, yi -, xi, yi +,, yj -, xj, yj +,, y ) i, j = i < j - + ( - ) Ft,, t ( x,, x ) ; (4 ) (,, ) (,, ), (5 )m <, Ft,, t ( x,, x ) = Ft,, t ( x,, x ) ; Ft,, t m ( x,, x m ) = (4 ) (5 ) lim F t,, t x +,, x + ( x,, x ) m + 3 3 {, =,, 3},,, N (, ), { } ( ) / exp - { } i = ( xi - ) ; 4,,, 4 ( ) { t } Ft ( x ), { t } ( ) E t xd Ft ( x ) D t E( t - E t ) = ( x - E t ) d Ft ( x ) { t }, t = X ( t) + j Y ( t), 3 E t EXt + je Y t D t E t - E,, t t

, E t D t t Y, Y =, Y = X i, X, X, iid( Xi i = ), P( Xi = - ) = q, P( Xi = ) = p, p + q =, E Y D Y Y, 4 E Y = E Xi i = = EXi i = EXi = p + ( - )q = p - q EX i = p + q = p + q = DXi = EX i - ( EXi ) = - ( p - q) E Y = ( p - q) D Y = - ( p - q) { t } Ft, t ( x, x ), { t } :t, t T r( t, t ) E( t, t ) = x x d Ft, t ( x, x ) R( t, t ) Cov( t, t ) = E{ (- E t ) (- E t ) } =( x { t }, - E t ) ( x r( t, t ) E( t, t ) R( t, t ) Cov( t, t ) t - E t ) d Ft, t ( x, x ) = E{ ( t - E t ) ( t - E t ) },, r ( t, t )R ( t, t )t t, : R ( t, t ) = r ( t, t ) - E t E { t }{ t - E t }, { t } { te j }, -,, { t }, { t } U ( -, )Ee j = E( t e j ) = E t Ee j = { t e j } ( ) E { ( t e j ) ( t e j ) } = E( t t e j- j ) = E( t t ) = r( t, t ), E t = R( t, t ) = r( t, t ) ;t = t, R ( t, t) = D t, t t 3

4 3 ( ){, }, N (, ),,,, E = D = r(, ) = E(, ) = E E = R(, ) = r (, ) - E E = ( )g( t) L ()( ), P(= - ) = P(= ) = :t, t g( t) { t }, L, E t = E(g( t) ) = g( t) E= D t = E(g( t) ) = g ( t) E = g ( t) R( t, t ) = r ( t, t ) = E{ (g( t ) ) (g( t ) ) } = g( t ) g( t ) E = g( t ) g( t ) g( t ) L (3 ) (, t, (, t (, ), t = ;, t = - ( t, t + tk t, t + t >, k =,,,, - t = t P( t) ( t Poisso ) : Pk ( t)p( t = k) = ( t) k k! e - t P((, t) = p ( t) + p ( t) + = e - t + ( t)! + ( t)4 4! P((, t) ) = e - t t + ( t)3 3! e - t si h t P( t = ) = e - t cos h t, P( t = - ) = e - t si h t + e - t cos h t + ( t)5 5! + E t = e - t cos h t + ( - )e - t si h t = e - t (cos h t - si h t) = e - t e - t = e - t : t, t, r ( t, t ) = e - t - t R ( t, t ) = r ( t, t ) - E t E t t { t } 3 = e - t - t - e - t - t e

D t = R( t, t) = - (e - t ) = - e - 4 t, :, (4 ) t = k = k t ( tr ) ke j k, k =,,,, N (, ) { t } k, k e j k t, E t = k = ( E k )e j k t = R ( t, t ) = r ( t, t ) = E{ t t } = E i = = i, k = i e j k t k = ke - j k t E( i k )e j( i t - k t ) = ke j( t - t ) k k =, { t },, { t }, X( t) = t + { X( t), tt},,, X( t)( x), Y ( t) = X ( t)x, X ( t) > x Y ( t) X ( t) 3X( t) m ( t) R ( t, t ), ( t), Y ( t) = X ( t) + ( t) 4 X ( t),, X ( t) Y ( t) = X ( t + ) - X( t) 5X ( t) = V t + b, t(, ), b, VN (, ), X( t) 33

,,,,,,,,,,,,, { X( t), tt}, tt, X ( t) DX ( t) <, { X ( t), tt },,, 3 ( ) X( t) = X + V t, tb, X V N (, ), X V, X ( t),, m X ( t) = E [ X( t) ] = E[ X + V t ] = E[ X ] + te[ V ] = r ( t, t ) = E[ X( t ) X( t ) ] = E[ ( X + V t ) ( X + V t ) ] = E[ X ] + E [ V ] t t = + t t { X( t), tt} ( ) Z( t) = Xcost + Ysit, t, X, Y N (, ),, Z ( t)x, Y Z ( t),, Z( t ), Z ( t ),, Z ( t ), Z ( t), mz ( t) = E[ Z ( t) ] = E[ Xcost + Ysit] = R( t, t ) = E [ Z( t ) Z( t ) ] = E [ ( Xcos t + Y sit ) ( Xcost + Y sit ] = (cos t cost + sitsit ) = cos( t - t ), Z( t) 34

N (3 ) Z ( t) = Xkej k t t, Xk,kN N (, k ) k =, j = -, e j x = cos x + jsi x Z ( t), m ( t), r ( t, t ) 4 { X ( t), tt} r ( t, t ), X ( t), r ( t, t ) = r ( t, t ) t, t T ( ) R ( t, t ) = E{ X( t ) X ( t ) } = E{ X ( t ) X( t ) } r ( t, t ) = r ( t, t ) ( ) = E{ X( t ) X ( t ) } = r( t, t ) X ( t),, r ( t, t ) = r ( t, t ) 5 { X ( t), tt} r( t, t ), t, t,, t T,,, i = i = j = j = r( ti, tj ) i r ( ti, tj ) i j i = j = j = i = r( ti, tj ) i j ( 3) j = = E i = E{ X( ti ) X ( tj ) } i j j = = E X ( ti ) j i = = E i = (,,, ) X( ti ) i X( ti ) j X ( ti ) i r ( t, t ) r ( t, t )r ( t, t ) r ( t, t ) r ( t, t )r ( t, t ) r ( t, t ) r( t, t )r( t, t ) (,,, ) r( ti, tj ) X ( tj ) j j = 6 { X ( t), tt} r ( t, t ), r ( t, t ) r ( t, t ) r ( t, t ) ( 4) 35

[ E X Y ] E[ X ] E[ Y ] r ( t, t ) r( t, t ) r( t, t ) 7 { X ( t), tt}, RX ( t, t ) ( ) R X ( t, t ) = R X ( t, t ) ( ) t, t,, t i, j = R ( t, t ) (3 ) R X ( t, t ) RX ( t, t ) RX ( t, t ) i j RX ( ti, tj ) ( i, j ) 8 TT r ( s, t),, () { X( t), tt }, r ( s, t) { X ( t ), tt}, r ( s, t), r ( s, t),, ( t),, t, t,, t T, i = j =,, exp - i = r ( ti, tj )( ti )( tj ) j = r ( ti, tj )( ti )( tj ) r ( t, t ) r ( t, t ) r ( t, t ) r ( t, t ) r ( t, t ) r ( t, t ) r ( t, t ) r( t, t )r( t, t ), : F( x, x,, x ; t, t,, t ), { F( x, x,, x ; t, t,, t ),, t, t,, t T},, { X ( t), tt }, { F( x, x,, x ; t, t,, t ),, t, t,, t T }( [ ] ), : E{ X( t) } = r ( s, t), E{ X( s) X ( t)} = r ( s, t) s, tt ( t) = ( t) - j( t) r( s, t) = A ( s, t) + j B( s, t) s, tt ( t), ( t), A ( s, t), B( s, t), 36 Q i = k = r( ti, tk ) ( ti ) ( tk )

= i = k = [ A ( ti, tk ) + j B( ti, tk ) ][ ( ti ) - j( ti ) ] [( tk ) + j ( t k ) ] Q,,, Q = i = k = i = k =, : A ( ti, tk ) [( ti )( tk ) + ( ti )( tk ) ] - B( tj, tk ) [( ti ) ( tk ) + ( tk ) ( ti ) ] exp{ ( - / 4) Q} (/ ) A( t, t )( / ) A ( t, t ) ( - / ) B( t, t )( - / ) B( t, t ) (/ ) A( t, t )(/ ) A( t, t ) ( - / ) B( t, t )( - / ) B( t, t ) (/ ) B( t, t )( / ) B( t, t ) (/ ) A( t, t ) ( / ) A ( t, t ) (/ ) B( t, t )( / ) B( t, t ) ( / ) A ( t, t ) ( / ) B ( t, t ), { X ( t), Y ( t), t T },, A ( s, t) = A ( t, s) E{ X ( t)} = E{ Y ( t) } = E{ X ( s) X( t) } = E{ Y ( s) Y ( t)} = (/ ) A( s, t) E{ X ( s) Y ( t)} = ( - / ) B( s, t) Z( t) = X ( t) + j Y ( t) E{ Z ( t)} = E{ X ( t)} + j E{ Y ( t) } = E{ Z ( s) Z( t)} = E{[ X ( s) + j Y ( s) ] [ X( t) - j Y ( t) ]} = E[ X( s) X ( t) ] + E[ Y ( s) Y ( t) ] - j E [ X( s) Y ( t) ] + j E[ X ( t) Y ( s) ] = (/ ) A ( s, t) + (/ ) A( s, t) + (j/ ) B( s, t) - (j/ ) B( t, s) A ( s, t) + j B( s, t) = r ( s, t) = r ( t, s) = A( t, s) - j B( t, s) B( s, t) = - B ( t, s), : E { Z( s) Z ( t)} = A ( s, t) + (j/ ) B( s, t) + (j/ ) B( s, t) = A ( s, t) + j B( s, t) = r( s, t) H (, F, P), 37

,, H, P(= ) =, = Schwrz,, H, H{ : E < }, E E E < E + E + E + { ( E ) ( E ) } <,, H, + H, H H : ( ) (, ) = (, ) ( ) (, ) = (, ) (3 ) (, ), (, ) = = (+, ) = E[ (+ ) ] (,)E( ), H = E( ) + E( ) = (, ) + (,) (, )E( ) H, (,) H,, H ( ), = = ; ( ) = ; (3 ) + + + = E + = E + Re E( ) + E () E + E( ) + E + + = ( + ) H :, H, ( )(, ), (, ) = = ; ( )(, ) = (, ) ; (3 )(, ) (, ) + (,) 38 (, ) - (, ) = - = - + - - + -

(,)H = (, ) + (, ) H, H : H, =,,, H, { }, L : L ( ) ( ) lim ; (, ) = ; (3 ) lim - = L, lim E - = ( 5) lim =, ( ) (3 )( 5), { C }, lim C = C, ( ) lim C = C; ( ) lim C = C, H lim =, L L = (P(= ) = ) E - P(= ) = = E (- ) + ( - ) = (- ) + ( - ) ( - + - ) L 3, L,,, ( - ) L + : ( + ) - ( + ) = (( - ) + ( - ) ( - )+ ( - ) ( ) L 4 ( H) 39

lim - m =, m L :( ), - m = ( - ) + (- m ) - + - m, m H { }, { }, L { } L 5, L, ( ) lim E = E, lim E = E(l.i.m ) ( l.i.m { } ) ( ) ( ) lim E( m ) = E( ),, m ( ): ( ) : E( m ) - E( ) lim E( m ) = E{ l.i.m, m lim E = E (l.i.mm ) } E - E = E( - ) E - ( ) = { E (m ) - ) + ( - )+ ( - ) (m - ) } (m - )+ ( - ) + ( - ) (m - ) m - + - + - m - (m, ),, : ( ) ( ), L 6 E( ) = E,, L,, ( ) ( ) E( ) = E L lim E( 7 Lo ve E( ) = E ) = lim lim E( L E ) = E( ) = E : c lim E( m ) = c, m ( ), C = E : lim E, m 4 m = lim ( E + E m, m - E( m ) - E( m ) )

= C + C - C - C = { }, 4, { } { t, tt }, { t } (), tt, t H, T R 8 { t }, t T, H, >, >, < t - t < E t - t < t - t < { t } t, lim tt t = t L { t }tt, { t } T L 9 t { tt }, t T, lim t = t L t,, : lim tt t, >, >, t, < t- t < t- t {, =,, } lim =, >, t, < t - t < -, t t, t t t, L ( ) t, t L, = L ( ) t lim t- t = t, t t L (3 ) t, t L, t + t L + L (4 ) t L t, lim E t = E tt lim t E ( t t) = E( ), t t L (5 ) t c, lim E( t t, t t t) = c, c = E 4

{ X ( ), =,, } r (, ) = E{ X( ) X ( ) }, {, =,, } Y ( ) = k X ( k), k = Y ( ) E{ Y ( ) Y ( m) } = E = k = m i = = k = i = k = m i = k ie { X ( k) X( i) } k i r ( k, i) k i X ( k) X( i) Lo ve, { Y ( ) }, lim m E { Y ( ) Y ( m ) } = c m k = i = k i r( k, i) = c 3 { t } t, { t } t, lim ( t + ) = t,, t, 3 { t }, t T, >, >, t - t < t - t { t } t, lim t = t tt L t t { t }tt, { t }T 8, { t } T, E t T 3 { t }t = t { t }r( t, s)( t, t ) :, ( 5) L t t lim t r( t, t ) = lim, t t t E( t t), t t 4

3 3 : ( ) { t } T ; = E( t t ) = r( t, t ) ( ) r( t, t ) ( t, t) ( ( t, t)tt ); (3 ) r ( t, t )T T ( )( ) 3, ( ) ( 3), ( ) ( 3), ( ), ( ), t T, t T ( 4) lim t= t t t lim t t t= t lim tt tt r( t, t ) = lim t t) t t E( t t = E( t t ) = r( t, t ) :( ), :, TT 3 4 { X ( t), tt}, lim t lim t E{ X( t + t) } = EX ( t) lim E{ X ( t + t) - X ( t) } = t EX ( t + t) = E(l.i.m t X ( t + t) ) = EX( t),,,, 4 t, t= d d t t = lim ( t + ) - ( t) t, t, t,, t,, t, 4 { t }, ttt l.i.m h ( t + h ) - ( t) h ( t + h) - ( t) h = t L t 43

{ t }t ( ), t{ t } t tt, { t } T, X ( t) = d X( t) d t = lim t X ( t + t) - X ( t) t = X( t) 4 X ( t) = si t,, E 4 <,, X ( t) = cos t,, E = E X ( t + t) - X( t) t si ( t + t) - si t t - X ( t) - cos t = E t ( si t(cos t - ) + cos t( si t - t) E si t(cos t - ) ( t) + E cos t(si t - t) ( t) ( + b) + b, cos +, si +, t E si t( t) 4 (t) + E cos t( t) 4 (t) 4( t) E 4 lim t E X ( t + t) - X( t) t - cos t, 4 3 rx ( s, t){ X( t ), tt }, ( s, t ) h, k, lim h k rx ( s + k, t + h) - rx ( s, t + h) hk - = rx ( s + k, t) - rx ( s, t) hk rx ( s, t)( s, t)( ) r s, t ( s, t) = lim h rx ( s + k, t + h) - rx ( s, t + h) - rx ( s + k, t) + rx ( s, t) hk 4 4 ( ) { X ( t), tt }, t = rx ( s, t) (, ) { X( t), tt}, l.i.m h X (+ h ) - X() h, lo ve, lim h k,, 44 E X (+ h) - X ( ) h X (+ k) - X() k

lim h k = r rx (+ h,+ k) - rx (,+ k) - rx (+ h, ) + rx (, ) hk, ),(, rx ( s, t)(, ) { X( t), tt }, T { (, ), T}r s, t ( s, t) r s, t ( s, t) = r ( s, t) s t,, : [,][, ] r ( s, t),, (, ), s r(, ) = t r (, ) = r (, ) = r (, ) = s t t s h = k, r ( h, h ) - r( h, ) - r(, h) + r (, ) lim h h = lim h h = r ( s, t) { (, ), T }, r s( s, t), r t( s, t), r s, t ( s, t), r t, s ( s, t) TT E { X ( s) X( t)} = r s ( s, t) E { X( s) X ( t)} = r t( s, t) E { X ( s) X ( t)} = r s, t ( s, t) = r t, s ( s, t) 4 5 O t 8 4 { X ( t), tt} t, X( t) t, X ( t) = Y, X ( t) = Z, Y = Z 3 X ( t)y ( t),, b, X( t) + by ( t), { X( t) + by ( t) }= X ( t) + by ( t) 4 X ( t), f( t), d d f( t) d X( t) [ f( t) X ( t) ] = X ( t) + f ( t) d t d t d t 4 6 X ( t), X ( t) X ( t), d X( t)/ d t,, Y ( t) = d X( t)/ d t = X ( t) (X( t) ) d X( t ) d t Y( t ) = X ( t) X ( t),,, Y ( t) ( ) Y ( t)m Y ( t) s 45

Y ( t) = d X ( t)/ d t = l.i.m t m y ( t) = E[ X ( t) ] = E l.i.m t = lim t E X( t + t) - X ( t) t X ( t + t) - X( t) t X( t + t) - X ( t) t = lim t m X ( t + t) - m X ( t) t = d m X ( t) d t, E d X( t) d t = d E[ X ( t) ] ( 6) d t E[ X( t) ], ( ) X( t)y ( t) rx Y ( s, t), r Y X ( s, t) rx Y ( s, t) = t r x ( s, t) ( 7), rx Y ( s, t) = E[ X ( s) Y ( t) ] = E X( s)l.i.m t r Y X ( s, t) = t r x ( s, t) ( 8) Y ( s) = l.i.m t X( t + t) - X ( t) t X ( s + s) - X ( s) s = t lim E[ X( s) X ( t + t - X ( s) X( t) ] t = t lim t [ r X ( s, t + t) - rx ( s, t) ] = t r X ( s, t) r Y X ( s, t) = s r X ( s, t) ( 7 )( 8), X( t)y ( t) rx Y ( s, t), r Y X ( s, t) rx ( s, t)s t 46 (3 ) Y ( t) ry ( s, t) r Y ( s, t) = E [ Y ( s) Y ( t) ] ry ( s, t) = [ rx ( s, t) ] st ( 9) X( s + h ) - X( s) = E l.i.m Y ( t) h = lim h (/ h ) E[ X ( s + h) Y ( t) - X ( s) Y ( t) ] = lim h (/ h ) E[ rx Y ( s + h, t) - rx Y ( s, t) ] = [ rx Y ( s, t) ] s r Y ( s, t) = [ r Y X ( s, t) ] t ( 7 )( 8) ( ), ( ) rx ( s, t) = [ rx ( s, t) ] s t ( ) ( )

, Y ( t), X ( t) 4 7 X ( t) rx () = exp{ - }, rx ( ) = e - = exp - ( ), exp{ } <, =, rx ( X ( t)t r X () = ), 3 - exp{ }, exp{ } < 4, =, r X ( ), r X ( ), X ( t) r() r ( ) O O - 3 4 4 8 { X ( t), tt}, { X ( t), tt },, { X ( t), tt } X ( t) = d X( t)/ d t ( X ( t) ), X ( ) ( t) = d X ( t)/ d t, X ( ) ( t) X ( m ) ( t) r ( x - ), X ( ) ( t) rx ( ) X ( ) ( t, t ) = rx X ( t, t ) t t rx ( ) X ( m) ( t, t ) = + m rx X ( t, t ) t t m m ( t, t ) t, t, rx ( t, t ) t t t = t 47

5 5 { X( t), tt } T = [, b], [, b] +, = t < t < t < < t = b ti = ti - ti -, t i [ ti -, ti ], ( i =,,, ) t = mx oi ( t i -, ti ) X( t i ) ( ti - ti - ) i = t,, t i, X ( t) [, b], lim E t b X ( t) d t = l.i.m t X( t i ) ti i = X( t i )ti i = b - X( t) d t = ( ) 5 { X ( t), tt} T = [, b], h( t)( tt ) (), X( t) h( t)[, b] b X ( t) h ( t) d t = l.i.m t h( t i ) X( t i ) ti i = ( 3) 5 3 { X ( t), tb}, h( t,) [, b],, [, b] Y ( b ) = Y ( ) 5 4 l.i.m b b X ( t) 48 X( t) h( t,) d t = l.i.m t X( t) h( t,)d t, X( t) h( t,)d t = l.i.m b b h( t i,) X( t i )ti i = X ( t) h ( t, ) d t 5 5 X( t) [, b] b b love, ( ) rx ( s, t) d sd t < ( 4)

X ( t i ) ti =,, i = E X( t i ) ti X( t k ) tk i = k = = i = k = rx ( t i, t k ) titk ti, tk, b b rx ( s, t) d sd t = C < 5 6 h ( t) X( t)[, b] b b h ( u ) h ( v) r x ( u, v) d ud v Loe ve, h( t) X( t)[, b] lim t i = h( t i ) X ( t i )ti lim t E = lim t i i = t j h ( t i ) X ( t i )ti i = b = b j = h( t j ) X( t j ) tj i = h( t i ) h( t j ) rx ( t i, t j ) titj h ( u ) h ( v) r x ( u, v) d ud v 5 7 h ( t, ) X ( t)[, b], b b h( u,) h ( v,) r x ( u, v) d ud v 5 8 ( ) X( t) [, b] X ( t)[, b], k, kx ( t)[, b] b kx ( t) d t = k b X ( t)y ( t)[, b], b k, k [ k X ( t) + k Y ( t) ] d t = k b X ( t) d t ( 5) X( t) d t + k b 3 X ( t)[, b], < c < b, b X ( t) d t = c 4 X ( t)[, b], X ( t) d t + b c Y ( t)d t ( 6) X( t) d t ( 7) 49

t Y ( t) = X( u ) d u tb Y ( t), [, b],, Y ( t) = X( t), Y ( t) Y ( t + t) - Y ( t) E - X ( t) t t + t = E{ [ (/ t) ] t X ( u) d u - t + t = E{ [ / t) X ( u) d u - X ( t) ] } t t + t = E{ / t) [ X( u) - X ( t) ] d u } t t + t { ( / t) t { mx u - t < t X ( u) d u ] - X( t) } E[ X ( u) - X ( t) ] d u} E[ X( u ) - X( t) ] }, t,, Y ( t) Y ( t) = X ( t) 5 X ( t)[, b], b X( b) - X( ) = :X ( t) [, b], t X( t) - X ( ) = X( t)[, b], b E[ b E [ Y ] = E[ = lim t m Y ( t) = m X ( s) d s 5 t i = X( t) b Y = X( t) d t] = b X ( t) d t ( 8) X ( u ) d u tb ( 9) X ( t) d t] = E l.i.m E[ X( t X( t) d t Y b = X ( t) d t b E[ Y b ] = b E [ X( t) ] d t t i = X ( t i ) ti b i ) ti ] = E[ X( t) ] d t = b m X ( t) d t X( s) d s = b b X( s) X ( t) d sd t E[ X( s) X ( t) ]d sd t

b = b r X ( s, t) d sd t = E[ Y ] - [ E( Y ) ] 3 b = b b r X ( t, s)d td s - E [ X( t) ] d t b b = b [ rx ( t, s) - m X ( t) m X ( s) ] d td s = b b RX ( t, s) d td s E [ X( s) ]d s t ( ){ X ( t), tt} [, b], Y ( t) = X( s) d s, t b, Y ( t) t r Y ( t, t ) = E[ t X( t) X ( s) d td s] t = t t ( ) X( t) Y ( t) rx Y ( t, t ) = 4 R Y ( t, t ) = t t t E[ X( t) X ( s) ] d td s = t rx ( t, s) d td s r Y X ( t, t ) = t rx ( s, t ) d s [ rx ( t, s) - m X ( t) m X ( s) ] d td s 5 9 ( ){ X( t), tt }, : ts, X( t)x( s) ; tt, X ( t) E[ X( t) ] =, E[ X ( t) ] = { X( t), tt} X ( t), t t E[ Y ( t) ] = E [ X ( s) d s] = E[ Y ( t) ] = E[ ( t X ( s) d s) ] Y ( t) = t X ( s) d s E[ X ( s) ] d s = = E l.i.m i X ( ti ) ( ti + - ti ) = E l.i.m = lim i i j j X( ti ) X( tj ) ( ti + - ti ) ( tj + - tj ) E [ X ( ti ) X( tj ) ] ( ti + - ti ) ( tj + - tj ) = lim E[ X ( ti ) ] ( ti + - ti ) i rx ( t, s) d s ti tj, X( ti )X( tj ) [, t] = t < t < < tn = t, 5

mx( ti + - ti ) N mx ( ti + - ti ), E[ Y ( t) ] = lim E[ X ( ti ) ] ( ti + - ti ) i = lim C( N N ) N = E[ X ( t) ]C, Y ( t) = t X( s) d t, X( t) = d Y ( t) d t E[ X d Y ( t) ( t) ] = E d t Y ( t + h) - Y ( t) Y ( t + k) - Y ( t) = E[lim h lim k ] h k = lim h lim k E {[ Y ( t + h ) - Y ( t) ][ Y ( t + k) - Y ( t) ]} hk lim h lim k hk { E[ Y ( t + h) - Y ( t) ] E[ Y ( t + k) - Y ( t) ] } / =, E [ X ( t) ] =, E [ X ( t) ] =,, X( t) = ( ), ( ) X( t) = Acos(t + ), A, mx ( t) =, rx ( ) = A cos T /, = t - sx ( t)t Y ( T ) = m Y ( T ) = T m X ( t) d t = T r Y ( T, T - ) = T - T = T - rx ( t, s)d td s A cos( t - s) d td s = ( A / ) [cos- cos( T - ) - cost + ] Y ( T ) = R Y ( T, T ) = r Y ( T, T ) = ( A / ) ( - cost ) X ( t) d t, Y ( T ) 6 ( RiemStieltjes) b I = f ( t) d X ( t) ( ) b I = X( t)d f ( t) ( ) 6 ( )f ( x) [, b], [, b], = t < t < < t = b, 5

- If ( t, t,, t ) = f ( ti + - f ( ti ) ) i = f t, t,, t,, { If ( t, t,, t ) }, Sup ( t, t,, t ) f [, b] 6 ( RiemStieltjes) If ( t, t,, t ) < { X( t), tt }, f ( t), T = [, b], [, b] ti = ti - ti - I I, i = t < t < < t = b, I = f( ti ) [ X ( ti ) - X( ti - ) ] i = I = X ( ti ) [ f ( ti ) - f ( ti - ) ] i = ti - titi i =,,, l.i.m mx t i i l.i.m mx t i i ( I ) = I ( I ) = I ti, I ( I ) f( t) X ( t) ( X ( t) f( t) ), I b b = f ( t) d X ( t) I = X( t) d f( x ) 6 3 ( ) I I b b f ( t) f ( s) E{d X( t) d X( s) } b = b b b f ( t) f ( s)d CX ( t, s) CX ( t, s) d f ( t) d f( s) ( 3), ( ), I f ( t)[, b], R X ( t, s) [, b] [, b] I RX ( t, s) [, b] [, b], f ( t) [, b], ( ) ( 3) 6 4 I (I ), I (I ), I = [ f( t) X ( t) ] b - I ( 4) 6 5 f ( u, s) [, b] [ c, d], { X ( t), t[ c, d ]} 53

( E[ X ( t) ] [, b]), b Y ( u ) = f( u, s) d X ( s) [, b], [, b] b d Y ( u ) d u = b f( u, s) d X ( s) c 6 6 b E [ I ] = f ( t) d E[ X( t) ] b E [ I ] = E[ X( t) ] d f( t) E [ I b ] = b E [ I b ] = b 6 7 f ( t) f ( s)d RX ( t, s) ] R X ( t, s) d f ( t) d f( s) X ( t) f( t) f ( t) X ( t), I I, b f ( t) d X ( t) = f ( t) X( t) b b X( t)d f ( t) = X( t) f ( t) - b b b - X ( t) d f ( t) ( 5) f ( t) d X ( t) { X, =,, }, P( ) = P( ) = -, { X } { X, =,, }, X,,, lim P{ X - X > } = X X, X X 3 { X } X, X 4 { X } X, X 5 N p, N,, N, X, X X X 54 :X Qi, X = Qi, Q i = L X X e X E [ X ] E[ X ] 6{ X( t), tt }, =,

7 X, =,,, f ( x ) = x < X X + e - ( x - ) x, x <, + x < 8 X( t) = A t + B, AB, m A m B, A B, : ( ) X( t) = A ; ( ) mx ( t) = m A ; (3 ) R X X ( t, t ) = A + B ; (4 ) X ( t) = A 9 { X( t), tt } rx () = e -, Y ( t) = X ( t) + X ( t ), r Y ( ) = (3-4 )e - r ( t, s){ ( t, t), tt}, r t ( t, s), r s ( t, s ), r t, s ( t, s), r st( t, s)tt E{ X( t) X ( s) } = r t ( t, s) E{ X( t) X ( s) } = r s ( t, s) X( t) [, b], X ( t) - X ( ) t = X ( u) d u tb X ( t) 5, mx =, rx ( ) = e - Y = T T - T X ( t) d t, : E{ X( t) X ( s) } = r t s ( t, s) = r st ( t, s) 5 ( ) m Y = ; ( ) R Y ( t, t ) = T - - e - 4T 8 T 3 X ( t)[, b], b X ( t) d t b O x( t) X( t) d t t 55

3 Poisso Poisso { Nt, t}, Nt (, t ] (, ), ( ) N t ; ( )s < t, NsN t ; (3 ) N t R + =, ) ; (4 ) s < t, Ns, t N t - Ns ( s, t] Nt N ( t) { N t, t},, ( s, t] Ns, t s { N t, t},,,, t < t s >, Nt, t Nt + s, t + s { N t, t} ( ) Poisso, : ( ) P( N = ) = ( ) (3 )s < t, Ns, t = N t - Ns ( t - s)poisso, P( Ns, t = k) = k [( t - s) ] e - ( t - s) k =,,, k!, ( 3) Poisso E Nt = t, () Poisso, ( ), ( )( 3) ( ), ( ) (3 ) Poisso f ( h), f o( h ):lim h h f h, o( h ) 56 3 { Nt, t} Poisso,,>, ( ) N = ; =,,

( ) ; (3 ) P( Nh = ) = h + o( h ) ; (4 ) P( Nh ) = o( h) 4 3 3, P ( t): P ( t)p( Nt = ) P ( t + h) = P( N t + h = ) = P( N t =, N t + h - N t = ) = P( N t = ) P( N t + h - N t = ) = P ( t) ( - h + o( h) ) ( ) ( 3) ( 4) P( N h = ) = - h + o( h ) h P ( ) = P( N = ) =, P ( t + h) = P( N t + h = ) h P ( t + h) - P ( t) h = - P ( t) + P ( t) = - P ( t) P ( t) P ( t) = - l P ( t) = - t + c P ( t) = Ke - t o( h) h P ( t) = e - t ( 3 ) = P( N t =, N t + h - N t = ) + P ( N t = -, N t + h - Nt = ) + P ( Nt + h =, ( ) = N t + h - N t ) P ( t) P ( h ) + P - ( t) P ( t) + o( h ) = ( - h) P ( t) + hp - ( t) + o( h) P ( t + h) - P ( t) h (3 ), = = - P ( t) + P - ) ( t) + P ( t) = - P ( t) + P - ( t) e t ( P ( t) + P ( t) ) = e t P - ( t) d d t e t P ( t) o( h) h = e t P - ( t) ( 3 ) 57

P ( ) = d d t e t P ( t) = P ( t) = ( t + c)e - t P ( t) = te - t P ( t) = e - t ( t) /!,, ( - ) : P - ( t) = e - t ( t) - / ( - )!, (3 ) d d t e t P ( t) = ( t) - ( - )! e t P ( t) = ( t) + c! P ( ) = P( N = ) =, c =, P ( t) = e - t ( t) /! 3 ( ) 5 N t P( t) Poisso,, [, t] k, k, k, P( ) k P( i ) i = o( t/ k) = ko( t/ k) = t ( k ) t/ k, N t ( ), B k, Nt Poisso, lim k t k k + o k, t/ k + o( t/ k) Poisso, t k = t + lim k to( t/ k) t/ k = t 6 Poisso, = 4 /, 9, 93, 3 5 P( N 5 =, N 5 = 5 ) = P( N 5 =, N 5 - N 5 = 4 ) = P( N 5 = )P( N = 4) = 4 5) e - 4 5 (4) 4 e - 4 55! 4! Poisso,,, Poisso ;,, Poisso ;:, ( ) Poisso, (( ), ( 58

),,,,,, Poisso, Poisso Poisso Poisso, X,, X - { X, } X ( X > t) Poisso [, t], P( X > t) = P( N t = ) = e - t, X / X, X : P( X > t X = s) = P(( s, s + t] X = s) = P(( s, s + t]) () = P((, t]) () = e - t X /, X X, X, X, =,, /,,, (, (),, S,, S = Xi, i = S (,), S, S f( t) = e - t ( t) - ( t) ( - )! : t t, Nt S t, P( S t) = p ( N t ) S f( t) = - j = e - t ( t) j j! = e - j = t ( t) j j! + j = t) j - e - t ( ( j - )! 59

= e - t ( t) - ( t ) ( - )! Poisso / { X, }, S, S = X + X + + X, { Nt, t} Poisso Poisso { N t, t}t, Poisso, X ( ( X Nt = ) Poisso,, t,, t, 3 ( X Nt = )U (, t)(, t] st, ( x Nt = ) P( x s N t = ) = P( x s, N t = ) P( N t = ) = = P(, s, ( s, t)) P( Nt = ) P(, s) P(( s, t]) P( N t = ) = P( N s = ) P( N t - s = ) P( N t = ) = se - s e - ( t - s ) te - t [, t] F( s), 9 Y, Y, Y, Y ( k ) Y,, Y k, k =,,, Y ( ), Y ( ), Y ( ) Y, Y,, Y Y i, i =,,, f, ( Y ( ), Y ( ),, Y ( ) ), = s t f( y, y,, y ) =! f ( yi ) (y < y < < y ) i = : ( ) ( Y ( ),, Y ( ) ( y,, y ), ( Y,, Y ) ( y,, y )! ; ( ) ( yi,, yi )( y,, y ), ( Y,, Y ) ( yi,, yi ) f ( yi )f( yi ) = f( y i ), ( Y ( ),, Y ( ) ) i =, Yi (, t],, f( y,, y ) =! t < y < < y < t 4 Nt =, s, s,, s (, t) ( S, S ) Nt = 6

= t < t < t f( t,, t ) =! / t, < t < t < t. < t t, P( ti - ti < Si ti, i Nt = ) =! = P( N t - t, t i i i =, N t, t - t i - i i =, i ; N t, t = )/ P( Nt = ) = ti e - t i i = = t i e - t i i = t i i = / t e - ( t - t ) e - ( t - t - t ) e - ( t - t ) / (e - t ( t) /!) e - t e i = t i / e - t t /! ti = ti - ti -, N t = ( S,, S ) f s,, s ( t,, t N t = ) = lim mxx i P( t i - ti < S iti,i N t = )/ t i i = =! / t ( ), (, t], S,, S,, U (, t) 5 Poisso, t, (, t ) Poisso, S, ( t - S ), i ( t - Si ), (, t ) Nt, t - t s t t N N t t ( t - S i ), E{ ( t i = i = - Si ) },, N t E ( t - Si ) Nt = i = = E ( t - Si ) Nt = i = = t - E Si N t i = N t =, Si, i =,, (, t ) U ( ),, U ( ), N t E i = ( t - Si ) Nt = E Si Nt i = = = = E U( i ) = E { i = i = Ui = t = t - t = t 6

N t E i = N t ( t - S i ) = E E ( t - Si ) N t = E N t t = t i = EN t = t, t, Poisso 6 i Xi ( i), N ( t), t, Xi, i, Si, i, t, X( t),, N( t) X ( t) = Xi e - ( t - s i ) i = Xi ( i), { Xi, i) } { N ( t), t }, { X( t), t} (5 ),, X ( t), N ( t), 3, N ( t) = (, t) E[exp{ sx( t)} N ( t) = ] = E exp sxi e - ( t - U i ) i = U, U (, t) E[exp{ sx( t) } N ( t) = ] = ( E [exp{ sxi e - ( t - ( u ) = E[e u X ]X = t ( se - y ) d y/ t E[exp{ sx( t) }] = e - = = e - t e t X ( t), t ( t)! U i ) } ] ) = exp{ t [( se - y ) - ] d y} ( 3 3) E[ X( t) ] = E [ X] ( - e - t )/ Vr[ X( t) ] = E [ X ] ( - e - t )/ Cov( X( t), X ( t + s) ), X ( t + s) = e - s X( t) + X ( s) X ( s)x ( s), X( t), X( s) ( t, t + s)t + s 6

Cov( X( t), X( t + s) = e - s Vr( X( t) ) (3 3 ) tx ( t) = e - s E[ X ] ( - e - t )/ lim E[exp{ sx( t) }] = exp{ t [( se - y ) - ] d y} Xi, lim E[exp{ sx( t) }] = exp t ( u) = - u = exp s d x - x = / - s - se - y - d y ( / (- s) ) / /, Xi, X( t) f( y) = e - y ( ( / ) y) / -, < y < ( 3 4), Xi, t Poisso, A( t), P{ A( t) > s X ( t) = y} = lim h P{ A ( t) > s y < X ( t) < y + h} = lim h = lim h P{ ye s < X( t - s) < ( y + h )e s, ( t - s, t) } P{ y < X( t) < y + h} f ( ye s )e s he - s + o( h) f ( y) h + o( h) = exp{ - y(e s - ) } ( 3 5), X( t) X ( t - s) ( 3 4) (3 5 ), X ( t) = y, A( t), ( s y), ( s y) = d P{ A ( t)s X( t) = y} d s P{ A ( t) > s X( t) = y} = ye s, t y ( ( y) = y),, t (X( t), ) 4, Poisso,, s, P( s), - P( s),, 4 63

7 Ni ( t) t i, i =,, N ( t)n ( t) Poisso, tp t( - p ),, EN ( t) = t P( s) d s p = t t P( s) d s Nt, N ( t), N ( t): P( N + t) =, N ( t) = m) = P( N ( t) =, N ( t) = m Nt = k) P( Nt = k) k = = P( N ( t) =, N ( t) = m Nt = + m) P( Nt = + m) [, t] s, P( s), 4 (, t), (, t) - p = t t P( s) d s, P( N ( t) =, N ( t) = m N t = + m ) + m m, p, P( N ( t) =, N ( t) = m N t = + m ) = ( + m P( N ( t) = N ( t) = m) = ( + m )!! m! p ( - p) m = e tp ( tp)!, e - t ( - p ) ( ) p ( - p) m ( t) + m ( + m)! e - t t( - p ) ) m m! 8 Poisso Poisso,, G t,, t s( st), t - s,, G, G( t - s), P( s) = G( t - s), st 7, N ( t) () E N ( t) = t G( t - s) d s = t Poisso N ( t) ( t )Poisso, EN ( t) = t ( - G( y) ) d y N ( t) N ( t) 64 9 { X ( t), t}{ X ( t), t} G( y) d y y O D x

Poisso W ( ) k X ( t) k, W ( ) X ( t), P( W ( ) k < W ( ) ), Poisso k Poisso W ( ) k W ( ) x) k - f ( ) k ( x) = e - x ( ( x) ( k - )! f ( ) ( y) = e - y ( y ) P( W ( ) k < W ( ) = f( x, y) d xd y D = { ( x, y) : x, y, y > x }f ( x, y) = f ( ) k ( x) f ( ) ( y), x D e - x ( x ) k - ( k - )! e - y d yd x = ( ) k + 3 Poisso Poisso, t ( )t 3 { Nt, t} Poisso, ( t), t, ( ) N = ( ) ( ) { Nt, t} (3 ) P( Nt + h - Nt ) = o( h) (4 ) P( Nt + h - Nt = ) = ( t) h + o( h) Nt + s P( N t + s - N t = ) m ( t) = t ( s) d s = exp{ - ( m( t + s) - m( t) ) }[ m( t + s) - m( t) ] /! =,, - N t m( t + s) - m( t)poisso Poisso, ( t), Poisso Poisso : ( t), t Poisso t ( t)/, ( t) Poisso, 3 ( ), ( ), (3 )Poisso, (4 ), P( N t + h - N t = ) = P(( t, t + h] ) = P(( t, t + h] )( t)/ + o( h) 65

( t) = h + o( h) = ( t) h + o( h) 3 X, X,, ( t) = f( t)/ ( - F ( t) ), f F X X > mx ( X, X - ) ( X ),, X Nt t, N t = I( x t) x, x { N t, t} Poisso, ( t), t t + h t Xi t t + h, ( Xi, i = ) P( X ( t, t + h) X > t) = ( t) h + o( h) Poisso A,,,, Nt Poisso,, Y ( t),, { Yt, t} N t Y ( t) = = N t 3 3 { Yt, t}poisso, Y t =, N t Poisso =, {, }, { N t, t}{, } N t 3 4 Yt = Poisso, = ( ) Y t ; ( ) Y t ( u) (Ee j u, j = Y ( t) = exp{ t(( u ) - ) } (3 )E <, E Yt = te, D Yt = te ( )t - ), ; < t < < t m, Y t N t = = N tk Y t k - Y t k - = = N + tk - k =,, mp { N t, t}{, }, Yt 66 (4 ) Y ( t) = Ee j u Y ( t)

(5 ) = E {e j u Y ( t ) N t = } P( N t ) = ) = = E = = E exp = exp(j u k ) Nt = e - t ( t)! k = j uk k = = ( Ee j u ) e - = = ( u ) = t ( t)! e - = exp{ t( ( u ) - ) } t ( t)! e - N t E{ Y t N t = } = E = E k = = E k N t = k = t ( t)! k Nt = = E k k = E Y t = E{ E ( Yt Nt ) } = E{ N te} = t E N t D( Y t N t = ) = D D( Y t N t ) = N t D D Y t DY t = E{ D( Y t N t ) } + D{ E ( Yt Nt ) } k = k Nt = = D k = D k = = E{ N t D+ D( Nt E ) } = td+ t( E ) = t( D+ E ) = te 3 5 t Nt Poisso, N t, Y t = [, t),,, Yt Poisso = 3 6 Nt Poisso, Y ( Y ), Xt, t), Y 67

EXt, DXt Xt X t ( u ) f Y ( y) = e - y ( y) (> ) N t Xt = Y Poisso, 3 4 = EXt = te Y = t/ E Y = /, E Y = /, 3 4 ( ) X t ( u ) = Ee j u X t DXt = t E Y = t/ ( u) = - j u = exp{ t( Y ( u) - ) } = exp t - j u - 3 N ( t)poisso, s < t, P( N ( s) = K N ( t) = ) { N ( t), t} Poisso, s >, E { N ( t) N ( t + s) } 3 3 Poisso, ( ) 8? ( )? 4 { N ( t), t} = Poisso, ( ) P( N ( ) ) ; ( ) P( N ( ) = N ( ) = 3) ; ( 3) P( N ( ) N ( )) 56 4, Poisso 6 N ( t) Poisso, N ( t) =, r( r) Sr fs r N ( t ) = ( s ) 7 { N i ( t), t}, i =,, Poisso, T, T 8Poisso N ( t), p, N ( t) N ( t)? N ( t) - N ( t)?? 9 Poisso N ( t), Poisso N ( t), N ( t)n ( t) N ( t) = N ( t) + N ( t)? N ( t) ( shock model)n ( t) t, Poisso k Y k, Y k ( k =,, ) X( t),, T E T( : E T = P( T > t)d t) 68

4 Poisso,, Poisso { X, =,, }, F( x ) T ( - ), S = Ti i = S =, Nt = sup{ S t} ( 4.) { N t, t}(, { S, =, }{ X, =, } ), X F( x), F( ) = P ( X = ) <,,, ( ), ( ), T ( =, ), { T, }N t (, t],,, ( ) T, T,, S T,, :?,,, S 69

= E T = xd F( x ) T F( ) <, <, S >, S S t,, ( 4 ) N t, N t = mx{ S t} N t N t m ( t ) E Nt { N t, t}t, T, F,, F( ) <, E T = xd F( x ) > : t >, N t (, t] m( t) = E Nt,, { Nt, t}, Nt P( N t = ) = F ( ) ( t) - F ( + ) ( t) m ( t) = F ( ) ( t) = F ( ) ( t)f( t), F( t) Ti, t N t t m ( t), ( N t ) ( S t) P( N t = ) = P( Nt ) - P( Nt + ) I = Nt = I, = 7 = P( S t) - P( S + t) = F ( ) ( t) - F ( + ) ( t) [, t], m( t) = EN t = E( = = = I ) = = P( I = ) = = EI P( S t) = = F ( ) ( t)( )

, Poisso Poisso Nt, T, T, E (), 4 F ( ) ( t) = P( S t) = P( N t ) = e - j = t ( t) j j! P( N t = ) = P( Nt ) - P( Nt + ) = F ( ) ( t) - F ( + ) ( t) = e - j = = e t ( t)! t ( t) j j! m ( t) = F ( ) ( t) = = = = j = = j = = = t e - j = e - = e - t t k = - e - j = + t ( t) j j! t ( t) j j! ( t) j ( j - )! e - t ( t) k k! t ( t) j j! 3 N { N t, t}, T < ( P ( T = ) =, ) ; N N, T, >, T {,,, }, P( T = k) = k = T N T, F, >, N, T F, F( )= t lim F( t) =, N, F, F() <, N N t t, Nlim t N t ( ), Nt 3 3 N, 7

P( N < ) = P( T = ) = P( ( T = ) ) = P( T = ) = N= N, P( T = ) =, P( N< ) =, P( N= ) = Nt N t m()lim t m( t) = lim E N t = E( lim N t t t ) =, t, N ( t), N ( t), N ( t) lim t t N ( t), S N( t ), N ( t) = 3, SN ( t) = S3, 3 t 3, S3 t ( t), S N( t ) : t t SN ( t ) + t N t SN ( t ) + 3 t, S N( t ) t < SN ( t ) +, SN ( t) Nt N ( t) t t N t < S N ( t ) +, SN ( t)/ N ( t) N ( t),, N ( t), SN ( t)/ N ( t) tn ( t), t Nt t SN ( t ) + N ( t) SN ( t ) N ( t) = S N( t) + ( t) + N N ( t) + N ( t) S N( t ) + N ( t), t/ N t, t ( ) :,, /, m( t)/ t /, Wld 7

3 3 X X, N X X,, =,,, { N = }X +, X +,, X, X,, N N =, X, X X +, X +, 3 4 ( )X, =,,, P{ X = } = P{ X = } =, =,, N = mi{ X + X = } N N, ( )X, =,,, P{ X = - } = P{ X = } = N = mi{ X + + X = },,, N 3 5 ( Wld ) X, X,,, N X, X,, E( N ) <, N E X = E [ N] E [ X] = Lebesgue N E X = I = N = N, N < X = = = E X I = X I = E[ X I ] = I = X, X,, X - I X - X N E X = = E[ X ] E[ I ] = = E[ X ] E [ I ] = = E[ X ] P{ N} = = E[ X ] E [ N] X,, 73

3 4 ( ), Wld E[ X + + X N ] = EN, N X + + X N =, E [ N] = Wld 3 4 () E[ X + + X N ] = E[ N] E [ X ] X + + X N = E [ X ] =, Wld, E [ N] = 3 6 X, X, N ( t), N ( t ) + { X } ( t N ( t) + ) N ( t) + = N ( t) = - X + + X - t, X + + X N > t { N ( t) + = } X,, X, X +, ; N ( t) + EX = <, E[ S N( t ) + ] = [ m( t) + ] ( 4 ), 3 7 ( ) E[ X + + X N ( t ) + ] = E[ X] E[ N ( t) + ] m ( t) t <, ( ) SN ( t ) + > t [ m ( t) + ] > t limif t m( t) t, M, { X, =,, } : X = X X M, =,,, M X > M ( 4 3) S = X i N ( t) = sup{ S t } M i =, SN ( t ) + t + M (4 ) M = E[ X ], [ m ( t) + ]Mt + M limsup t m( t) t M 74

S S, N ( t)n ( t)m ( t)m( t), M (4 3 )( 4 5) limsup t limsup t m( t) t m( t) t M ( 4 4) ( 4 5) =,, M M, (4 5 ) 4,, Blckwll,, ( W Feller, A Itroductio to Problity Theory d Its Appli ctios, Vol.I d, Wiley, New York,957 d 966.) { N ( t), t}, X, X, F, 4 ( Blckwell) ( )F,, t m( t + ) - m( t) / ( )F, d, E[d ]d/, F, /,, g ()lim t [ m ( t + ) - m( t) ] ( 4 6),, /, g( + b) = t lim [ m( t + + b) - m ( t) ] = t lim [ m( t + + b) - m ( t + ) + m( t + ) - m ( t) ] = g( b) + g( ), g( + b) = g( ) + g( b)() c c = /, g( ) = c, > x = m( ) - m( ) x = m( ) - m( ) x = m( ) - m ( - ) lim x = c 75

,, c = / x + + x lim lim m ( ) F, d, (4 6 ) d,, d,, d, lim E [ d ], d/, Blckwell ( ), = c = c lim P{d } = d/ h [, ] >, m ( )m ( )h( t) ( - ) t >, = lim m ( ) = lim = = m ( ) h h ( )t, h( t) ; ( ) h( t) ; (3 ) h( t) d t < m ( ) 4 ( )F, h( t), lim t t + h( t - x) d m( x) = h( t) d t m ( x ) = = + F ( x) = F( t) d t = m ( ),,Blckwell :Blckwell,, lim m( t + ) - m ( t) lim t m ( t + ) - m( t) t lim d m ( t) lim t d t = = =, t 76

g( t) ( t ), : t t g( t) = h( t) + h ( t - x) d m( t) lim t g ( t), t SN ( t) 4 3 { N ( t), t} F( x ), m ( t) = E N ( t), F( x) = - F( x ), SN ( t ) s P( SN ( t) s) = F( t) + F( t - y) d m( y) st P( SN ( t ) s) = P( S s, S + > t) = = F( t) + P( S s, S + > t) = = = F( t) + s = = F( t) + F( t - y) d F ( y) s = F( t) + P( S s, S + > t S = y) d F ( y) F( t - y) d( = F ( y) ) = F( t) + s F( t - y) d m( y) ( ) P( SN ( t ) = ) = F( t), F, f, y > d m( y) = f ( y) d y =, d FS N( t) ( y) = F( t - y) d m ( y), < y < m ( y) = F ( y) = = P( ( y, y + d y) ) = = P( ( y, y + d y)) f S N( t) ( y) d y = P(y, y + d y), > t - y) 4 4 ( ) = d m ( y) F( t - y) 77

, : Z ; Y ;, Z ;, Y ; ( Z, Y ),,, { Z } { Y } ;Z Y,,, H Z, G Y, F Z + Y,, P( t) = P{ t } ( )E[ Z + Y ] <, F, lim P( t) = E( Z ) t E [ Z ] + E[ Y ] ( 4 7), t ( t ) y > t P( t) = P{ t SN ( t ) = } P{ S N( t ) = } + + P{ t S N( t ) = y}d FsN ( t ) ( y) P{ t SN ( t ) = } = P{ Z > t Z + Y > t} = H( t)/ F( t) P{t SN ( t ) = y} = P{ Z > t - y Z + Y > t - y} = H( t - y)/ F( t - y), 4 3 t P( t) = H( t) + H( t - y) d m( y) m ( y) = = F ( y)h( t), + t, H( t), + H( t)d t P( t) F Q( t) = P{t } = - P( t), = Q( t) E[ Y ] E[ Z] + E[ Y ] E[ Z ] E[ Z ] + E[ Y ], H( t) d t = E [ Z] < (4 7 ),,, Y ( t) t, A ( t)[, t], Y ( t) = S N( t) + A ( t) = t - SN ( t) Y ( t) t, A( t)t, A ( t) t Y 78 - t

( t), ( ) F, <, lim P( Y ( t)x ) = lim P( A ( t)x) = t x F( y) d y/ ( 4 8),, t x, t, x, F, (4 7 ) lim t P( A ( t)x) = Emi ( X, x) / EX = P( mi ( X, x) > y) d y/ EX = F( y) d y/, x, mi( x, X ), lim P( Y ( t)x ) = lim P(t ) t t = E [ mi ( x, X) ]/ E[ X] (4 8 ) x = F( y) d y/ 4 { N t, t}, f ( t) = ( t) k - e - t / ( k - )! ( t ), S Nt { N ( t ), t}{ N ( t ), t}, f g { Nt, t}, N t { N ( i ) t, t}, i =, Poisso 79

5 M rkov Mrkov : ( )Mrkov, Mrkov ; ( )Mrkov, Mrkov ; (3 )Mrkov Mrkov Mrkov Mrkov 96 Mrkov, Kolmogorov Feller Doob, t Xt, Xs ( s > t) Xu ( u < t), Xt Mrkov { X, =,,, }, E,, {,,, }X = i, i, i, j Pij i, i,, i -, i, j P( X + = j X = i, X - = i -,, X = i, X = i ) = Pij Mrkov, Mrkov Mrkov Pij i j,, P Pi j, Pij, i, j, Pij =, i =,,, j = P P P P j P P P P j P = Pi Pi Pi Pij, Mrkov, Mrkov, Mrkov P i + X = i, X + Mrkov, P 8

( ), j ( j ), j + p, j - q = - p,, X X Mrkov, p k = j +, Pjk = q k = j -, o p = q =, ( )Mrkov E = {, },,, p ;, q, X, { X, } Mrkov, P = - p p q - q (3 ),,, Y, Y : P( Y = k) = pk k =,,,, pk = k X, X + = X - + Y X, { X, } Mrkov, P = Y X = p p p p3 p4 p p p p3 p4 p p p p3 p p p P = P( X P = P( X = X = ) = P( Y = ) = p = X = ) = P( Y = ) = p P = P( X + = X = ) = P( X - + Y = X = ) = P( Y = ) = p P = P( X + = X = ) = P( X - + Y = X = ) = P( Y = ) = p (4 ), 4, T,,, 3, 4,, 3, 3,,, 4, 3 X, { X, T } Mrkov, 8

P = 3 3 3 3 3 3, E = {,,3,4}, P Pij i j, P =,, ; P3 = 3,, 3 3 ; P43 =, 4, 3, 5 4, ; 4,, (5 )X, 3 / 3 5 3 / 3 3 3 E = {,,, } i, ( ), Pij = P( X + = j X = i) = pi j = i +, ri j = i, qi j = i - pi + ri + qi = ie i =, P =, { X, T } Mrkov, q r p P = q r p, k,,, X i, pi j = i +, 8 Pij = qi j = i -,

pi + qi =, i =,,, P = Mrkov P = q p q p k N - k, X Mrkov E = {,,, N}, P = q p q p q - p - 3 Mrkov Pi P( X P = ( Pij ) P( X = i, X = i,, X = i ) = i) i =,, = P( X = i, X = i,, X - = i - ) P( X = i X = i,, X - = i - ) = P( X = i, X - = i - ) P( X = i X - = i - ) = P( X = i,, X - = i - ) Pi - i = Pi Pi, i Pi, i Pi - i ( ) Chpm - Kolmogorov Mrkov m P ( ij m ) P ( m ) i j = P( X + m = j X = i) Mrkov,, P ( m ) ij,, i, m + m j P ( ) ij i j P ( ), Mrkov P ( ) ( Chpm - Kolmogorov ) m,, P ( m + ) = P ( m ) P ( ) P ( ) = PP ( - ) P ( ) ij = P( X = i X = i) = P( X = j, X = k X = i) k = 83

= P( X = k X = i) P( X = j X = i, X = k) k = = Pi k P ( k j - ) k = ( ) P ( ) = PPP,, Pij, X = i Pij, pi = P( X = i) ie E, { pi }, ie, pi pi = ie, Mrkov { pi }, ie P ( ) Beroulli, : E = {, }, X, Mrkov, P( X = ) = - p = q, P( X = ) = p( < p < ) P = q p P = P = q P = P = p q p P ( ) = q p q p ( ),, =, = E = {,}Mrkov, P = P P P P = = P - - 4 ( = 7,= 4) P = - - = 7 3 4 6 6 39 P ( ) = PP = 5 48 5749 45 P ( 4 ) = ( P ( ) ) = 5668 433 P(4 ) = P 4 = 5749 (3 ) E = {,, } Mrkov { X, }, P = p q r p, q, r >, p + q + r = Mrkov, 84