(X

n

)

0

F

0

= {∅, Ω}

F

n

= σ(X

1

, X

2

, . . . , X

n

)

n = 1, 2, . . .

Y

n

n ≥ 1

Y

n

F

n−1

(M

n

), (N

n

)

M

n

=

n

k=1

X

k

Y

k

, N

n

=

n

k=1

X

k

, n = 1, 2, . . . .

(M

n

)

(F

n

)

!

"

#

$

(N

n

)

%

n

|Y

n

| ≤ 1

&

E max

1≤k≤n

|M

k

|

2

≤ 4E|N

n

|

2

, n = 1, 2, . . . .

(X

n

)

P (X

n

= 1) =

1

3

= 1 −P (X

n

= −1)

n = 1, 2, . . .

'

&

(F

n

) = σ(X

1

, X

2

, . . . , X

n

)

(

τ = inf{n ≥ 3 : X

n

≥ X

n−1

≥ X

n−2

},

σ = inf{n ≥ 2005 : X

1

= X

2

+ X

3

+ . . . + X

n

},

η = inf{n ≥ 5 : X

n

≥ −X

n

+ X

n+1

},

γ = inf{n ≥ 3 : X

n

+ X

n−1

> X

n+1

}.

(ε

n

)

P (ε

n

= ±1) =

1

2

n = 1, 2, . . .

τ

τ = inf{n ≥ 4 : ε

1

+ ε

2

+ ε

3

= 2005 + ε

4

+ ε

5

+ . . . + ε

n

}.

%

τ

)

(σ(ε

1

, ε

2

, . . . , ε

n

)),

n = 1, 2, . . .

%

P (τ < ∞) = 1

Eτ.

(X

n

)

P (X

n

=

1

2

) = P (X

n

=

3

2

) =

1

2

τ

τ = inf{n : X

1

X

2

. . . X

n

<

1

100

}.

%

τ

)

(σ(X

1

, X

2

, . . . , X

n

)),

n = 1, 2, . . .

%

P (τ < ∞) = 1

(X

n

)

&

n ≥ 1

X

n

*$

&

n,

1

n

+

"

(α

n

)

S

n

=

n

k=1

X

k

− α

k

,

L

2

-