Асимптотические оценки. Мицик М.Ф. - 5 стр.

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
( )
λ
2
F
. 
)(xS
 [a;b] 
, 
( )
caSxS −≤)(

b
≤
≤
+
xa
δ
, 
−
>
0c
.  1 
λ

( )
( )
( )
caS
eF
−
≤
λ
λ
02
,

( )
λ
2
F

aS
e
λ
.

( )
λ
1
F
:
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
xd
xS
xf
dx
d
e
xS
exf
ed
xS
xf
F
a
a
xS
a
a
xS
xS
a
a
∫∫
+
+
+
′
−
′
=
′
=
δ
λ
δ
λ
λ
δ
λλλ
λ
1
|
1
(10)

δ
+
=
ax


( )
aS
e
λ

∞
→
λ
, 
( )
0)( <−+ aSaS δ
.
. 
[ ]
δ
+
aa;

( )
0
<
′
xS

0
1
>
S
, , 
( )
1
SxS
−
≤
′

[ ]
δ
+
∈
aax ;
. 
( ) ( ) ( )
ξδ SaxaSaS
′
−=−+ )(
, 
( )
δξ +∈aa;
,
() ( )
axSaSaS −−≤−+
1
)( δ

[ ]
δ
+
∈
aax ;
.

( )
( )
( )
′
=
xS
xf
dx
d
xf
1
:
( )
11
Mxf ≤

[ ]
δ
+
∈
aax ;
. ,
( )
( )
( )
( )
( )
( ) ( )( )
≤≤
′
∫∫
+
−−
+
xdexfexd
xS
xf
dx
d
e
a
a
aSxSaS
a
a
xS
δ
λλ
δ
λ
1
( )
λλ
δ
λ
1
1
1
1
1
c
S
M
xdeM
a
a
axS
=<≤
∫
+
−−

∞
→
λ
.
 (10) 
( )
( )
( )
( )
( )
+
′
−
=
−2
1
λ
λ
λ
λ
O
aS
af
eF
aS

∞
→
λ
.

( )
λ
2
F
 (9)  (8).
 5.       

+∞
→
p
,
( ) ( )
dxexfpF
px−
∞
∫
=
0
,
                                     F2 (λ ) .                   S (x)                                           [a;b]
                                                    ,    S ( x ) ≤ S (a ) − c                               a +δ ≤ x ≤ b,
c >0−                                    .                1                                                       λ
                                                   F2 (λ ) ≤        0   e λ ( S ( a )−c ) ,
                                         F2 (λ )                                                                              e λS (a ) .
                       F1 (λ )                                                         :
                         f (x )                 f ( x )e λ S ( x ) a +δ 1                                           d  f (x ) 
                 a +δ                                                                          a +δ
F1 (λ ) =          ∫    λS ′( x )
                                  d e λ S (x)
                                             (=     )
                                                  λS ′( x ) a λ
                                                                     | −                         ∫    eλ S (x )                    d x (10)
                                                                                                                    dx  S ′( x ) 
                   a                                                                             a

                                                                        x = a +δ
                            e λS ( a )            λ →∞,                             S (a + δ ) − S (a ) < 0 .
                                                                    .                            [a; a + δ ]
S ′( x ) < 0                                                                             S1 > 0 ,                    ,
S ′( x ) ≤ − S1                   x ∈ [a; a + δ ] .
             S (a + δ ) − S (a ) = ( x − a )S ′(ξ ) ,                            ξ ∈ (a; a + δ ) ,
             S ( a + δ ) − S (a ) ≤ − S1 ( x − a )                             x ∈ [a; a + δ ] .
                                                                                  d  f (x ) 
                                                                f1 ( x ) =                       
                                                                                  dx  S ′( x ) 
                                                                                                                          :

              f1 ( x ) ≤ M 1                     x ∈ [a; a + δ ] .                                           ,

                       d  f (x ) 
a +δ                                                           a +δ

                                                                ∫ f (x ) e
           λ S (x)
 ∫     e                           d x e −λS (a ) ≤                             λ ( S ( x )− S ( a ))
                                                                                                            dx ≤
                       dx  S ′( x ) 
                                                                         1
 a                                                              a

            a +δ
                                                 M 1 c1
≤ M 1 ∫ e −S1 ( x −a )λ d x <                       =                    λ →∞.
             a
                                                 S1λ λ
                                                                         (10)
                                    f (a )             
             F1 (λ ) = e λ S ( a )             + O λ−2        ( )                        λ →∞.
                                    − λS ′(a )         
                         F2 (λ )                    (9)                                                            (8).
                  5.
                                                                                                      ∞

                                                              p → +∞ , F ( p ) = ∫ f ( x ) e dx ,
                                                                                            − px

                                                                                                      0

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