设为首页
加入收藏
期刊导航
网站地图
首页
期刊
数学与物理
地球与环境
信息通讯
经济与管理
生命科学
工程技术
医药卫生
人文社科
化学与材料
会议
合作
新闻
我们
招聘
千人智库
我要投稿
办刊
期刊菜单
●领域
●编委
●投稿须知
●最新文章
●检索
●投稿
文章导航
●Abstract
●Full-Text PDF
●Full-Text HTML
●Full-Text ePUB
●Linked References
●How to Cite this Article
PureMathematics
n
Ø
ê
Æ
,2021,11(12),2023-2030
PublishedOnlineDecember2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1112226
@
Ï
Ê
Ž
½
n
3
‘
Å
i
Ä
¥
A^
ëëë
®
é
Ü
Œ
Æ
§
ê
n
†
‰
Æï
Ä
§
®
Â
v
F
Ï
µ
2021
c
11
7
F
¶
¹
^
F
Ï
µ
2021
c
12
8
F
¶
u
Ù
F
Ï
µ
2021
c
12
15
F
Á
‡
Diaconis
Ú
Fill
|
^
@
Ï
Ê
Ž
½
n
,
‰
Ñ
Z
+
þ
‘
Å
i
Ä
Â
ñ
²
-
©
Ù
„
Ý
O
ž
Ñ
y
†
Ø
§
©
Ø
=Å
ù
‡
†
Ø
§
…
|
^
Markov
Ø
ª
Ú
@
Ï
Ê
Ž
½
n
§
•
‰
Ñ
Z
+
þ
‘
Å
i
Ä
Â
ñ
²
-
©
Ù
„
Ý
O
"
'
…
c
‘
Å
i
Ä
§
@
Ï
Ê
Ž
½
n
§
r
²
-
é
ó
§
‘
Å
›
›
§
‘
Å
ü
N
TheApplicationoftheEarlyStopping
TheoreminRandomWalk
PanZhao
InstituteofFundamentalandInterdisciplinarySciences,BeijingUnionUniversity,Beijing
Received:Nov.7
th
,2021;accepted:Dec.8
th
,2021;published:Dec.15
th
,2021
Abstract
Byusingtheearlystoppingtheorem,DiaconisandFillmadeamistakewhendealing
withtheconvergencetostationarityforarandomwalk.Inthepaper,wenotonly
©
Ù
Ú
^
:
ë
.
@
Ï
Ê
Ž
½
n
3
‘
Å
i
Ä
¥
A^
[J].
n
Ø
ê
Æ
,2021,11(12):2023-2030.
DOI:10.12677/pm.2021.1112226
ë
correctthemistake,butalsogivethespeedestimationofconvergencetostationarity
fortherandomwalk,byusingtheMarkovinequationandtheearlystoppingtheorem.
Keywords
RandomWalk,TheEarlyStoppingTheorem,StrongStationaryDuality,
StochasticallyDominate,StochasticallyMonotone
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.
Ú
ó
Ú
Ì
‡
(
J
Diaconis
Ú
Fill[1]
J
Ñ
r
²
-
é
ó
(SSD)
´
ï
Ä
ê
¼
ó
Â
ñ
„
Ý
˜
‡
k
å
ó
ä
.
·
‚
Œ
±
Ï
L
•
N
´
ï
Ä
é
ó
ó5
O
ê
¼
ó
Â
ñ
²
-
©
Ù
Â
ñ
„
Ý
.SSD
ä
k
é
õ
-
‡
A
^
,
ä
N
Œ
ë
•
©
z
[2–5].
é
u
G
˜
m
•
l
Ñ
8
œ
/
,Diaconis
Ú
Fill[1]
‰
Ñ
˜
‡
E
SSD
ó
•{
.
A
O
/
,
é
k
•
S
G
˜
m
,Diaconis
Ú
Fill
3
[1]
¥
‰
Ñ
˜
‡
N
´
¦
SSD
ó
œ
/
.
3
d
œ
/
¥
,
3
ž
m
_
ê
¼
ó
´
‘
Å
ü
N
b
e
,
¦
‚
3
Ó
˜
‡
G
˜
m
e
E
SSD
ó
.
Diaconis
Ú
Fill[6]
ò
ù
«
A
Ï
œ
/
í
2
Œ
ê
,
S
G
˜
m
Z
+
þ
l
Ñ
ž
m
ê
¼
ó
,
E
Ù
SSD
ó
.
¦
‚
|
^
@
Ï
Ê
Ž
½
n
,
Ï
L
ï
Ä
é
ó
ó
Â
¥
ž
©
Ù
5
O
ê
¼
ó
Â
ñ
²
-
©
Ù
Â
ñ
„
Ý
.
Ù
¥
,
¦
‚
3
1
3
!
‰
Ñ
Z
+
þ
‘
Å
i
Ä
ù
‡
~
f
¥
,
Ñ
y
†
Ø
,
©
Ø
=Å
ù
‡
†
Ø
,
…
|
^
Markov
Ø
ª
Ú
@
Ï
Ê
Ž
½
n
,
•
‰
Ñ
‘
Å
i
Ä
Â
ñ
²
-
©
Ù
„
Ý
O
.
-
X
=(
X
n
)
n
≥
0
•
±
E
=
{
0
,
1
,...
}
•
G
˜
m
l
Ñ
ž
m
ê
¼
ó
,
=
£
Ý
•
P
,
d
[7]
•
,
P
´
‘
Å
ü
N
…
=
é
?
¿
y
∈
E
,
P
z
≥
y
P
(
x,z
)
'
u
x
´
ü
N4
O
.
w
,
,
P
´
‘
Å
ü
N
,
d
u
é
?
¿
y
∈
E
,
P
z
≤
y
P
(
x,z
)
'
u
x
ü
N4
~
.
©
-
E
=
{
0
,
1
,...
}
,E
∗
=
E
∪{∞}
.
-
X
= (
X
n
)
n
≥
0
•
½
Â
3
V
Ç
˜
m
(Ω
,
F
,
P
)
þ
l
Ñ
ž
m
ê
¼
ó
,
Ð
©
©
Ù
•
π
0
,
=
£
Ý
•
P
.
·
‚
{
ü
P
•
X
∼
(
π
0
,P
).
·
‚
Ä
k
5
Q
ã
Diaconis
Ú
Fill[6]
¥
k
'
SSD
ó
•
3
5
9
A^
5
Ì
‡
(
J
.
½½½
nnn
1.
[6]
X
J
-
X
∼
(
π
0
,P
)
•
˜
‡
Ø
Œ
,
š
±
Ï
,
H
{
l
Ñ
ž
m
ê
¼
ó
,
²
-
©
Ù
•
π,
G
˜
m
•
E
.
½
Â
P
ž
m
_
•
←−
P
(
x,y
)=
π
(
y
)
π
(
x
)
P
(
y,x
)
,
-
H
(
x
)=
P
y
∈
E
:
y
≤
x
π
(
y
),
x
∈
E
∗
;
g
(
x
)=
π
0
(
x
)
π
(
x
)
,x
∈
E
.
@
o
,
'
u
'
é
Ý
:Λ(
x
∗
,x
)=
I
(
x
≤
x
∗
)
π
(
x
)
H
(
x
∗
)
,x
∗
∈
E
∗
,x
∈
E,
•
3
±
E
∗
•
DOI:10.12677/pm.2021.11122262024
n
Ø
ê
Æ
ë
G
˜
m
SSD
ó
X
∗
∼
(
π
∗
0
,P
∗
)
¿
©
7
‡
^
‡
´
±
e
ü
‡
^
‡
Ó
ž
¤
á
(a)
π
0
(
x
)
π
(
x
)
'
u
x
ü
N4
~
;
(b)
←−
P
´
‘
Å
ü
N
.
d
ž
,SSD
ó
X
∗
∼
(
π
∗
0
,P
∗
)
d
±
e
•
˜
û
½
:
π
∗
0
(
x
∗
) =
H
(
x
∗
)
h
π
0
(
x
∗
)
π
(
x
∗
)
−
π
0
(
x
∗
+1)
π
(
x
∗
+1)
i
,x
∗
∈
E,
π
∗
0
(
∞
) =lim
x
→∞
π
0
(
x
)
π
(
x
)
;
P
∗
(
x
∗
,y
∗
) =
H
(
y
∗
)
H
(
x
∗
)
h
P
y
∗
(
←−
X
1
≤
x
∗
)
−
P
y
∗
+1
(
←−
X
1
≤
x
∗
)
i
,x
∗
∈
E
∗
,y
∗
∈
E,
P
∗
(
x
∗
,
∞
) =
1
H
(
x
∗
)
lim
y
→∞
P
y
(
←−
X
1
≤
x
∗
)
,x
∗
∈
E
∗
.
½
Â
C
•
k
π
n
−
π
k
:= max
A
⊂
E
|
π
n
(
A
)
−
π
(
A
)
|
,
Ù
¥
π
n
´
X
n
©
Ù
.
@
Ï
Ê
Ž
½
n
Œ
±
Ï
L
©
Û
SSD
ó
Â
¥
ž
©
Ù
5
O
ê
¼
ó
Â
ñ
²
-
©
Ù
Â
ñ
„
Ý
.Diaconis
Ú
Fill[6]
ò
§
ä
N
A^
½
n
1
¥
ê
¼
ó
,
X
e
í
Ø
.
ííí
ØØØ
1.
[6]
é
u
½
n
1
¥
ó
(
X
∗
,
X
),
-
T
∗
•
X
∗
Ä
g
Â
¥
{
x
∗
,x
∗
+1
,...
}∪{∞}
Â
¥
ž
,
@
o
k
π
n
−
π
k≤
(1
−
H
(
x
∗
))+
H
(
x
∗
)
P
{
T
∗
>n
}
.
du
é
ó
ó
Â
¥
ž
©
ÙØ
N
´
O
Ž
,Diaconis
Ú
Fill[6]
‰
Ñ
X
e
‘
Å
Œ
'
Ú
n
,
^
§
5
O
é
ó
ó
Â
¥
ž
©
Ù
.
ÚÚÚ
nnn
1.
[6]
e
P
1
Ú
P
2
•
½
Â
3
E
∗
þ
=
£
¼
ê
,
K
X
e
ü
‡
^
‡
d
:
(a)
é
?
¿
0
≤
x
1
≤
x
2
≤∞
,y
∈
E
∗
,
k
P
0
≤
z
≤
y
P
1
(
x
1
,z
)
≥
P
0
≤
z
≤
y
P
2
(
x
2
,z
);
(b)
®
•
V
Ç
ÿ
Ý
π
(1)
0
Ú
π
(2)
0
÷
v
:
é
?
¿
y
∈
E
∗
,
k
P
0
≤
z
≤
y
π
(1)
0
(
z
)
≥
P
0
≤
z
≤
y
π
(2)
0
(
z
),
@
o
•
3
½
Â
3
ƒ
Ó
V
Ç
˜
m
þ
ê
¼
ó
X
(
i
)
=(
X
(
i
)
n
)
n
≥
0
∼
(
π
(
i
)
0
,P
i
)
,i
=1
,
2
,
¦
é
?
¿
n
≥
0
,
k
X
(1)
n
≤
X
(2)
n
.
^
‡
(a)
½
(b)
¤
á
ž
,
¡
P
2
‘
Å
›
›
P
1
,
P
•
P
1
≤
P
2
.
ÚÚÚ
nnn
2.
[6]
-
P
1
Ú
P
2
•
½
Â
3
E
∗
þ
=
£
¼
ê
,
X
J
(a)
P
1
½
P
2
´
‘
Å
ü
N
;
(b)
é
?
¿
x,y
∈
E
∗
,
k
P
0
≤
z
≤
y
P
1
(
x,z
)
≥
P
0
≤
z
≤
y
P
2
(
x,z
).
DOI:10.12677/pm.2021.11122262025
n
Ø
ê
Æ
ë
@
o
,
P
1
≤
P
2
.
b
ü
‡
ê
¼
ó
X
(
i
)
∼
(
π
(
i
)
0
,P
i
),
i
=1
,
2,
÷
v
é
?
¿
y
∈
E
∗
,
k
P
0
≤
z
≤
y
π
(1)
0
(
z
)
≥
P
0
≤
z
≤
y
π
(2)
0
(
z
),
¿
…
P
1
≤
P
2
,
@
o
d
‘
Å
Œ
'
Ú
n
1
,
P
(
T
2
>n
)
≤
P
(
T
1
>n
)
,n
≥
0
.
Ù
¥
,
T
i
•
X
(
i
)
Ä
g
Â
¥
{
x,x
+1
,...
}∪{∞}
Â
¥
ž
,
Ï
d
,
X
J
Ž
|
^
‘
Å
Œ
'
Ú
n5
O
é
ó
ó
Ä
g
Â
¥
ž
©
Ù
,
'
…
´
Ï
é
˜
‡
´
u
©
Û
ê
¼
ó
,
¦
é
ó
ó
‘
Å
›
›
§
.
[6]
¥
¤
Þ
k
'
‘
Å
i
Ä
~
f
,
3
Ï
éù
‡
é
ó
ó
‘
Å
›
›
ê
¼
ó
ž
Ñ
y
†
Ø
.
[6]
¥
¤
Þ
‘
Å
i
Ä
.
X
e
.
-
X
∼
P
•
l
Ñ
ž
m
{
ü
‘
Å
i
Ä
,
G
˜
m
•
E
=
{
0
,
1
,
2
,
···}
.
-
0
<p<
1
,q
:= 1
−
p
,
¿
…
,
é
?
¿
0
<x<
∞
,
k
P
(
x,x
−
1) =
q,P
(
x,x
) = 0
,P
(
x,x
+1) =
p
;
P
(0
,
0) =
q,P
(0
,
1) =
p.
(1)
du
P
(
x,x
+1)+
P
(
x
+1
,x
)
≤
1
,
T
ó
´
‘
Å
ü
N
.
´
•
T
ó
š
±
Ï
.
b
0
<p<
1
2
,
K
T
ó
~
ˆ
,
²
-
©
Ù
•
:
π
(
x
) = (1
−
p/q
)(
p/q
)
x
,x
∈
E.H
(
x
∗
) =
P
y
≤
x
∗
π
(
y
) = 1
−
(
p/q
)
x
∗
+1
.
d
[6]
•
,
b
ó
X
l
0
Ñ
u
,
K
d
½
n
1
•
,
é
ó
ó
•
´
l
0
Ñ
u
)
«
ó
,
¿
…
=
£
Ý
P
∗
X
e
:
é
?
¿
0
<x
∗
<
∞
,
k
P
∗
(
x
∗
,x
∗
−
1) =
H
(
x
∗
−
1)
H
(
x
∗
)
p,P
∗
(
x
∗
,x
∗
)
∗
= 0
,P
∗
(
x
∗
,x
∗
+1) =
H
(
x
∗
+1)
H
(
x
∗
)
q
;
P
∗
(0
,
0) = 0
,P
∗
(0
,
1) =1
.
½
Â
=
£
Ý
P
0
•
:
é
?
¿
0
<x<
∞
,
k
P
0
(
x,x
−
1) =
p,P
0
(
x,x
) = 0
,P
0
(
x,x
+1) =
q
;
P
0
(0
,
0) = 0
,P
0
(0
,
1) = 1
.
[6]
¥•
Ñ
:
é
ó
ó
‘
Å
›
›
±
P
0
•
=
£
Ý
‘
Å
i
Ä
,
´
·
‚
u
y
ù
´
†
Ø
.
Ï
•
P
0
(0
,
0)
≤
P
∗
(1
,
0),
¿Ø
÷
v
‘
Å
Œ
'
Ú
n
1
¥
(a).
e
5
,
·
‚
Ä
k
Ï
é
é
ó
ó
‘
Å
›
›
‘
Å
i
Ä
,
,
^
[6]
¥
•{
5
O
¤
Ï
é
‘
Å
i
Ä
Ä
g
Â
¥
ž
©
Ù
,
?
O
‘
Å
i
Ä
Ä
g
Â
¥
ž
©
Ù
,
l
3
C
¿Â
e
‰
Ñ
‘
Å
i
Ä
Â
ñ
„
Ý
O
.
,
,
·
‚
•
Œ
|
^
Markov
Ø
ª
5
O
¤
Ï
é
‘
Å
i
Ä
Ä
g
Â
¥
ž
©
Ù
,
l
|
^
@
Ï
Ê
Ž
½
n
‰
Ñ
‘
Å
i
Ä
Â
ñ
„
Ý
O
,
X
e
(
J
.
-
0
<<
1
,p/q
:= 1
−
,
d
x
e
L
«
Œ
u
½
u
x
•
ê
.
·
‚
Ì
‡
½
n
X
e
.
½½½
nnn
2.
b
‘
Å
i
Ä
X
=
£
Ý
P
•
(1),
¿
…
0
<p
≤
1
3
,>
0
.
X
J
n
=
d
c
2
e
,
Ù
¥
c>
1,
@
o
k
π
n
−
π
k≤
1
√
c
1+
1
2
ln
c
+
√
6
.
DOI:10.12677/pm.2021.11122262026
n
Ø
ê
Æ
ë
½½½
nnn
3.
b
‘
Å
i
Ä
X
=
£
Ý
P
•
(1),
¿
…
0
<p<
1
2
,>
0
.
X
J
n
=
d
c
2
e
,
Ù
¥
c>
1,
@
o
k
π
n
−
π
k≤
1
c
√
c
+
1
4
(ln
c
)
2
−
1
2
ln
c
.
2.
½
n
2
y
²
½
Â
=
£
Ý
P
1
•
:
é
?
¿
0
<x<
∞
,
k
P
1
(
x,x
−
1) =
1
3
,P
1
(
x,x
) = 0
,P
1
(
x,x
+1) =
2
3
;
P
1
(0
,
0) =
1
3
,P
1
(0
,
1) =
2
3
.
Ï
•
é
?
¿
x
≥
0,
k
P
1
(
x,x
+1)+
P
1
(
x
+1
,x
)
≤
1
,
¤
±
P
1
´
‘
Å
ü
N
.
Ï
•
0
<p
≤
1
3
,
¤
±
é
?
¿
x,y
∈
E
,
k
P
0
≤
z
≤
y
P
1
(
x,z
)
≥
P
0
≤
z
≤
y
P
∗
(
x,z
).
d
‘
Å
Œ
'
Ú
n
2
•
,
P
∗
‘
Å
›
›
P
1
,
=
P
1
≤
P
∗
.
·
‚
e
¡
Ï
é
˜
‡•
N
´
©
Û
{
ü
é
¡
‘
Å
i
Ä
ý
é
Š
L
§
,
¦
§
P
1
›
›
,
,
Ò
Œ
±
d
[6]
¥
•{
,
=
|
^
[8]
¥
Berry-Esseen
½
n
,
5
O
§
Ä
g
Â
¥
ž
©
Ù
.
b
Õ
á
Ó
©
Ù
‘
Å
C
þ
S
{
Y
i
:
i
≥
1
}
÷
v
P
{
Y
i
=
−
1
}
=
P
{
Y
i
= 1
}
=
1
3
;
P
{
Y
i
= 0
}
=
1
3
.
½
Â
V
0
:= 0
,V
n
:=
P
n
i
=1
Y
i
,
@
o
(
V
n
)
n
≥
0
´
½
Â
3
Z
þ
{
ü
é
¡
‘
Å
i
Ä
,
§
=
£
Ý
P
v
X
e
:
é
?
¿
x,y
∈
Z
,
k
P
v
(
x,x
−
1) =
P
v
(
x,x
+1)=
1
3
;
P
v
(
x,x
) =
1
3
.
w
,
,
V
n
Ï
"
Ú
•
©
O
•
E
(
V
n
) = 0,
D
(
V
n
) =
2
n
3
.
5
¿
(
|
V
n
|
)
n
≥
0
´
±
E
•
G
˜
m
l
Ñ
ž
m
ê
¼
ó
,
§
=
£
Ý
P
|
v
|
X
e
:
é
?
¿
0
<x<
∞
,
k
P
|
v
|
(
x,x
−
1) =
P
|
v
|
(
x,x
+1) =
1
3
,P
|
v
|
(
x,x
) =
1
3
,
0
<x<
∞
;
P
|
v
|
(0
,
1) =
2
3
,P
|
v
|
(0
,
0) =
1
3
.
Ï
•
P
1
´
‘
Å
ü
N
,
¿
…
é
?
¿
x,y
∈
E
,
k
P
0
≤
z
≤
y
P
|
v
|
(
x,z
)
≥
P
0
≤
z
≤
y
P
1
(
x,z
).
d
‘
Å
Œ
'
Ú
n
2
•
,
P
1
‘
Å
›
›
P
|
v
|
,
=
P
|
v
|
≤
P
1
.
q
P
1
≤
P
∗
,
¤
±
P
|
v
|
≤
P
∗
.
-
T
∗
,T
|
v
|
©
O
•
é
ó
ó
Ú
(
|
V
n
|
)
n
≥
0
Ä
g
Â
¥
{
x
∗
,x
∗
+1
,...
}
Â
¥
ž
,
K
é
?
¿
n
≥
0,
k
P
(
T
∗
>n
)
≤
P
(
T
|
v
|
>n
)
.
e
5
·
‚
òY
[6]
¥
y
²
g
´
,
|
^
@
Ï
Ê
Ž
½
n
,
‰
Ñ
‘
Å
i
Ä
Â
ñ
²
-
©
Ù
„
Ý
DOI:10.12677/pm.2021.11122262027
n
Ø
ê
Æ
ë
O
.
À
x
∗
=
d
b/
e−
1
,
1
−
H
(
x
∗
)
≤
e
−
b
.
-
Z
•
I
O
‘
Å
C
þ
,
C
≤
0
.
7655
•
[8]
¥
Û
~
ê
.
du
D
(
V
n
)=
2
n
3
,
0
:=
P
n
i
=1
E
|
Y
i
|
3
(
√
D
(
V
n
))
3
=
q
3
2
n
,
À
n
=
d
c/
2
e
,x
∗
=
d
b/
e−
1
,
K
d
[8]
¥
Berry-Esseen
½
n
,
k
P
(
T
∗
>n
)
≤
P
(
T
|
v
|
>n
)
≤
P
{|
V
n
|
<x
∗
}
≤
P
|
Z
|
<x
∗
r
3
2
n
+2
C
0
≤
P
|
Z
|
<
b
q
2
c
3
+
r
6
c
≤
b
+
√
6
√
c
.
Ï
d
,
k
π
n
−
π
k≤
e
−
b
+
b
+
√
6
√
c
.
X
J
c>
1,
Ï
L
À
b
=
1
2
ln
c
,
·
‚
k
π
n
−
π
k≤
1
√
c
1+
1
2
ln
c
+
√
6
.
3.
½
n
3
y
²
½
Â
=
£
Ý
P
2
•
:
é
?
¿
0
<x<
∞
,
k
P
2
(
x,x
−
1) =
p,P
2
(
x,x
) = 0
,P
2
(
x,x
+1) =
q
;
P
2
(0
,
0) =
p,P
1
(0
,
1) =
q.
d
½
n
2
y
²
•
,
P
2
≤
P
∗
.
-
e
X
=(
e
X
n
)
n
≥
0
•
l
0
Ñ
u
{
ü
é
¡
‘
Å
i
Ä
,
Ù
=
£
Ý
e
P
•
:
é
?
¿
0
<x<
∞
,
k
e
P
(
x,x
−
1) =
1
2
,
e
P
(
x,x
) = 0
,
e
P
(
x,x
+1)=
1
2
;
e
P
(0
,
0) =
1
2
,
e
P
(0
,
1) =
1
2
.
Ï
•
0
<p<
1
2
,
¤
±
é
?
¿
x,y
∈
E
,
k
P
0
≤
z
≤
y
e
P
(
x,z
)
≥
P
0
≤
z
≤
y
P
2
(
x,z
).
d
‘
Å
Œ
'
Ú
n
2
•
,
e
P
≤
P
2
.
q
P
2
≤
P
∗
,
¤
±
e
P
≤
P
∗
.
-
T
∗
,
e
T
©
O
•
é
ó
ó
Ú
e
X
Ä
g
Â
¥
{
x
∗
,x
∗
+1
,...
}
Â
¥
ž
,
K
é
?
¿
n
≥
0,
k
P
(
T
∗
>n
)
≤
P
(
e
T>n
)
.
d
Markov
Ø
ª
•
,
P
(
e
T>n
)
≤
E
e
T
n
.
e
¡
·
‚
Ï
L
O
Ž
E
e
T
,
5
O
P
(
T
∗
>n
).
w
,
,
e
P
´
Ø
Œ
,
š
±
Ï
.
d
[7],
½
Â
e
P
(
i,j
) :=
e
p
ij
,
e
p
(
k
)
n
:=
P
k
j
=0
e
p
nj
Ú
e
F
(
i
)
i
:= 1
,
e
F
(
i
)
n
:=
1
e
p
n,n
+1
n
−
1
X
k
=
i
e
p
(
k
)
n
e
F
(
i
)
k
,n>i
≥
0
.
DOI:10.12677/pm.2021.11122262028
n
Ø
ê
Æ
ë
d
[[7],
½
n
2]
•
,
e
P
´
~
ˆ
…
=
P
∞
n
=0
e
F
(0)
n
=
∞
.
N
´
¦
,
é
?
¿
n
≥
0
,
k
e
F
(0)
n
=
1
,
P
∞
n
=0
e
F
(0)
n
=
∞
.
Ï
d
,
e
P
´
~
ˆ
.
Ï
•
e
P
Ø
Œ
,
š
±
Ï
,
~
ˆ
,
d
[[7],
½
n
5]
•
,
E
0
e
σ
x
∗
=
x
∗
−
1
X
k
=0
e
v
k
.
Ù
¥
,
e
v
k
=
P
k
j
=0
e
F
(
j
)
k
e
p
j,j
+1
,
e
σ
x
∗
= inf
{
n
≥
1 :
e
X
n
=
x
∗
}
.
5
¿
e
T
=
e
σ
x
∗
,
Ï
d
E
e
T
=
E
e
σ
x
∗
.
Ï
•
e
F
(
i
)
i
= 1,
¤
±
,
e
F
(
i
)
i
+1
=
1
e
p
i
+1
,i
+2
i
X
k
=
i
e
p
(
k
)
i
+1
e
F
(
i
)
k
=
1
e
p
i
+1
,i
+2
e
p
i
+1
,i
e
F
(
i
)
i
= 1
.
Ó
n
,
Œ
¦
:
e
F
(
i
)
i
+2
= 1.
Ï
d
,
8
B
Œ
,
e
F
(
i
)
n
= 1
,
1
≤
i
≤
n
.
E
0
e
σ
x
∗
=
x
∗
−
1
X
k
=0
e
v
k
=
x
∗
−
1
X
k
=0
k
X
j
=0
e
F
(
j
)
k
e
p
j,j
+1
=
x
∗
−
1
X
k
=0
2(
k
+1) = (
x
∗
)
2
+
x
∗
.
Ï
d
,
E
0
e
T
=
E
0
e
σ
x
∗
= (
x
∗
)
2
+
x
∗
.
À
x
∗
=
d
b/
e−
1
,
1
−
H
(
x
∗
)
≤
e
−
b
.
À
n
=
d
c/
2
e
,
k
k
π
n
−
π
k≤
1
−
H
(
x
∗
)+
H
(
x
∗
)
P
0
(
T
∗
>n
)
≤
e
−
b
+
E
0
e
T
n
≤
e
−
b
+
(
x
∗
)
2
+
x
∗
n
=
e
−
b
+
(
d
b
e
)
2
−d
b
e
d
c
2
e
≤
e
−
b
+
b
2
+
b
c
.
X
J
c>
1,
Ï
L
À
b
=
1
2
ln
c
,
·
‚
k
π
n
−
π
k≤
1
c
√
c
+
1
4
(ln
c
)
2
−
1
2
ln
c
.
Ä
7
‘
8
®
½
g
,
‰
Æ
Ä
7
]
Ï
‘
8
(1194022),
®
é
Ü
Œ
Æ
<
â
r
`
À
O
y
(BPHR2020EZ01)
Ú
ZB10202001.
ë
•
©
z
[1]Diaconis,P.andFill,J.A.(1990)StrongStationaryTimesviaaNewFormofDuality.
The
AnnalsofProbability
,
18
,1483-1522.https://doi.org/10.1214/aop/1176990628
DOI:10.12677/pm.2021.11122262029
n
Ø
ê
Æ
ë
[2]Diaconis, P. andMiclo, L. (2009)On Timesto Quasi-Stationarity for Birthand DeathProcess-
es.
JournalofTheoreticalProbability
,
22
, 558-586. https://doi.org/10.1007/s10959-009-0234-6
[3]Diaconis,P. andSaloff-Coste,L.(2006) SeparationCut-Offs forBirthandDeath Chains.
The
AnnalsofAppliedProbability
,
16
,2098-2122.https://doi.org/10.1214/105051606000000501
[4]Fill, J.A. (2009)The Passage Time Distribution for a Birth-and-DeathChain:Strong Station-
aryDualityGivesaFirstStochasticProof.
JournalofTheoreticalProbability
,
22
,543-557.
https://doi.org/10.1007/s10959-009-0235-5
[5]Fill,J.A.andKahn,J.(2013)ComparisonInequalitiesandFastest-MixingMarkovChains.
TheAnnalsofAppliedProbability
,
23
,1778-1816.https://doi.org/10.1214/12-AAP886
[6]Diaconis,P.andFill,J.A.(1990)ExamplesfortheTheoryofStrongStationaryDualitywith
Countable StateSpaces.
ProbabilityintheEngineeringandInformationalSciences
,
4
, 157-180.
https://doi.org/10.1017/S0269964800001522
[7]
x
¬¬
,
o
Ü
,
Ü
{
Ÿ
,
ë
.
l
Ñ
ž
m
ü
)
L
§
O
O
K
[J].
®
“
‰
Œ
ÆÆ
(
g
,
‰
Æ
‡
),2015,51(3):227-235.
[8]Shiganov,I.S.(1986)RefinementoftheUpperBoundoftheConstantintheCentralLimit
Theorem.
JournalofSovietMathematics
,
35
, 2545-2550.https://doi.org/10.1007/BF01121471
DOI:10.12677/pm.2021.11122262030
n
Ø
ê
Æ