Теория вероятностей и математическая статистика. Билялов Р.Ф. - 105 стр.

UptoLike

ВУЗ: 

КФУ | Казань

Составители: 

Рубрика: 

P = (p
ij
) =
p
11
··· p
1N
p
21
··· p
2N
· ··· ·
· ··· ·
· ··· ·
p
N1
··· p
NN
,
N
X
j=1
p
ij
= 1,
p
ij
|
t
(2) =
P (ξ
t+2
= j|ξ
t
= i) :
P (ξ
t+2
= j|ξ
t
= i) =
=
N
X
k=1
P (ξ
t+1
= k|ξ
t
= i)P (ξ
t+2
= j|ξ
t
= i, ξ
t+1
= k) =
=
N
X
k=1
P (ξ
t+1
= k|ξ
t
= i)P (ξ
t+2
= j|ξ
t+1
= k) =
N
X
k=1
p
ik
p
kj
= (P
2
)
ij
.
p
ij
|
t
(s) = P (ξ
t+s
= j|ξ
t
= i) = (P
s
)
ij
,
q = (q
1
, q
2
, ..., q
N
), q
i
= P (ξ
0
= i),
P (ξ
t
= i) =
N
X
k=1
P (ξ
0
= k)P (ξ
t
= i|ξ
0
= k) =
N
X
k=1
q
k
p
ki
= (qP
t
)
i
.
∃ s P
s
lim
t→∞
P
t
=
q
∗
1
··· q
∗
N
· ··· ·
· ··· ·
· ··· ·
q
∗
1
··· q
∗
N
,
q
∗
= (q
∗
1
, q
∗
2
, ..., q
∗
N
)
q
∗
P = q
∗
,
N
P
i=1
q
∗
i
= 1, q
∗
i
≥ 0.
ðàññìàòðèâàòü òîëüêî îäíîðîäíûå öåïè Ìàðêîâà. Ìàòðèöà
                                       
                          p11 · · · p1N
                         p21 · · · p2N 
                                        X
                         ·   · · · ·    N
           P = (pij ) = 
                         ·
                                        ,
                                             pij = 1,
                             ··· ·      j=1
                         ·   ··· ·     
                                     pN 1 · · ·   pN N
íàçûâàåòñÿ ìàòðèöåé âåðîÿòíîñòåé ïåðåõîäà. Ñîñ÷èòàåì pij |t (2) =
P (ξt+2 = j|ξt = i) :
                      P (ξt+2 = j|ξt = i) =
                N
                X
            =         P (ξt+1 = k|ξt = i)P (ξt+2 = j|ξt = i, ξt+1 = k) =
                k=1

      N
      X                                                            N
                                                                   X
  =         P (ξt+1 = k|ξt = i)P (ξt+2 = j|ξt+1 = k) =                   pik pkj = (P 2 )ij .
      k=1                                                          k=1

Àíàëîãè÷íî pij |t (s) = P (ξt+s = j|ξt = i) =             (P s )ij ,
                                                       ïîýòîìó, åñëè ââå-
ñòè âåêòîð íà÷àëüíûõ âåðîÿòíîñòåé q = (q1 , q2 , ..., qN ), qi = P (ξ0 = i),
òî
                       N
                       X                                        N
                                                                X
      P (ξt = i) =           P (ξ0 = k)P (ξt = i|ξ0 = k) =             qk pki = (qP t )i .
                       k=1                                      k=1

   Òåîðåìà. Ïóñòü ∃ s , òàêîå, ÷òî ó                      Ps   âñå ýëåìåíòû áîëüøå
íóëÿ, òîãäà                                                   
                                            q1∗   ···    qN∗
                                           ·     ···    ·     
                                                              
                               lim P = 
                                    t
                                           ·     ···    ·     ,
                                                               
                              t→∞
                                           ·     ···    ·     
                                            q1∗   ···    qN∗

ãäå q ∗ = (q1∗ , q2∗ , ..., qN
                             ∗ )  òàê íàçûâàåìûé âåêòîð ñòàöèîíàðíûõ ñî-

ñòîÿíèé. Âåêòîð ñòàöèîíàðíûõ ñîñòîÿíèé îïðåäåëÿåòñÿ ñëåäóþùè-
                                     P
                                     N
ìè óñëîâèÿìè: q ∗ P = q ∗ ,             qi∗ = 1, qi∗ ≥ 0. Òåîðåìó ïðèìåì áåç
                                    i=1
äîêàçàòåëüñòâà.

                                            105

Страницы