1



   

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
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  
X
1

 Marcinkiewicz khi
m  p  X
1
   1< p <2      
martingale 

2 

              



 





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
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
Tr
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



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= 0.
t         
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     


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
              

c
u tit (modulated averages)
3


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
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


p, . 
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

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
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. 
qu

t trong mô hình hi qui tuyn tính vi
nhic lp cùng phân phi, vi  ta  


 





 









 4


Tôi xin cchân thành ti các thy giáo thuc t b môn Toán
ng dng  Khoa Toán Tin   tôi trong
sut quá trình hc tp và góp ý cho lu       
Nguy-PGS. TS Phm u -
c bit tôi mun t lòng bic ti TS.
Nguyn , cán b thuc - , c

  này.
 



5



-











1.1.1. 


, i- 
P( 

) = 







, 


, i-



E X
+







.
8

1.1.3. .
  



         











1.3. .
{








 (h.c.c).
1.4. 
i) 





 





}, {

}; 0 < 

 sao cho:
9













 (h.c.c).
iii) lmogo
.






 (h.c.c) ;   E






.
  - Zygmund   








10

a)




b)




c)












  
   



    



  




, 





 

, 
   



     


















. 
 



















 










 . 




) là hu hn vi mi
     

         


 suy ra









.
 

} là mt dãy s n vô cùng. Gi  là
mm, kh vi trên , sao cho 




t















. Rõ ràng,
 






.
Theo chú ý  trên, và s d-Steiltjes, vi mi
 ta có





















, thì rõ ràng tích phân ca M hi t. Nu tích phân
hi t, bi s u ca  và gi thit 




, chúng ta có








 











13

































 , qui
 







 
(i)
 và 






.
(ii)
 và 

















14































    










































Vì 

 


.







hi t hu chc chn vi mi  (thm chí vi c ).
























hi tng hp
  



































 













Chú ý 1. Khi  

, tphn  và  cho ta s
ng ca lut mnh s ln Marcinkiewicz-Zymund ([10
15

6). Mt ví d   ra rng hp tng quát, khi  u kin





 























hi tng thy nu
và 



















i xng hoc 










.
 







   




       

. Thì














là dãy bin ngc lp
cùng phân phi, qui tâm vi 






2.2.
Chú ý 1. Vi các trng s b chn vi tng phân kì, lut s ln có trng s
   i các bin ng  c lp cùng phân phi vi











, theo [13].
2. Lin và Weber [8] không 









  







là mt dãy khác không các s ph  t









. Vi  các m
i)




.
ii)
Dãy




           
{

 vi 




, vì vy 
T- trong





 







     ,     









.







, sao
cho 

 








,
 





 sao cho




















 và 


















 (h.c.c) 




cùng phân phi (không cn thit phc
lp) vi 






 











ani ([7].




vi 







  {

} 










(i) 

 


.
(ii) Dãy 






Chú ý 1. Dãy 



 ch ra rng khi  m (i) ca H qu
không suy ra  




(h.c.c) 





cùng phân phi ( không cn thic l






.


{







 




c lp cùng
phân phi, qui tâm , vi 







u kin  suy ra  



và









u kin ca H quc mt
kt qu ma Chen, Zhu (1996) ([6]),   c
chng minh;  










 





có tng phân
kì, và vi  





cùng phân phi, qui tâm vi 





















([9]) chúng ta có
 




 







)



suy ra .
    

} là mt dãy s ph    t 


























{

} ,  










.
Chng minh.  theo chng minh ca phn  cnh lí 2.2,
và rõ ràng . Chng minh ca   vi chng minh
 
21

2.2 và M 2.5 gii quyng hp
 
 









g, 
Chú ý















} 
       



 (h.c.c)    


 

.










c

22

(iii) Trung bình










        




 
 {

 
 
(i),(ii), và (iv) 







. S(iii) xem chú ý 2
sau 
Bây 



































 














.
23

Vi mi 






vi  và 

 vi . Theo


 
















c 











.
Chú ý 
[13] 
     [13]         








hình h


 







trong Chen, Zhu và Fang [6].
24




iii) 


là
(i) 
 
{

, 
lí 2.9, sao cho trung bình có trng s trong  là không hi t hu chc chn v
0. Vì vy gi thii xng là không th b qua trong phu cnh lí 2.9.
Mnh lí ch ra rng hi xng,  







 (h.c.c)

H qu 2.10. Gi ( là mt không gian xác sut, gt ánh x
b . Gi  








c  i mi 


, kh
:
(i) Trung bình
























i xng hoc 














(iii): 










  theo
(i) cho (iii).
(iv): Vi 











 











hi t hu chc chn vi
mi Marcinkiewicz và Zygmunund ([10]