查看“︁史拉斯基定理”︁的源代码

{{NoteTA |G1=Math}}

在[[概率論]]，'''史拉斯基定理'''將[[實數|實數列]][[極限 (數列)|極限]]的若干代數性質推廣到[[隨機變數|隨機變量]]序列。<ref>{{cite book |first=Arthur S. |last=Goldberger |title=Econometric Theory |location=New York |publisher=Wiley |year=1964 |pages=[https://archive.org/details/econometrictheor0000gold/page/117 117]–120 |url=https://archive.org/details/econometrictheor0000gold |language = en}}</ref>

定理得名自[[尤金·史拉斯基]]。<ref>{{Cite journal
  | last = Slutsky | first = E. | author-link = 尤金·史拉斯基
  | year = 1925
  | title = Über stochastische Asymptoten und Grenzwerte
  | language = de
  | journal = Metron
  | volume = 5 | issue = 3
  | pages = 3–89
  | jfm = 51.0380.03
  }}</ref>史拉斯基定理有時歸功於[[哈拉爾德·克拉梅爾]]。{{noteTag|在{{Cite book
  | last      = Gut | first = Allan
  | title     = Probability: a graduate course
  | url      = https://archive.org/details/probabilitygradu0000guta | publisher = Springer-Verlag
  | year      = 2005
  | isbn      = 0-387-22833-0
  }}，p.249的評注11.1中，此定理稱為[[哈拉爾德·克拉梅爾|克拉梅爾]]定理。}}

==敘述==
設<math>(X_n), (Y_n)</math>為[[隨機元素|隨機]][[标量 (数学)|純量]]、[[向量]]或[[矩陣]]序列。若<math>(X_n)</math>[[依分佈收斂]]至[[随机元素]]<math>X</math>，且<math>(Y_n)</math>[[依概率收斂]]至常數<math>c</math>，則

* <math>(X_n + Y_n) \ \xrightarrow{d}\ X + c ;</math>
* <math>(X_nY_n) \ \xrightarrow{d}\ Xc ;</math>
* <math>(X_n/Y_n) \ \xrightarrow{d}\ X/c,</math> &nbsp; 若''c''可逆，

其中<math>\xrightarrow{d}</math>表示[[依分佈收斂]]。

===說明===
#<math>(Y_n)</math>趨向於常數的條件不能省略。假如允許趨向於非退化的隨機元，則定理不再成立。例如，設<math>X_n \sim {\rm Uniform}(0,1)</math>，<math>Y_n = -X_n</math>，則對所有<math>n</math>，皆有和<math>X_n + Y_n = 0</math>。再者，<math>Y_n \xrightarrow{d} {\rm Uniform}(-1,0)</math>，但<math>X_n + Y_n</math>並不依分佈收斂至<math>X + Y</math>，其中<math>X \sim {\rm Uniform}(0,1)</math>，<math>Y \sim {\rm Uniform}(-1,0)</math>，<math>X</math>和<math>Y</math>獨立。<ref>{{cite web |first=Donglin |last=Zeng |title=Large Sample Theory of Random Variables (lecture slides) |work=Advanced Probability and Statistical Inference I (BIOS 760) |url=https://www.bios.unc.edu/~dzeng/BIOS760/ChapC_Slide.pdf#page=59 |publisher=University of North Carolina at Chapel Hill |date=Fall 2018 |at=Slide 59 |language=en |access-date=2021-07-31 |archive-date=2013-02-03 |archive-url=https://web.archive.org/web/20130203040223/http://www.bios.unc.edu/~dzeng/BIOS760/ChapC_Slide.pdf#page=59 |dead-url=no }}</ref>
# 若將定理中，所有「依分佈收斂」改成「依概率收斂」，則結論仍然成立。

==證明==
引用以下引理：若<math>(X_n)</math>依分佈收斂至<math>X</math>，且<math>(Y_n)</math>依概率收斂至常數<math>c</math>，則[[聯合分佈|聯合向量]]<math>(X_n, Y_n)</math>依分佈收斂到<math>(X, c)</math>。<ref>{{cite book|last=van der Vaart|first=Aad W.|title=Asymptotic statistics|year=1998|publisher=Cambridge University Press|location=New York|isbn=978-0-521-49603-2 |language = en}}</ref>

現對上述依分佈的收斂使用[[連續映射定理]]。由<math>f(x, y) = x+y</math>，<math>g(x, y) = xy</math>，<math>h(x, y) = xy^{-1}</math>定義的函數<math>f, g, h</math>皆為連續函數（為使<math>h</math>連續，要求<math>y</math>可逆），故由連續映射定理，史拉斯基定理成立。

==參見==
* [[隨機變數的收斂]]

== 註 ==
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==參考資料==
{{reflist}}


{{DEFAULTSORT:Slutsky's Theorem}}
[[Category:渐近理论 (统计学)]]
[[Category:機率論定理]]
[[Category:统计定理]]