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In probability theory, Slutsky’s theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables. The theorem was named after Eugen Slutsky. Slutsky’s theorem is also attributed to Harald Cramér.〔Slutsky's theorem is also called Cramér’s theorem according to Remark 11.1 (page 249) of Allan Gut. ''A Graduate Course in Probability.'' Springer Verlag. 2005.〕 ==Statement== Let , be sequences of scalar/vector/matrix random elements. If ''X''''n'' converges in distribution to a random element ''X''; and ''Y''''n'' converges in probability to a constant ''c'', then * * * provided that ''c'' is invertible, where denotes convergence in distribution. Notes: # In the statement of the theorem, the condition “''Y''''n'' converges in probability to a constant ''c''” may be replaced with “''Y''''n'' converges in distribution to a constant ''c''” — these two requirements are equivalent according to this property. # The requirement that ''Yn'' converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid. # The theorem remains valid if we replace all convergences in distribution with convergences in probability (due to this property). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Slutsky's theorem」の詳細全文を読む スポンサード リンク
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