ABSTRACT
To introduce stochastic ordering we begin with the notion of bounded in probability of a sequence of random elements, say { X n : n = 1 , 2 , … } $ \{X _{\scriptscriptstyle n} : n = 1, 2, \ldots \} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/inline-math9_1.tif"/> . We say that X n $ X _{\scriptscriptstyle n} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/inline-math9_2.tif"/> is bounded in probability if, for all ϵ > 0 $ \epsilon> 0 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/inline-math9_3.tif"/> , there exists a K > 0 $ K> 0 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/inline-math9_4.tif"/> such that lim sup n → ∞ P ( ‖ X n ‖ > K ) < ϵ , $$ \begin{aligned} \limsup _{\scriptscriptstyle n\rightarrow \infty } P ( \Vert X _{\scriptscriptstyle n} \Vert> K ) < \epsilon , \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/um335.tif"/>
