u n = U n i f o r m [ 0 , n ] where u n ( [ a , b ] ) = ( b − a ) / n.I'm trying to show there isn't such probability measure u such that u n ⇒ u.Using proof by contradiction,Suppose there exists such a distribution u, then by the definition of weak convergence, lim n → ∞ u n ( ( − ∞ , x ] ) = u ( ( − ∞ , x ] )I want to show such lim n → ∞ u n ( ( − ∞ , x ] ) doesn't exist, but confused how to proceed.Any help would be much appreciated!

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u          n        =  U  n  i  f  o  r  m  [  0  ,  n  ] where       u          n        (  [  a  ,  b  ]  )  =  (  b  −  a  )      /    n.I&#039;m trying to show there isn&#039;t such probability measure   u such that       u          n        ⇒  u.Using proof by contradiction,Suppose there exists such a distribution u, then by the definition of weak convergence,       lim          n      →      ∞            u          n        (  (  −  ∞  ,  x  ]  )  =  u  (  (  −  ∞  ,  x  ]  )I want to show such       lim          n      →      ∞            u          n        (  (  −  ∞  ,  x  ]  ) doesn&#039;t exist, but confused how to proceed.Any help would be much appreciated!

rioolpijpgp · Accepted Answer

You can easily argue that       u    n    (  K  )  →  0 for any compact set   K. If it does not seem obvious, you can first show that       u    n    (  I  )  →  0 for any interval   I.Now suppose       u    n   converges to a probability measure   u weakly. Recall (Prove if you haven&#039;t already) that any probability measure on       R   is tight. In other words, there exists a compact set   K  ⊆      R   such that   u  (  K  )  ≥      1    2  . But this is not possible (Why?).

u <mrow class="MJX-TeXAtom-ORD"> n </mrow> </msub> = U n i

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