Consider a random variable X with E ( X 2 ) &lt; ∞. How do I show that n P ( | X | &gt; ε n ) → 0 for each ε &gt; 0? I don't even know where to start, and would like any hint.As X ∈ L 2 , { X } is an absolutely continuous collection, wich means that there is an N ∈ N such that ∫ | X | &gt; N X d P &lt; εWhere can I go from there?

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Consider a random variable   X with   E  (      X    2    )  &amp;lt;  ∞. How do I show that   n  P  (      |    X      |    &amp;gt;  ε      n    )  →  0 for each   ε  &amp;gt;  0? I don&#039;t even know where to start, and would like any hint.As   X  ∈      L    2  ,   {  X  } is an absolutely continuous collection, wich means that there is an   N  ∈      N   such that      ∫                  |            X              |            &amp;gt;      N        X    d  P  &amp;lt;  εWhere can I go from there?

Trey Ross · Accepted Answer

n  P  (      |    X      |    &amp;gt;  ε      n    )  =      1          ϵ      2        ⋅  (      ϵ    2    n  )  ⋅  P  (      X    2    &amp;gt;      ϵ    2    n  )  ≤      1          ϵ      2            ∫                  X        2            ≥              ϵ        2            n            X    2    d  P  →  0

Consider a random variable X with E ( X 2 </msup> ) < <mi

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