in my probability theory course, we defined a sequence of random variables ( X i ) i = 1 ∞ to be tight if for all ϵ &gt; 0, there is a constant M s.th. P ( | X n | &gt; M ) &lt; ϵ for all n ∈ N .I have seen the following criteria for tightness/non-tightness and I was wondering whether they are true or not:1. ( X i ) i = 1 ∞ tight if there exists M s.th. lim n → ∞ P ( | X n | &gt; M ) = 02. ( X i ) i = 1 ∞ not tight if for all M lim n → ∞ P ( | X n | &gt; M ) &gt; 0I am quite sure that the first one is right (we get the condition that the probability is smaller than ϵ for all but finitely many n and can take the maximum of all the remaining M necessary for the finitely many n). For the second one I am not so sure, I was thinking that maybe we need some uniform bound, i.e. limn n → ∞ P ( | X n | &gt; M ) ≥ ϵ &gt; 0. I know that the criterion is right if the limit is equal to 1.

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in my probability theory course, we defined a sequence of random variables   (      X    i        )          i      =      1              ∞       to be tight if for all   ϵ  &amp;gt;  0, there is a constant M s.th.   P  (      |        X    n        |    &amp;gt;  M  )  &amp;lt;  ϵ for all   n  ∈      N  .I have seen the following criteria for tightness/non-tightness and I was wondering whether they are true or not:1.   (      X    i        )          i      =      1              ∞       tight if there exists M s.th. lim                n      →      ∞        P  (      |        X    n        |    &amp;gt;  M  )  =  02.   (      X    i        )          i      =      1              ∞       not tight if for all M lim                n      →      ∞        P  (      |        X    n        |    &amp;gt;  M  )  &amp;gt;  0I am quite sure that the first one is right (we get the condition that the probability is smaller than ϵ for all but finitely many n and can take the maximum of all the remaining M necessary for the finitely many n). For the second one I am not so sure, I was thinking that maybe we need some uniform bound, i.e. limn                n      →      ∞        P  (      |        X    n        |    &amp;gt;  M  )  ≥  ϵ  &amp;gt;  0. I know that the criterion is right if the limit is equal to 1.

seraphinod · Accepted Answer

Your first proposal is not equivalent to tightness. A family consisting of just copies of a single N(0,1) random variable is tight but does not satisfy that definition. However your first proposal is sufficient, as your argument shows.As for your second proposal, if you replace lim with lim sup (which is necessary because the limit you&#039;re asking for doesn&#039;t generally even exist) then you recover the negation of the first proposal. This is a common pitfall;   ¬      (          lim              n        →        ∞                    a      n        =    a    )   is really equivalent to       lim sup          n      →      ∞            |        a    n    −  a      |    &amp;gt;  0, not       lim          n      →      ∞            a    n    ≠  a.

in my probability theory course, we defined a sequence of random variables ( X i

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