论文信息 - When is the Student $t$-statistic asymptotically standard normal?

When is the Student $t$-statistic asymptotically standard normal?

Let X, X i , i ∈ N, be independent, identically distributed random variables. It is shown that the Student t-statistic based upon the sample {X i } n i =1 is asymptotically N(0,1) if and only if X is in the domain of attraction of the normal law It is also shown that, for any X, if the self-normalized sums U n := Σn i=1 X i /(Σ n i=1 X i 2 ) 1/2 , n ∈ N, are stochastically bounded then they are uniformly subgaussian that is, sup n E exp(λU n 2 ) 0.

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