Solved – Limiting moments and asymptotic moments of a statistic

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  1. From Casella's Statistical Inference:

    Definition 10.1.7 For an estimator $T_n$, if $lim_{nto infty} k_n Var T_n = tau^2 < infty$, where ${k_n}$ is a sequence of
    constants, then $tau^2$ is called the limiting variance or
    limit of the variances of $T_n$.

    Definition 10.1.9 For an estimator $T_n$, suppose that $k_n(T_n – tau(theta)) to n(0, sigma^2)$ in distribution. The parameter
    $sigma^2$ is called the asymptotic variance or variance of
    the limit distribution     of $T_n$.

    • I was wondering if both definitions depend on the choice of the
      sequence $k_n$, because I suspect for some choice of the sequence
      $k_n$, the convergence may fail, while for some other choice of the
      sequence $k_n$, the convergence may succeed. Then are the two
      definitions not well defined, because aren't they supposed to be not
      dependent on the choice of the sequence $k_n$?

      For example, in Lyapunov CLT, $frac{1}{s_n} sum_{i=1}^{n} (X_i – mu_i) xrightarrow{d} mathcal{N}(0,;1)$ where $ s_n^2 = sum_{i=1}^n sigma_i^2 $. According to the above definition of asymptotic variance, $T_n = sum_{i=1}^n X_i$, $tau(theta) = sum_{i=1}^n mu_i$ (should tau(theta) be independent of sample size $n$?), and the asymptotic variance of $sum_{i=1}^n X_i$ is $1$ (this is hard to believe, because the variance $sigma_i^2$ of $X_i$ can be any as long as it is finite)?

    • Can the limiting distribution in the definition of the asymptotic
      variance to be other than a Normal distribution?

    • When will the limiting variance and the asymptotic variance be the
      same?

  2. Similarly but more generally,

    • how can we define limiting moments and
      asymptotic moments?

    • Is the limiting distribution in the definition of an asymptotic
      moment required to be a Normal distribution?

    • When will the limiting moment and the asymptotic moment coincide?

    For example, those two concepts for means: limiting mean and
    asymptotic mean?

Thanks and regards!

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Asymptotic Moments
Let ${X_n}$ be a sequence of random variables with cumulative distribution function $F_n(x)$. If this sequence converges in distribution to a random variable $X$, $lim_{nrightarrow infty}F_n(x) = F(x)$ for every point of continuity of $F(x)$, then the asymptotic moments of ${X_n}$ are the moments of the limiting distribution $F(x)$.

Limiting moments
Let ${X_n}$ be a sequence of random variables with cumulative distribution function $F_n(x)$. For every moment $M_{n,r}$ of $F_n(x)$ that exists, the limiting moment is defined as $M_r equiv lim_{nrightarrow infty}M_{n,r}$.

When the two coincide?
If

1) $M_r equiv lim_{nrightarrow infty}M_n(r)$ is finite (i.e. if the limiting moment is a real number)
2) There exists $delta > 0 : Eleft(|X_n|^{r+delta}right) < M < infty;; forall n$

then, if $X_n rightarrow_d X$, the limiting moment $M_r$ will be the asymptotic moment also.

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