Limiting moments and asymptotic moments of a statistic

Question

1. From Casella's Statistical Inference:

   Definition 10.1.7 For an estimator $T_n$, if $\lim_{n\to \infty} k_n Var T_n = \tau^2 < \infty$, 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!

Answer 1

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 $X$, $\lim_{n\rightarrow \infty}F_n(x) = F(x)$ 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 $M_r \equiv \lim_{n\rightarrow \infty}M_{n,r}$.

When the two coincide?
If

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

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