# Solved – Limiting moments and asymptotic moments of a statistic

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!

Contents

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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