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To close this one, as whuber's comment pointed out, let a multivariate distribution be symmetric in the sense $$(X_1, \ldots, X_m) \sim_d (-X_1, \ldots, -X_m)$$ Suppose also that moments exist. …

The Wikipedia equation uses the biased, maximum likelihood estimator for the sample variance (divide by $n$), while, as you say the function from the R-packages uses the bias-corrected formula (divide …

Both are strictly stationary processes, the first has the "infinite/undefined moments" issue, while for the Logistic process, the moments exist. … These are more primitive concepts than the moments. So for example if the moments do not exist, this does not mean that the concept of "location" is also undefined. …

In general, for $k,\ell$ moments, set $i=k-1, j=\ell-1$ in the variance-covariance expression. …

The moments of a distribution are defined as expected values of specific functions of the random variable following this distribution. Raw moments $\mu'_k \equiv E(X^k)$. … Central moments: $\mu_k \equiv E[X- E(X)]^k$. Fundamentally $k$ is taken to be an integer, but there is also theory developed for fractional moments (i.e. with $k$ not being an integer). …

Since $X_n$ are positive random variables, we do not need the absolute value. We have $$\{\mathbb{E}X_n\}=O(a_n) \implies \lim_{n\to \infty}\frac{\mathbb{E}X_n}{a_n} < K \in \mathbb R_{++}$$ Then, s …

As given in A. Dasgupta (2008) Asymptotic Theory of Statistics and Probability, ch 3, p. 42, The reference to Serfling is Serfling R.J. (1980) Approximation Theorems of Mathematical Statistics.

Indeed, it is a known erratum of this book (see its website in the errata .pdf), that the specific lemma does not state the moment-boundedness condition $$\exists \; \delta : E(|z_n|^{s+\delta}) < M …