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My question is about the usage of Taylor expansions when dealing with asymptotics in local polynomial fitting. The expressions that set me of are of the type: $$ g(X_i) = \sum_{j=0}^{p} \frac{g^{(j)}(x_0)}{j!}(X_i-x_0)^j + o_p((X_i-x_0)^p), $$ where $x_0$ is fixed and $X_i$ is a random design point. I have two questions concerning these type of expressions. First, the $o_p$ symbol stands (as far as I understand) for convergence in probability to 0, but there is no sequence to converge. There in fact is only 1 random variable $X_i$ ($i$ is a fixed index, in a sample of e.g. size $n$). So my question is how to make sense out of this? My second question concerns the order of the remainder term. The above fits to the real analysis intuition where we have $$ f(x) = \sum_{j=0}^{p} \frac{f^{(j)}(x_0)}{j!}(x-x_0)^j + o(|x-x_0|^p), $$ for $x\to x_0$. However, e.g. in Brockwell, Davis: Time Series: Theory and Methods (1987), chapter 6 - asymptotic theory, where ``Taylor Expansion in Probability'' is treated (p. 194/195) we have $$ g(X_n) = \sum_{j=0}^{p-1} \frac{g^{(j)}(x_0)}{j!}(X_n-x_0)^j + o_p(r_n^p), $$ with $\{X_n\}$ a sequence of rv's such that $X_n = x_0 + O_p(r_n)$, and $0<r_n\to 0$ as $n\to\infty$. Thus the sum contains only $p$ terms, instead of $p+1$ as above. Nevertheless the $o_p$ symbol here makes sense as an actual sequence is involved. So do I include $p$ or $p+1$ terms in the sum above? I appreciate any comments - if my points are not clear I can try to clarify$\ldots$

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