# What is a “polynomially bounded” function, and why is this a requirement of the The Delta Method?

I am reading a paper "A note on the Delta Method" by Gary Oehlert, JASA, 1992.

I am trying to estimate the variance of a function of a random variable, but first I want to understand the limitations of using the Delta Method Taylor series.

Oehlert states:$^1$

when the function we are approximating is polynomially bounded in the random variables, then the naive Taylor series approximation will yield the correct asymptotic approximation..."

How can I tell if a function is polynomially bounded? Why is this a requirement of using the naive Taylor series approximation?

Is the alternative to a naive Taylor seires a higher-order Taylor series or a different approximation?

$^1$ p 28, middle of right col, last paragraph before Examples

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For the definition of polynomially bounded, see this math.se question. –  MånsT Jun 5 '12 at 7:14
At first glance it appears to me that polynomial boundeness is not a necessary condition (requirement) but a sufficient condition for the approximation in the Theorem to have accuracy $O(n^{−\beta})$. This is actually clarified by the author in Example 3. An approximation of order $O(n^{−(q+1)/2})$ is presented in $(3)$ where the assumptions are: $g$ and its first $q+1$ derivatives are bounded in a neighbourhood of $w$. I do not believe there is a practical general method to check polynomial boundeness. You can check, for example, how their ratio/difference behaves in the region of interest. –  user10525 Jun 5 '12 at 10:43
@MånsT thanks for the link. I posted a 100 point bounty, ending in seven days, requesting more details. –  David Jun 5 '12 at 14:31
@David, it doesn't look like you've posted a bounty. –  Macro Jun 5 '12 at 20:39
@Macro The bounty is posted on the linked question at math.SE math.stackexchange.com/q/109352/3733 –  David Jun 5 '12 at 20:42