Taylor expansion in Hoeffding's Lemma proof

Question

Hoeffding's Lemma proof uses Taylor expansion with this statement:

From Taylor's theorem, for some $ 0\leq \theta \leq 1$

$ L(h) = L(0) + h L'(0) + \frac{1}{2} h^2 L''(h\theta) \leq \frac{1}{8}h^2 $

Why does it use $0$ in the first two terms and $h\theta$ in the last? But as I know they must be same in the Taylor.

Thomas Lumley · Accepted Answer · 2022-05-27 07:43:02Z

11

This is the mean-value form of Taylor's theorem:

$$f(x)=f(0)+xf'(0)+\frac{x^2}{2}f''(c)$$ where $c$ is between $0$ and $x$

Take $x=h$ and $c=h\theta$

answered May 27 at 7:43

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$\begingroup$ thanks. Is it related to Lagrange error? and can you explain a good document for mean-value form of Taylor's theorem? $\endgroup$
– Tavakoli
May 27 at 9:03
$\begingroup$ Any decent Calculus textbook. $\endgroup$
– whuber ♦
May 27 at 13:34
$\begingroup$ The derivation is further down on the Wikipedia page I linked: en.wikipedia.org/wiki/… $\endgroup$
– Thomas Lumley
Jun 1 at 21:25

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