1$ and let $(Z_{n})_{n \in \mathbb{N}}$ be a sequence of random variables with

$$( {\bf E}
|Z_{n}|^{p})^{1/p} \leq c_p \cdot n^{-\alpha}$$ for all $p \geq 1$ and $n \in \mathbb{N}$. Then for every $\varepsilon > 0$ there exists a finite and non-negative random variable $\eta_{{\varepsilon}}$ such that $$|Z_{n}| \leq \eta_{\varepsilon}\cdot n^{-\alpha + \varepsilon}$$ almost surely for all $n \in \mathbb{N}$.

The Burkholder-Davis-Gundy inequality and the Borel-Cantelli-type Lemma allow one to show that the Itô-Taylor scheme of order $\gamma$ has pathwise convergence rate $\gamma- \varepsilon$ for smooth and bounded coefficients with bounded derivatives, thereby extending the classical mean-square convergence analysis in [@KP]. Then a localization argument is applied to avoid the boundedness assumptions. Roughly speaking, this localization argument works as follows: A fixed sample path $X_t(\omega), \, t \in [0,T],$ of the SDE solution is bounded, i.e. stays in some open set $B(\omega)$. However for the SDE $$dY_t =\widetilde{a}(Y_t)\,dt + \sum_{j=1}^m \widetilde{b}_j(Y_t)\, d W^{(j)}_t, \qquad Y_0=x_0$$ with smooth and bounded coefficients $\widetilde{a}$, $\widetilde{b}_j$ with bounded derivatives, which coincide with the ones of the original SDE on $B(\omega)$, the solution sample path $Y_t(\omega), t \in [0,T]$, coincides with $X_t(\omega), t \in [0,T]$. Asymptotically this also holds for the corresponding sample paths of 