In Sandra Cerrai, Normal deviations from the averaged motion for some reaction–diffusion equations with fast oscillating perturbation (Journal de Mathématiques Pures et Appliquées, Volume 91(6), 2009, pp.614-647) the author proves Lemma 4.2 (see p. 629) by induction. The thesis of the lemma is that for all $j$ positive integer there exists $C_{T,j}>0$ s.t. $$\left|\int_{[s, t]^j} \mathbb{E} \prod_{i=1}^j \vartheta_{\alpha, \beta}\left(t-r_i\right) \Psi_{\epsilon, h}\left(r_i / \epsilon\right) d r_1 \cdots d r_j\right| \leqslant c_{T, j}\left(1+|x|^j+|y|^j\right) \epsilon^{\frac{j}{2}}\left(\frac{(t-s) \alpha \wedge 1}{\alpha}\right)^{\frac{(1-2 \beta) j}{2}}|h|_H^j,$$ where $H$ is a Hilbert space, $\alpha>0$ and $\beta \in[0,1 / 3)$. For any $x, y, h \in H, r \geqslant 0$, $\epsilon>0$, and $\Psi_{\epsilon, h}(r):=\left\langle F\left(\bar{u}^x(\epsilon r), v^y(r)\right)-\bar{F}\left(\bar{u}^x(\epsilon r)\right), h\right\rangle_H, \quad \vartheta_{\alpha, \beta}(r):=e^{-r \alpha} r^{-\beta}.$ Here $\bar u^x(t)$ and $v^y(t)$, $t \leq T$ are some stochastic processes in $H$ following some stochastic differential equation starting from $x \in H$ and $y \in H$ respectively.
At p.642 the author estimates the term $I_{2,\epsilon}$ using the inductive hypothesis (see 641) and gets: $$I_{2, \epsilon} \leqslant c_{T, n}\left(1+|x|_H^{2 n}+|y|_H^{2 n}\right)|h|_H^{2 n} \epsilon^{-n} \alpha^{-n(1-2 \beta)}[(t-s) \alpha \wedge 1]^{n(1-2 \beta)}.$$ Note that $I_{2,\epsilon}$ was defined at p. 641 as $$\begin{align} I_{2,\epsilon}& = \sum_{j=1}^{n-1} \int_{\frac{s}{\epsilon}}^{r_{2 j}} \cdots \int_{\frac{s}{\epsilon}}^{r_2} \prod_{i=1}^{2 j} \vartheta_{\alpha, \beta}\left(t-\epsilon r_i\right)\left|\mathbb{E} \prod_{i=1}^{2 j} \Psi_{\epsilon, h}\left(t, r_i\right)\right| d r_1 \cdots r_{2 j} \\ & \quad \quad \quad \quad \times \int_{\frac{s}{\epsilon}}^{\frac{t}{\epsilon}} \int_{\frac{s}{\epsilon}}^{r_{2(n-j)}} \cdots \int_{\frac{s}{\epsilon}}^{r_2} \prod_{i=1}^{2(n-j)} \vartheta_{\alpha, \beta}\left(t-\epsilon r_i\right)\left|\mathbb{E} \prod_{i=1}^{2(n-j)} \Psi_{\epsilon, h}\left(t, r_i\right)\right| d r_1 \cdots r_{2(n-j)}. \end{align}$$ However how can the author do that? The inductive hypothesis (hence the thesis of the lemma with $j<2n$) is with the modulus outside the integrals and the terms inside the integral can be negative or positive being scalar products (a part from this it is ok as with a change of variable we get rid of $1/\epsilon$ in the intervals of integration and we get $\epsilon^{-2n}$ and then with the inductive hypothesis, if it can be used, we get $\epsilon^{n}$ so in turn we have $\epsilon^{-n}$). Does anyone has any ideas?