Let $X_i$ be a continuous, real-valued martingale such that $E[X_i] < \infty$ and $E[X_i | H_i] = X_{i-1}$ where $H_i = \{X_1, X_2, \dots, X_{i-1}\}$. Also, let $Y_i = X_i - X_{i-1}$ be a martingale difference sequence (MDS) such that $E[Y_i] < \infty$ and $E[Y_i | H_i] = 0$.
If $Y_i$ is bounded as $|Y_i| \leq a$ then we can use the Azuma-Hoeffding inequality to show a concentration on the sum of the MDS as $P\left(\sum_{i=1}^n Y_i \geq t\right) \leq 2 \exp\left(-\frac{t^2}{2na^2}\right)$.
My question is, if we assume $Y_i$ is bounded as before, can we show a similar concentration on the sum of squared MDS, specifically, $P\left(\sum_{i=1}^n Y_i^2 \geq t\right)$?
We may have problems, even in the case where $Y_i$ is an i.i.d. sequence such that $\mathbb P\left\{Y_1=1\right\}=\mathbb P\left\{Y_1=-1\right\}=1/2$: in this case, the probability we want to estimate is $0$ or $1$ according to the cases $t\gt n$ or not.
However, using Azuma's inequality, we can find bounds for $\mathbb P\left\{\sum_{i=1}^n\left(Y_i^2-\mathbb E\left[Y_i^2\mid H_{i-1}\right]\right)\geqslant t\right\}$, since the increments are bounded by $2a^2$.