$E[\exp(t \Delta_{N})]\le\exp(t^2(b-a)^2/8)$ and $E[\Delta_{N}\lvert F_{N-1}]=0$, why is $E[\exp(t \Delta_{N})\lvert F_{N-1}]\le\exp(t^2(b-a)^2/8)$ Let $\Delta_{N}$ be bounded by $[a,b]$, where $$\Delta_{N}:=E[f(X_{1},...,X_{M})\lvert F_{N}]-E[f(X_{1},...,X_{M})\lvert F_{N-1}]$$
with $F_{N}:=\sigma(X_{1},...,X_{N})$
When proving the bounded differences lemma I came across the following statement with very little explanation:
Since $$E[\exp(t \Delta_{N})]\leq \exp(t^2(b-a)^2/8)$$ and $$E[\Delta_{N}\lvert F_{N-1}]=0 \; (*)\; \; ,$$
we obtain
$$E[\exp(t \Delta_{N})\lvert F_{N-1}]\leq \exp(t^2(b-a)^2/8)$$
I am aware that line $(*)$ is merely a consequence of Hoeffding's Inequality, but I do not understand how can conclude the inequality with respect to the sigma algebra $F_{N-1}$. Any ideas?
 A: If you just want a proof you can find one here. Otherwise, here is some additional information about the result.
It seems you are just confused with line $(*)$. This follows from the fact that this sequence is a martingale difference. We can prove this using the properties of conditional expectations. First, note that,
$$F_{N-1}\subset F_{N} \subset \mathcal{F}$$
Then consider,
$$\mathbb{E}[\Delta_N|F_{N-1}]=\mathbb{E}[\mathbb{E}[f(X_{1},...,X_{M})\lvert F_{N}]-\mathbb{E}[f(X_{1},...,X_{M})\lvert F_{N-1}]|F_{N-1}]\\=\mathbb{E}[\mathbb{E}[f(X_{1},...,X_{M})\lvert F_{N}]|F_{N-1}]-\mathbb{E}[\mathbb{E}[f(X_{1},...,X_{M})\lvert F_{N-1}]|F_{N-1}]\\=\mathbb{E}[f(X_{1},...,X_{M})\lvert F_{N-1}]-\mathbb{E}[f(X_{1},...,X_{M})\lvert F_{N-1}]=0$$
Where we use the law of total expectation. With this out of the way we can prove Hoeffding's lemma conditional on $F_{N-1}$.
Note that the lemma gives us the bound,
$$\mathbb{E}[e^{s\Delta_N}]\leq \exp(s\mathbb{E}[\Delta_N]+\frac{s^2(b-a)^2}{8})$$
So conditional on $F_{N-1}$ we hope the first term will drop out to give the result that you have. Indeed, the conditional version of the lemma gives,
$$\log\mathbb{E}[e^{s\Delta_N}|F_{N-1}]\leq s\mathbb{E}[\Delta_N | F_{N-1}] + \frac{s^2(b-a)^2}{8} \\\iff \mathbb{E}[e^{s\Delta_N}|F_{N-1}]\leq \exp(\frac{s^2(b-a)^2}{8})$$
Just as desired.
Proving it in the conditional case requires a little bit of work, so I will leave this link that has a proof of the conditional lemma on pg. 4.
