Showing Limit of Martingale Exists How could I go about showing a limit for a martingale $M_n=(\lambda_0)^{S_n}$ where $P[X_1 = 1]=p$,  $P[X_1 = -1]=1-p$, and where  $S_n=X_1 +...+X_n$, exists? 
The text says that $X_1, X_2,...$ is a sequence of i.i.d. r.v. and that $\lambda_0$ is the root solution to $E[\lambda^{X_1}]=1$. The first part of the questions asks to show that $M_n := (\lambda_0^ {S_{n}})$ is a martingale which I did and then it asks to show that it's limit exists. 
I started by finding $\lambda =\frac{1 \pm |1-2p|}{2p}$ but after this I'm not really sure how to use this with the Martingale Convergence Theorem to show that $lim_{n \to \infty } M_n$ does indeed exist in $\Bbb R$
 A: We can assume that $\lambda=\frac{1-p}p$, as otherwise $M_n$ is identically $1$, so $M_n = \left(\frac{1-p}p\right)^{S_n}$. First note that $M_n\geqslant 0$, so $|M_n| = M_n^- = M_n$. For integrability, we have
$$
\mathbb E[M_n] = \mathbb E[\lambda^{S_n}] = \sum_{k=-n}^n \mathbb P(S_n=k)\lambda^k \leqslant (2n+1)\cdot\max\{\lambda^{-n},\lambda^n\}\cdot\sum_{k=-n}^n\mathbb P(S_n=k)<\infty.
$$
For the martingale condition we have
\begin{align}
\mathbb E[M_{n+1}\mid\mathcal F_n] &= \mathbb E[\lambda^{S_{n+1}}\mid\mathcal F_n]\\
&= \mathbb E[\lambda^{S_n}\lambda^{X_{n+1}}\mid\mathcal F_n]\\
&= \mathbb E[\lambda^{X_{n+1}}]\mathbb E[\lambda^{S_n}]\\
&= \lambda^{S_n}\\
&= M_n.
\end{align}
To use Doob's martingale convergence theorem, we must show that $$\sup_n \mathbb E[\lambda^{S_n}]<\infty. $$
Note that $M_n$ is the probability generating function $\mathbb E[z^{S_n}]$ of $S_n$ evaluated at $z=\lambda$. I suspect the inequality above only holds when $\lambda<1$, that is, $p<1/2$. But I could be wrong; I will defer to others for a definitive answer to this question.
