I'm trying to do the following exercise:
I have a martingale $Z_n=A^{S_n}Q_A^{-n}$
where $A>1$, $Q_A=\frac{1}{2}(A+A^{-1})$ and $S_n=X_1+\cdots+X_n$ with $X_k$ r.v.'s iid such that $P(X_k=1)=P(X_k=-1)=1/2$.
It's asked to say whether it converges in $L^p$ for some $p$ or not.
My attempt:
Being a positive martingale it converges a.e. and thanks to the strong law of large numbers it's readily seen that it converges a.e. to $0$.
So, if it converges in $L^p$, it converges to $0$.
From a theorem about convergence in $L^p$ of martingales I know that: If $\sup_nE[|Z_n|^p]<\infty$ then it converges a.s. in $L^p$.
I'm stucked here. Maybe it's useful to use the Doob Inequality but I don't figure out how.
Any help it's appreciated. Thanks!
EDIT: I'm having a dilemma: if I say that, for $E[Z_n]=1$ for all $n$ then $0 \neq 1=||Z_n||_1\leq ||Z_n||_p$ for all $p $ then I can conclude that $Z_n$ does not converge in $L^p$ for any p.
But for the theorem "If $\sup_nE[|Z_n|^p]<\infty$ then it converges a.s. in $L^p$" it seems that $\sup_nE[|Z_n|]<\infty$ so it converges in $L^1$
Where's my error?
It seems that you are trying to use the theorem backwards. Once you know that, in this setting, there is convergence in $L^p$ if and only if there is convergence to $0$ in $L^p$, you just have to compute $E(Z_n^p)$ and to check whether $E(Z_n^p)\to0$ or not.
Hint: Let $Z_n(A)=A^{S_n}Q_A^{-n}$, then for every $p$, $Z_n(A)^p=Z_n(A^p)R_p(A)^n$ with $$R_p(A)=Q_{A^p}(Q_A)^{-p},$$ hence $E(Z_n(A)^p)=R_p(A)^n$ and $Z_n(A)\to0$ in $L^p$ if and only if $R_p(A)\lt1$. Can you compute $R_p(A)$ and conclude?