Show that $\sup\limits_nE(n/(a+S_n))$ is finite, where $(S_n)$ is a random walk with positive steps

Question

I think there is a martingale theorem somewhere to use, but I'm stuck with that problem could you help me please?

Answer 1

For each $a>0$, first choose $\textbf x$ in $(0,a]$, small enough to ensure that $P(\xi>\textbf x)\ne0$. Then, for every $k$, $\xi_k\geqslant \textbf x\eta_k$, where each $$\eta_k=\mathbf 1_{\xi_k>\textbf x}$$ is Bernoulli with parameter $$\textbf p=P(\eta_k=1)=P(\xi_k>\textbf x)=1-P(\eta_k=0)$$ Consider $\textbf b=a/\textbf x$, hence $\textbf b\geqslant1$. For every $n$, $S_n\geqslant(\eta_1+\cdots+\eta_n)\textbf x$ hence $$\frac{n\textbf x}{a+S_n}\leqslant\frac{n}{\textbf b+\eta_1+\cdots+\eta_n}=\int_0^1nt^{\textbf b-1}t^{\eta_1}\cdots t^{\eta_n}dt$$ where the equality stems from the fact that, for every positive $s$, $$\frac1s=\int_0^1t^{s-1}dt$$ Taking expectations in both sides and using $E(t^{\eta_k})=\textbf pt+1-\textbf p$ in the RHS, one gets $$\textbf xe_n\leqslant\int_0^1nt^{\textbf b-1}(\textbf pt+1-\textbf p)^ndt$$ with $$e_n=E\left(\frac{n}{a+S_n}\right)$$ The upper bound $t^{\textbf b-1}\leqslant \textbf p^{1-\textbf b}(\textbf pt+1-\textbf p)^{\textbf b-1}$, guaranteed by the condition $\textbf b>1$, yields $$\textbf xe_n\leqslant \textbf p^{1-\textbf b}\int_0^1(n+\textbf b)(\textbf pt+1-\textbf p)^{n+\textbf b-1}dt=\textbf p^{-\textbf b}\left.(\textbf pt+1-\textbf p)^{n+\textbf b}\right|_0^1\leqslant \textbf p^{-\textbf b}$$ This proves that $e_n$ is bounded uniformly on $n$, with

$$\sup_ne_n\leqslant\frac1{\textbf p^{a/\textbf x}\textbf x}$$

For example, if $P(\xi>a)\ne0$, then $$\sup_ne_n\leqslant\frac1{aP(\xi>a)}$$