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Suppose $X_t$ is a continuous time symmetric nearest neighbor walk on $\mathbb{Z}$ starting at $0$. I'm looking for an upper bound of the form $\mathbb{P}(X_t\geq y)\leq \exp(-c\frac{y^2}{t})$. Let $N$ be Poisson with parameter $t$ and $\epsilon_j$ be i.i.d. $\pm 1$ fair coin flips independent of $N$. Then for $y>0$ we can use independence and Azuma's inequality to write $$\mathbb{P}(X_t\geq y)\leq \mathbb{P}(\sum_{j=1}^N\epsilon_j\geq y)=\sum_{n=0}^{\infty}\mathbb{P}(\sum_{j=1}^n\epsilon_j\geq y)\mathbb{P}(N=n)\leq\sum_{n=1}^{\infty}\exp(-\frac{y^2}{2n})\frac{e^{-t}t^n}{n!}.$$ I'm not sure how to simplify this to the desired form or even if it is possible. Hints or references are welcome.

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