= Reflection principle for simple symmetric random walk
Let $S_0=0$ be a <simple symmetric random walk>, let $T_a$ be its first hitting time of the positive integer $a$, and reverse every increment after $T_a$. The resulting path is again a simple symmetric random walk. Consequently,
$$
\mathbb P\left(\max_{k\leq n}S_k\geq a\right)
=\mathbb P(S_n\geq a)+\mathbb P(S_n\geq a+1).
$$
The continuous analogue is the existing <Reflection principle (Wiener process)>; its tail formula has no discrete one-unit lattice correction.
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