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Concentration of a modified random walk

Let $\varepsilon$ be a number in $(0, 1)$, consider the following random walk on the real line $X^{(0)}, X^{(1)}, \dots$, where $X^{(0)}=0$ If $X^{(t)} > 0$, then with probability $.5$, $X^{(t+1)} …
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