This research derives bounds on the expected value and probability of a random walk exceeding barriers, suggesting new insights into random process behaviors.
This paper investigates the time N until a random walk first exceeds some specified barrier. Letting Xᵢ, i ≥ 1, be a sequence of independent, identically distributed random variables with a log-concave density or probability mass function, we derive both lower and upper bounds on the probability P(N > n), as well as bounds on the expected value $E[N].$ On barriers of the form a + b √k, where a is nonnegative, b is positive, and k is the number of steps, we provide additional bounds on $E[N].$
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Ross et al. (2025) studied this question.
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