Cumulative Sum of Large Numbers

Estimating passage probability in a cumulative sum problem

This is a medium-difficulty probability question that combines discrete stochastic processes with asymptotic reasoning. It asks you to estimate the likelihood that a cumulative sum passes through a specific interval en route to a large target, a skill that appears in quant interview loops when firms want to see whether you can handle problems where exact enumeration is infeasible.

The core challenge is recognizing that as the target grows large (here, 1000), the process behaves approximately like a continuous random walk, and the probability of hitting a particular narrow interval depends on the step distribution and the geometry of the problem. Candidates typically need to set up the stopping condition, reason about which states are reachable, and then apply renewal theory or a limiting argument to estimate the desired probability without computing every path.

• Renewal processes and the renewal equation
• Discrete vs. continuous approximation of random walks
• Asymptotic behaviour of sums over many steps
• Overshoot and interval-crossing probabilities