Renewal Theory

Renewal theory studies processes with repeated events and the times between successive events, known as interarrival times. It generalizes Poisson processes by allowing the times between events to follow any distribution, not just the exponential distribution.

The core concept is the renewal function m(t), which gives the expected number of renewals in the interval [0,t]. A key result, the Elementary Renewal Theorem, states that m(t)/t approaches 1/μ as t approaches infinity, where μ is the mean interarrival time.

Renewal processes generalize Poisson processes by allowing non-exponential distributions of interarrival times. This flexibility makes them suitable for modeling complex real-world phenomena where the memoryless property of exponential distributions doesn't hold.

Applications in machine learning include survival analysis, reliability modeling, customer lifetime value estimation, and maintenance scheduling algorithms.

In recommender systems, renewal processes help model repeat purchase behavior by capturing patterns in time intervals between purchases. In healthcare, they're used for predicting hospital readmissions and modeling disease recurrence. Reinforcement learning algorithms use renewal theory concepts to analyze reward processes and develop efficient exploration strategies.