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This study is focused on periodic Fisher markets where items with time-dependent and stochastic values are regularly replenished and buyers aim to maximize their utilities by spending budgets on these items. Traditional approaches of finding a market equilibrium in the single-period Fisher market rely on complete information about buyers' utility functions and budgets. However, it is impractical to consistently enforce buyers to disclose this private information in a periodic setting. We introduce a distributed auction algorithm, online proportional response, wherein buyers update bids solely based on the randomly fluctuating values of items in each period. The market then allocates items based on the bids provided by the buyers. Utilizing the known Shmyrev convex program that characterizes market equilibrium of a Fisher market, two performance metrics are proposed: the fairness regret is the cumulative difference in the objective value of a stochastic Shmyrev convex program between an online algorithm and an offline optimum, and the individual buyer's regret gauges the deviation in terms of utility for each buyer between the online algorithm and the offline optimum. Our algorithm attains a problem-dependent upper bound contingent on the number of items and buyers under stationary inputs in fairness regret. Additionally, we conduct analysis of regret under various non-stationary stochastic input models to demonstrate the algorithm's efficiency across diverse scenarios. The online proportional response algorithm addresses privacy concerns by allowing buyers to update bids without revealing sensitive information and ensures decentralized decision-making, fostering autonomy and potential improvements in buyer satisfaction. Furthermore, our algorithm is universally applicable to many worlds and shows the robust performance guarantees.