A (1±ε)-approximation for maximum subarray sum in the sliding window model can be maintained with O(ε^{-1}(log n)^2) bits of space and O(ε^{-1} log n) update time, which is asymptotically optimal.
Programming pearls: Algorithm design techniques.Communications of the ACM, 27(9):865–873, 1984
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Approximate Maintenance of Maximum Subarray Sum in the Sliding Window Model
A (1±ε)-approximation for maximum subarray sum in the sliding window model can be maintained with O(ε^{-1}(log n)^2) bits of space and O(ε^{-1} log n) update time, which is asymptotically optimal.