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Simul."],"published-print":{"date-parts":[[2025,1,31]]},"abstract":"<jats:p>In this work, we propose a method to construct a uniform error bound for the SK predictor. In investigating the asymptotic properties of the proposed uniform error bound, we examine the convergence rate of SK\u2019s predictive variance under the supremum norm in both fixed and random design settings. Our analyses reveal that the large-sample properties of SK prediction depend on the design-point sampling scheme and the budget allocation scheme adopted. Appropriately controlling the order of noise variances through budget allocation is crucial for achieving a desirable convergence rate of SK\u2019s approximation error, as quantified by the uniform error bound, and for maintaining SK\u2019s numerical stability. Moreover, we investigate the impact of noise variance estimation on the uniform error bound\u2019s performance theoretically and numerically. 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