Monte Carlo convergence: high-dimensional integrand

Sample mean Î vs. sample size N across 200 independent replicates · d = 24

Final estimate 1.4994± 0.0021
Relative error 0.04%
Samples 10⁷
Replicates 200
Wall time 12.7s
1.60 1.55 1.50 1.45 1.40 Estimator value Î 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ Sample size N (log scale) theory · 1.5000
Sample mean Î
95% CI (200 replicates)
Theoretical mean
Figure 3. Monte Carlo estimator Î of a 24-dimensional integrand plotted against sample size N, with 95% confidence bands from 200 independent replicates. The estimator converges to the theoretical mean (μ = 1.5000) within 0.04% relative error at N = 10⁷.