Linear Congruential Generators and Their Statistical Limits

Summary

This article explains linear congruential generators (LCGs), deterministic algorithms that produce pseudo-random sequences for uses such as Monte Carlo simulation and risk modeling. Each value is generated from the previous one using a multiplier, increment, and modulus; a seed makes the sequence reproducible. The article describes how parameter choices affect the sequence’s period and gives the Hull-Dobell conditions for reaching the maximum period.

A Python implementation is used to generate values, followed by visual checks of their distribution and dependence: a histogram, a scatter plot of successive values, and an autocorrelation plot. The article presents these as basic diagnostics rather than proof that a generator is suitable for demanding financial simulations. A long period alone does not guarantee good statistical behavior, and poorly chosen parameters can create patterns or weak distributional properties. It suggests considering other generator families when simulation accuracy is important.

Key ideas

• An LCG deterministically generates pseudo-random values from a seed and three parameters.
• The multiplier, increment, and modulus determine the sequence’s period and behavior.
• Hull-Dobell conditions characterize parameter choices that produce a maximum period.
• Histograms, successive-value plots, and autocorrelation plots offer basic sequence diagnostics.
• A full period does not ensure good statistical quality, so parameter selection and generator choice matter.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.