## Correlation-inducing variance reduction in regenerative simulation.(English)Zbl 0872.65123

Let $$\mathcal X = \big\{\mathbf X(t), t\geq0\big\}$$ be a multidimensional regenerative process in continuous time, and let $$\big\{T_k\mid k\geq0\big\}$$ be a sequence of regenerative points associated with $$\mathcal X$$. Let $$\tau_k= T_k-T_{k-1}, k\geq1$$, be the length of cycle of the process $$\mathcal X$$. Under mild regularity conditions the process $$\mathcal X$$ has a limiting distribution; that is $$\mathbf X(t)\to\mathbf X$$ as $$t\to\infty$$. Suppose that we are interested in estimating the expected value of $$f\big(\mathbf X(t)\big)$$ in the long run, where $$f$$ is a real-valued integrable function. The authors use (Shelder’s) strongly consistent point estimator of $$r(f)=\lim_{t\to\infty} E\big(f(\mathbf X(t))\big)$$ $\widehat r\big(n;f\big)={\frac{\overline Y(n)}{\overline \tau(n)}}= {\frac{n^{-1}\sum_{k=1}^{n} Y_k(f)}{n^{-1}\sum_{k=1}^{n} \tau_k}},$ where $Y_k(f)=\int_{T_{k-1}}^{T_k} f\big(\mathbf X(u)\big) du,\qquad 1\leq k\leq n,$ for a regenerative process in continuous time, and $Y_k(f)=\sum_{i=T_{k-1}}^{T_k-1} f\big(\mathbf X(i)\big)\qquad 1\leq k\leq n,$ for a regenerative process in discrete time. The main result is a proposal of a variance reduction technique that can be applied to regenerative simulations. The main difference from the classical approach is that the proposed technique induces correlation between consecutive nonoverlapping pairs of regenerations. It is shown analytically, that under mild conditions the proposed technique is superior to conventional techniques which use independent random numbers between regenerations. Two examples are provided to illustrate that the variance reduction is significant.
Reviewer: J.Antoch (Praha)

### MSC:

 65C99 Probabilistic methods, stochastic differential equations 62F10 Point estimation 62J10 Analysis of variance and covariance (ANOVA)
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### References:

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