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By M. A. Crane, A. J. Lemoine (eds.)

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36 Moreover 3 because we now have independent and identically distributed observations, we can use results from classlcal statistics to estimate E{YI}/E[~ I] . 3. 2 Regenerative Processes in ContinuousTime A regenerative process { ~ (t)3 t ~ 0) in K dimensions is a stochastic process which starts afresh probabilistically at an increasing sequence 0 ~R I < R2 . . of random epochs on the time axis Thus; between any two consecutive regeneration epochs the portion [ X (t)3 Rj ~ t < Rj + 1 ] R. 3 and [ 0 3 ~) .

For each cycle j ~ where Y. is 3 3 3 the sum of f ( X i ) over the jth cycle and ~j is the length of the 3. n jth cycle. Compute the sample statistics I n 7 I% , ~ = - j=l n , r = j~1 IIn performing these calculations; particularly those of Sll 3 s12 ~ and soo; it is wise to use double-precision arithmetic in order to [~sur~ the desY~ed degree of accuracy in computing the sums. 44 n n 2 i Sll - se2 n-I j=l S12 i n j=l - 2 s . 5. J = n ii n n j=l j=l n-I ^ = n Sll - 2 r s 1 2 " ^2 + r s22 Form the confidence interval ^ r± where z 5 = ~-i(i - 5) z~ s TNand is the standard normal distribution function.

J jth is the sum of cycle. 6) converges to n ~ ~ 5. 3) and the law of large numbers. ~). 6)" + %)/n represent an initial "transient" in the simulation and ~0 = ~I - i 50 YO and • and Yj~ j ~ I . ~). ~) is not very restrictive, and does not pose an obstacle to application of the regenerative method. Let us now return to the main problem of interest here~ namely that of estimating the value of E[f( ~ )) based on the simulation output. 7) [(Yj; C~j) , j >_ I} , estimate r -= E[YI)/E[a I} . 36 Moreover 3 because we now have independent and identically distributed observations, we can use results from classlcal statistics to estimate E{YI}/E[~ I] .

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