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Now, as a rough approximation, jAn mj c1 n1=a 1 for some positive constant c1 . Then we nd that n ! c2 10k= 1 1=a : We can say that given this many samples, k digit accuracy is ``highly likely.'' For example, assume we would like two-digit accuracy in An , and suppose c2 % 1. Then the number of samples n necessary to achieve this accuracy is shown in Table 3.1. This table shows that as a 3 1, the number of samples necessary to obtain convergence in the sample mean explodes. Thus it is not feasible in any reasonable amount of time to observe steady state in such a simulation as we have de ned it. Over any reasonable time scale, such a simulation is always in transient state. 3.3.2 High Variability at Steady-State

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Equation (3.6) shows that, even at steady state, the sample mean will be distributed according to a heavy-tailed distribution, and hence will show high variability. Thus the likelihood of an erroneous measurement of m is still nonnegligible. Equivalently, the simulation still behave erratically. To see this more clearly, let us de ne a swamping observation as one whose presence causes the estimate of m to be at least twice as large as it should be. That is, if we happen to encounter a swamping observation in our simulation, the observed mean of the input will have a relative error of at least 100%. In a simulation consisting of n inputs, a swamping observation must have value at least nm. Let us assume that the inputs are drawn from a Pareto distribution. Such a distribution has m ka= a 1 . Then the probability pnm of observing a value of nm or greater is pnm k P X > nm nka= a 1 a a 1 na a ;

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TABLE 3.1 Number of Samples Necessary to Achieve Two-Digit Accuracy in Mean as a Function of a a 2.0 1.7 1.5 1.2 1.1 n 10,000 72,000 1,000,000 1012 1022

... Preface -- 1 -- 2 -- 3 -- 4 -- 5 -- 6 -- 7 -- 8 -- 9 -- A -- B -- C -- Refs Front -- Contents -- Help

P(swamping observation)

and the probability p of observing such a value at least once in n trials is p 1 1 pnm n : Figure 3.6 shows a plot of p as a function of a for n 105. (The gure is not signi cantly different for other values of n, say 106 or 107.) It shows that even in a relatively long simulation, the probability of a swamping observation is not negligible; when a is below about 1.3, such an observation could occur more often than once in a hundred simulations. The probability declines very rapidly for a < 1:1 not because the variability of the simulation is declining, but because of the way we have de ned the swamping observation: in terms of the distributional mean. When a 1, the mean is in nite, and so it becomes impossible to observe a value greater than the mean. Taken together, Table 3.1 and Fig. 3.6 also provide some insight into the value of a above which it may be possible to obtain convergent, consistent simulations. The table shows that simulation convergence becomes impractical when a is somewhere in the region between 1.7 and 1.5; and the gure shows that simulations become erratic at steady state in approximately the same region. As a result, we can conclude that the dif culties inherent in simulations with heavy-tailed inputs are likely to be particularly great when a is less than about 1.7; and that when a is greater than or equal to about 1.7 it may be feasible (given suf cient computing effort) to obtain consistent steady state in simulation.

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