Page 224 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes with  being the variance of X and Cov(X ;X ) being the lag-i autocovariance. Because of the
2
n 1 1+i
2
dependence,  often is much bigger than  . We thus approximate  , the variance of X for
2
2
n n
any sufficiently large n by
 2
2
  Var (X )  .
n n
n
Again, this approximation reduces the unknowns to be estimated from the function { 2 n : n  1} to
the single parameter  .
2
In continuous time, we have the related asymptotic variance formula

2
  limt 2 t ,
t
where (under the assumption that {X(t) : t  0} is a stationary process)


2
dt
t
X
X
  2 0  Cov ( (0), ( )) .
2
In continuous or discrete time, a critical assumption here is that the asymptotic variance  is
actually finite. The asymptotic variance could be infinite for two reasons: (i) heavy-tailed
distributions and (ii) long-range dependence. In our context, we say that X or X(t) has a heavy-
n
tailed distribution if its variance is infinite. In our context, we say that there is long-range
dependence (without heavy-tailed distributions) if the variance Var(X ) or Var(X(t)) is finite,
n
but nevertheless the asymptotic variance is infinite because the autocovariances Cov(X ;X ) or
j j+k
Cov(X(t); X(t + k)) do not decay quickly enough as k   ; e.g., see Beran (1994), Samorodnitsky
and Taqqu (1994) and Chapter of Whitt (2002).
Assuming that    , we can apply CLT’s for weakly dependent random variables (involving
2
other regularity conditions, e.g., see Section 4 of Whitt (2002)) to deduce that X (as well as X )
n
t
is again asymptotically normally distributed as the sample size n increases, i.e.,
]
n 1/2 [X    N (0, 2 )asn   ,
n
so that the asymptotic variance  plays the role of the ordinary variance  in the classical IID
2
2
setting.
We thus again can use the large-sample theory to justify a normal approximation. The new
(1 — )100% confidence interval for  based on the sample mean X is
n

[X  z ( / n ),X  z ( / n )],

n  /2 n  /2
which is the same as for independent replications except that the asymptotic variance  replaces
2
2
the ordinary variance  .
The formulas for the confidence interval relative width, w (), and the required run length,
r
n (, ), are thus also the same as for independent replications in (1) except that the asymptotic
r

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