93 research outputs found
Large deviation asymptotics and control variates for simulating large functions
Consider the normalized partial sums of a real-valued function of a
Markov chain, The
chain takes values in a general state space ,
with transition kernel , and it is assumed that the Lyapunov drift condition
holds: where , , the set is small and dominates . Under these
assumptions, the following conclusions are obtained: 1. It is known that this
drift condition is equivalent to the existence of a unique invariant
distribution satisfying , and the law of large numbers
holds for any function dominated by :
2. The lower error
probability defined by , for , ,
satisfies a large deviation limit theorem when the function satisfies a
monotonicity condition. Under additional minor conditions an exact large
deviations expansion is obtained. 3. If is near-monotone, then
control-variates are constructed based on the Lyapunov function , providing
a pair of estimators that together satisfy nontrivial large asymptotics for the
lower and upper error probabilities. In an application to simulation of queues
it is shown that exact large deviation asymptotics are possible even when the
estimator does not satisfy a central limit theorem.Comment: Published at http://dx.doi.org/10.1214/105051605000000737 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Computable exponential bounds for screened estimation and simulation
Suppose the expectation is to be estimated by the empirical
averages of the values of on independent and identically distributed
samples . A sampling rule called the "screened" estimator is
introduced, and its performance is studied. When the mean of a
different function is known, the estimates are "screened," in that we only
consider those which correspond to times when the empirical average of the
is sufficiently close to its known mean. As long as dominates
appropriately, the screened estimates admit exponential error bounds, even
when is heavy-tailed. The main results are several nonasymptotic,
explicit exponential bounds for the screened estimates. A geometric
interpretation, in the spirit of Sanov's theorem, is given for the fact that
the screened estimates always admit exponential error bounds, even if the
standard estimates do not. And when they do, the screened estimates' error
probability has a significantly better exponent. This implies that screening
can be interpreted as a variance reduction technique. Our main mathematical
tools come from large deviations techniques. The results are illustrated by a
detailed simulation example.Comment: Published in at http://dx.doi.org/10.1214/00-AAP492 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Efficiency and marginal cost pricing in dynamic competitive markets with friction
This paper examines a dynamic general equilibrium model with supply friction. With or without friction, the competitive equilibrium is efficient. Without friction, the market price is completely determined by the marginal production cost. If friction is present, no matter how small, then the market price fluctuates between zero and the "choke-up" price, without any tendency to converge to the marginal production cost, exhibiting considerable volatility. The distribution of the gains from trading in an efficient allocation may be skewed in favor of the supplier, although every player in the market is a price taker.Dynamic general equilibrium model with supply friction, choke-up price, marginal production cost, welfare theorems
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