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The class of semimartingales is closed under optional stopping, localization, change of time and absolutely continuous change of probability measure (see Girsanov's Theorem). If X is an R m valued semimartingale and f is a twice continuously differentiable function from R m to R n, then f(X) is a semimartingale. This is a consequence of Itō's ...
Heavy traffic approximations are typically stated for the process X(t) describing the number of customers in the system at time t.They are arrived at by considering the model under the limiting values of some model parameters and therefore for the result to be finite the model must be rescaled by a factor n, denoted [3]: 490
The proof can also be phrased in the language of stochastic processes so as to become a corollary of the powerful theorem that a stopped submartingale is itself a submartingale. [2] In this setup, the minimal index i appearing in the above proof is interpreted as a stopping time .
An equivalent, but subtly different way to define the Chinese restaurant process, is to let new customers choose companions rather than tables. [4] Customer + chooses to sit at the same table as any one of the seated customers with probability +, or chooses to sit at a new, unoccupied table with probability +.
The martingale central limit theorem generalizes this result for random variables to martingales, which are stochastic processes where the change in the value of the process from time t to time t + 1 has expectation zero, even conditioned on previous outcomes.
Now consider a random walk X that starts at 0 and stops if it reaches –m or +m, and use the Y n = X n 2 – n martingale from the examples section. If τ is the time at which X first reaches ±m, then 0 = E[Y 0] = E[Y τ] = m 2 – E[τ]. This gives E[τ] = m 2. Care must be taken, however, to ensure that one of the conditions of the theorem ...
in which taking the limit first with respect to n gives 0, and with respect to m gives ∞. Many of the fundamental results of infinitesimal calculus also fall into this category: the symmetry of partial derivatives, differentiation under the integral sign, and Fubini's theorem deal with the interchange of differentiation and integration operators.
For example, if the renewal process is modelling the numbers of breakdown of different machines, then the holding time represents the time between one machine breaking down before another one does. The Poisson process is the unique renewal process with the Markov property , [ 1 ] as the exponential distribution is the unique continuous random ...