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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
Cayley's theorem (group theory) Central limit theorem (probability) Cesàro's theorem (real analysis) Ceva's theorem ; Chasles' theorem, Chasles' theorem ; Chasles' theorem (algebraic geometry) Chebotarev's density theorem (number theory) Chen's theorem (number theory) Cheng's eigenvalue comparison theorem (Riemannian geometry)
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.
The existence of such stable limit-cycle can be stated as a theorem. Theorem (Existence of a periodic orbit) [ 5 ] : If p 0 {\displaystyle p_{0}} is a hyperbolic fixed point of
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 .
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 ...
In statistics, asymptotic theory, or large sample theory, is a framework for assessing properties of estimators and statistical tests.Within this framework, it is often assumed that the sample size n may grow indefinitely; the properties of estimators and tests are then evaluated under the limit of n → ∞.