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The difference lies in the : for stochastic boundedness, it suffices that there exists one (arbitrary large) to satisfy the inequality, and is allowed to be dependent on (hence the ). On the other hand, for convergence, the statement has to hold not only for one, but for any (arbitrary small) δ {\displaystyle \delta } .
Continuous stochastic process: the question of continuity of a stochastic process is essentially a question of convergence, and many of the same concepts and relationships used above apply to the continuity question. Asymptotic distribution; Big O in probability notation; Skorokhod's representation theorem; The Tweedie convergence theorem ...
There is a symmetric statement for submartingales with bounded expectation of the positive part. A supermartingale is a stochastic analogue of a non-increasing sequence, and the condition of the theorem is analogous to the condition in the monotone convergence theorem that the sequence be bounded from below.
The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence. [1]
Stochastic dominance (already mentioned above), denoted , means that, for every possible outcome x, the probability that yields at least x is at least as large as the probability that yields at least x: for all x, [] [].
Continuous stochastic process – Stochastic process that is a continuous function of time or index parameter; Dini continuity; Direction-preserving function - an analogue of a continuous function in discrete spaces. Microcontinuity – Mathematical term; Normal function – Function of ordinals in mathematics
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. [1] The term 'random variable' in its mathematical definition refers to neither randomness nor variability [ 2 ] but instead is a mathematical function in which
Uniform integrability is an extension to the notion of a family of functions being dominated in which is central in dominated convergence.Several textbooks on real analysis and measure theory use the following definition: [1] [2]