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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 } .
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]
It is important to note that the convergence in Doob's first martingale convergence theorem is pointwise, not uniform, and is unrelated to convergence in mean square, or indeed in any L p space. In order to obtain convergence in L 1 (i.e., convergence in mean), one requires uniform integrability of the random variables .
When X n converges in r-th mean to X for r = 2, we say that X n converges in mean square (or in quadratic mean) to X. Convergence in the r-th mean, for r ≥ 1, implies convergence in probability (by Markov's inequality). Furthermore, if r > s ≥ 1, convergence in r-th mean implies convergence in s-th mean. Hence, convergence in mean square ...
Let be a domain (an open and connected set) in .Let be the Laplace operator, let be a bounded function on the boundary, and consider the problem: {() =, = (),It can be shown that if a solution exists, then () is the expected value of () at the (random) first exit point from for a canonical Brownian motion starting at .
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, [] [].
A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set. [4] [5] The set used to index the random variables is called the index set.
Stochastic computing is a collection of techniques that represent continuous values by streams of random bits. Complex computations can then be computed by simple bit-wise operations on the streams. Complex computations can then be computed by simple bit-wise operations on the streams.