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In a sense, this means that the sequence must be bounded, with a bound that gets smaller as the sample size increases. This suggests that if a sequence is o p ( 1 ) {\displaystyle o_{p}(1)} , then it is O p ( 1 ) {\displaystyle O_{p}(1)} , i.e. convergence in probability implies stochastic boundedness.
In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than ...
The following rules describe situations when one random variable is stochastically less than or equal to another. Strict version of some of these rules also exist. A ⪯ B {\displaystyle A\preceq B} if and only if for all non-decreasing functions u {\displaystyle u} , E [ u ( A ) ] ≤ E [ u ( B ) ] {\displaystyle \operatorname {E} [u(A ...
[58] [59] If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence. [ 55 ] If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process .
This sequence is then called a sequence of stochastic numbers. [23] The algorithms typically rely on pseudorandom numbers, computer generated numbers mimicking true random numbers, to generate a realization, one possible outcome of a process. [24]
One may think of supermartingales as the random variable analogues of non-increasing sequences; from this perspective, the martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There are symmetric results for submartingales, which are ...
Suppose that , [,] is given, and we wish to compute .Stochastic computing performs this operation using probability instead of arithmetic. Specifically, suppose that there are two random, independent bit streams called stochastic numbers (i.e. Bernoulli processes), where the probability of a 1 in the first stream is , and the probability in the second stream is .
X is said to be a Feller-continuous process if, for any fixed t ∈ T and any bounded, continuous and Σ-measurable function g : S → R, E x [g(X t)] depends continuously upon x. Here x denotes the initial state of the process X, and E x denotes expectation conditional upon the event that X starts at x.