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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.
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 ...
In other words, if X n converges in probability to X and all random variables X n are almost surely bounded above and below, then X n converges to X also in any rth mean. [10] Almost sure representation. Usually, convergence in distribution does not imply convergence almost surely.
Stochastic (/ s t ə ˈ k æ s t ɪ k /; from Ancient Greek στόχος (stókhos) 'aim, guess') [1] is the property of being well-described by a random probability distribution. [1] ...
a sample function of a stochastic process is a bounded function of ; and a sample function of a stochastic process X {\displaystyle X} is an increasing function of t ∈ T {\displaystyle t\in T} . where the symbol ∈ can be read "a member of the set", as in t {\displaystyle t} a member of the set T {\displaystyle T} .
Stochastic dominance is a partial order between random variables. [1] [2] It is a form of stochastic ordering.The concept arises in decision theory and decision analysis in situations where one gamble (a probability distribution over possible outcomes, also known as prospects) can be ranked as superior to another gamble for a broad class of decision-makers.
The Ornstein–Uhlenbeck process is an example of a Gaussian process that has a bounded variance and admits a stationary probability distribution, in contrast to the Wiener process; the difference between the two is in their "drift" term. For the Wiener process the drift term is constant, whereas for the Ornstein–Uhlenbeck process it is ...
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 .