Search results
Results from the WOW.Com Content Network
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.
Lévy’s continuity theorem: The sequence {X n} converges in distribution to X if and only if the sequence of corresponding characteristic functions {φ n} converges pointwise to the characteristic function φ of X. Convergence in distribution is metrizable by the Lévy–Prokhorov metric.
[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 .
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 ...
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]
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.
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 .
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.