Search results
Results from the WOW.Com Content Network
The order in probability notation is used in probability theory and statistical theory in direct parallel to the big O notation that is standard in mathematics.Where the big O notation deals with the convergence of sequences or sets of ordinary numbers, the order in probability notation deals with convergence of sets of random variables, where convergence is in the sense of convergence in ...
Initial work pointed towards the affirmative answer. For example, if a group G is finitely generated and the order of each element of G is a divisor of 4, then G is finite. . Moreover, A. I. Kostrikin was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, there exists a largest o
A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.
[2] Keanini's research is focused on molecular hydrodynamics, fluid dynamics, applied math, and statistical mechanics, with a primary interest in stochastic processes, applications of Green's function methods, and the utilization of fluid mechanics. In 1998, he published an online set of course notes in Advanced Viscous Flow which corresponded ...
Alternative proofs exist only making use of the fact that X is càdlàg, adapted, and the set {H · X t: |H| ≤ 1 is simple previsible} is bounded in probability for each time t, which is an alternative definition for X to be a semimartingale. A continuous linear extension can be used to construct the integral for all left-continuous and ...
The word "bounded" makes no sense in a general topological space without a corresponding metric. Boundary is a distinct concept; for example, a circle (not to be confused with a disk) in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. A bounded set is not necessarily a closed set and vice
Note that the vanilla Azuma's inequality requires symmetric bounds on martingale increments, i.e. .So, if known bound is asymmetric, e.g. , to use Azuma's inequality, one need to choose = (| |, | |) which might be a waste of information on the boundedness of .
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 ...