On Limit Theorems for Brownian Motions on Unbounded Fractal Sets

2000 ◽  
pp. 227-237
Author(s):  
Masatoshi Fukushima
Keyword(s):  
Limit Theorems ◽  
Brownian Motions ◽  
Fractal Sets

NONCOMMUTATIVE BROWNIAN MOTIONS INDEXED BY PARTIALLY ORDERED SETS

2010 ◽  
Vol 13 (04) ◽  
pp. 629-661 ◽  
Author(s):  
ANNA KULA ◽  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Limit Theorems ◽  
Ordered Set ◽  
Positive Cone ◽  
Symmetric Matrices ◽  
Ordered Sets ◽  
Brownian Motions ◽  
Partially Ordered ◽  
Independent Increments ◽  
Lorentz Cone ◽  
The Given

We construct noncommutative Brownian motions indexed by partially ordered subsets of Euclidean spaces. The noncommutative independence under consideration is the bm-independence and the time parameter is taken from a positive cone in a vector space ([Formula: see text], the Lorentz cone or the positive definite real symmetric matrices). The construction extends the Muraki's idea of monotonic Brownian motion. We show that our Brownian motions have bm-independent increments for bm-ordered intervals. The appropriate version of the Donsker Invariance Principle is also proved for each positive cone. It requires the bm-Central Limit Theorems related to intervals in the given partially ordered set of indices.

Limit theorems and price changes in financial markets

1998 ◽  
Vol 77 (5) ◽  
pp. 1353-1356
Author(s):  
Rosario N. Mantegna, H. Eugene Stanley
Keyword(s):  
Financial Markets ◽  
Limit Theorems ◽  
Price Changes

Limit theorems for long-memory flows on Wiener chaos

Bernoulli ◽  
2020 ◽  
Vol 26 (2) ◽  
pp. 1473-1503 ◽  
Author(s):  
Shuyang Bai ◽  
Murad S. Taqqu
Keyword(s):  
Long Memory ◽  
Limit Theorems ◽  
Wiener Chaos
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