On convergence of moving average series of martingale differences fields taking values in Banach spaces

ABSTRACT

Let be a field of martingale differences taking values in a Banach space, is an absolutely summable field of real numbers such that the moving average series converges almost surely. Let T_n = ∑_k⪰nZ_k be the tail corresponding series which converges to 0 a.s. The paper provides conditions under which and for every field of positive constants {b_n, n⪰1} such that b_n ⩽ b_m for all n⪯m. These results are applied to obtain some results about the convergence of Quadratic Chaos.

KEYWORDS:

• Convergent of series of random field
• field of random variables
• p-uniformly smooth Banach spaces
• strong law of large numbers
• tail series of random field

MATHEMATICS SUBJECT CLASSIFICATION:

• 60B11
• 60B12
• 60F15
• 60G42
• 60G60