Abstract. The notion of bundle convergence for single (ordinary) sequences in von Neumann algebras and their $L_2$-spaces was introduced by Hensz, Jajte and Paszkiewicz in 1996. We adopted this notion for double sequences in 2001. In the present paper, we prove a new two-parameter SLLN for double sequences of orthogonal vectors in an $L_2$-space, and this SLLN is an intermediate one between those two Strong Laws of Large Numbers (in abbreviation: SLLN) proved in [4, Theorems 1 and 2]. One of our tools is a more precise variant of the Rademacher--Menshov inequality in noncommutative setting, which may be useful in other cases.

AMS Subject Classification (1991): 46L10, 46L53, 60B12

Keyword(s): von Neumann algebra, Gelfand-Naimark-Segal representation theorem, bundle convergence, orthogonal vectors, Rademacher-Menshov inequality, L_2, SLLN in noncommutative-spaces

Received September 13, 2002, and in revised form December 30, 2003. (Registered under 5809/2009.)