Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Pete Daffer, Mike Torbett, Jim Nickila; Was Einstein left‐handed?. Am. J. Phys. 1 February 1996; 64 (2): 109–110. https://doi.org/10.1119/1.18124 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAmerican Association of Physics TeachersAmerican Journal of Physics Search Advanced Search |Citation Search

Convergence properties of weighted sums of functions in D([0, 1]; E) (E a Banach space) are investigated. We show that convergence in the Skorokhod J1-topology of a sequence (xn) in D([0, 1]; E) does not imply convergence of a sequence (xn) of averages. Convergence in the J1-topology of a sequence (xn) of averages is shown, under the growth condition ∥ xn ∥ ∞ = o(n), to be equivalent to the convergence of (xn) in the uniform topology. Convergence of a sequence (xn,) is shown to imply convergence of the sequence (xn) of averages in the M1 and M2 topologies. The strong law of large numbers in D[0, 1] is considered and an example is constructed to show that different definitions of the strong law of large numbers are nonequivalent.

AbstractThis article presents laws of large numbers for random elements of D(IR). The proofs bring forth the similarities between processes restricted to compact sets of D (IR) and processes with increasing trajectories. The results appear to generalize existing results and cover the independent identically distributed case studied by Ranga Rao.