Reverse Martingales and Approximation Operators

✍ Scribed by R.A. Khan

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 356 KB
• Volume: 80
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1995.1023

✦ Synopsis

Let (\left{\zeta_{\ldots}, \mathscr{F}{n}, n \geqslant m \geqslant 1\right}) be a reverse martingale such that the distribution of (\xi{n}) depends on (x \in I \subset R=(-x, x)) for each (n \geqslant m), and (\breve{\zeta}{n} \xrightarrow{a . x} x). For a continuous bounded function (f) on (R) let (L{n}(f, x)=E f\left(\xi_{n}\right)) be the associated positive linear operator. The properties of (\xi_{n}) are used to obtain the convergence properties of (L_{n}(f, x)). and some more details are given when (\xi_{n}) is a reverse martingale sequence of (\mathrm{Z})-statistics. Lipschitz properties for a subclass of these operators resulting from an exponential family of distributions are also given. It is further shown that this class of operators of convex functions preserves convexity also. An example of a reverse supermartingale related to the Bleimann-Butzer-Hahn operator is also discussed. ‘ 1995 Academic Press. lnc.