Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support

Stop Rule Inequalities for Uniformly Bounded Sequences of Random Variables

Theodore P. Hill and Robert P. Kertz
Transactions of the American Mathematical Society
Vol. 278, No. 1 (Jul., 1983), pp. 197-207
DOI: 10.2307/1999311
Stable URL: http://www.jstor.org/stable/1999311
Page Count: 11
  • Read Online (Free)
  • Download ($30.00)
  • Cite this Item
If you need an accessible version of this item please contact JSTOR User Support
Stop Rule Inequalities for Uniformly Bounded Sequences of Random Variables
Preview not available

Abstract

If $X_0, X_1,\ldots$ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and if $V(X_0,X_1,\ldots)$ is the supremum, over stop rules $t$, of $EX_t$, then the set of ordered pairs $\{(x, y): x = V(X_0, X_1,\ldots, X_n)$ and $y = E(\max_{j\leqslant n}X_j)$ for some $X_0,\ldots, X_n\}$ is precisely the set $$C_n = \{(x, y): x \leqslant y \leqslant x\big(1 + n(1 - x^{1/n})\big); 0 \leqslant x \leqslant 1\};$$ and the set of ordered pairs $\{(x, y): x = V(X_0,X_1,\ldots)$ and $y = E(\sup_n X_n)$ for some $X_0, X_1,\ldots\}$ is precisely the set $$C = \bigcup_{n = 1}^\infty C_n$$. As a special case, if $X_0, X_1,\ldots$ is a martingale with $EX_0 = x$, then $E(\max_{j\leqslant n} X) \leqslant x + nx(1 - x^{1/n})$ and $E(\sup_n X_n) \leqslant x - x\ln x$, and both inequalities are sharp.

Page Thumbnails

  • Thumbnail: Page 
197
    197
  • Thumbnail: Page 
198
    198
  • Thumbnail: Page 
199
    199
  • Thumbnail: Page 
200
    200
  • Thumbnail: Page 
201
    201
  • Thumbnail: Page 
202
    202
  • Thumbnail: Page 
203
    203
  • Thumbnail: Page 
204
    204
  • Thumbnail: Page 
205
    205
  • Thumbnail: Page 
206
    206
  • Thumbnail: Page 
207
    207