E^{(\nu)}R_c&=&\int_0^{\infty}P^{(\nu)}[R_c\geq t]dt\\ &\leq&\int_0^{(4\log c)/\nu^2}P^{(\nu)}[R_c\geq t]dt+\int_{(4\log c)/\nu^2}^{\infty}P^{(\nu)}[R_c\geq t]dt\\ &\leq &(4\log c)/\nu^2+\int_{(4\log c)/\nu^2}^{\infty}P^{(\nu)}[\tilde{W}_t\leq -\nu t/4]dt\\ &\leq &(4\log c)/\nu^2 + \int_0^{\infty}\bigg(\int_{-\infty}^{-\nu t/4}\frac{1}{\sqrt{2\pi t}}\exp(-\frac{x^2}{2t})\bigg)dt\\ &=&(4\log c)/\nu^2 + \int_0^{\infty}\bigg(\int_{\nu \sqrt{t}/4}^{\infty}\frac{1}{\sqrt{2\pi}}\exp(-\frac{u^2}{2})\bigg)dt\\ &\leq &(4\log c)/\nu^2 + \int_0^{\infty}\frac{4}{\nu\sqrt{t}}\exp(-\nu^2t/32)dt\\ &=&(4\log c)/\nu^2 +\frac{4}{\nu}\Gamma(3/2)(\nu^2/32)^{-3/2}<\infty
Embed the image to your blog in HTML, Markdown, Textile, BBCode.
Recent Referers:
  1. http://formula.s21g.com/?E%5E%7B(%5Cnu)%7DR_c%26%3D%26%5C...
  2. https://mkprob.hatenablog.com/entry/20120920/1348137802
  3. http://formula.s21g.com/?E%5E%7B(%5Cnu)%7DR_c%26%3D%26%5C...
  4. http://d.hatena.ne.jp/mkprob/touch/20120920/1348137802
  5. http://d.hatena.ne.jp/mkprob/20120920/1348137802