Exercise3.5.11(Robins&Siegmund(1973))
Exercise3.5.11
ν>0, c>1 に対して次のような 
-stopping time を定義する．
%3Dc%5C%7D.png)
このとき
%7DR_c%3D%5Cfrac%7B2%5Clog%20c%7D%7B%5Cnu%5E2%7D..png)
[証明]
)%5C%5C%0A%26%3D%26%5Clim_%7Bt%5Cto%5Cinfty%7DE[1_%7B%5C%7BR_c%5Cleq%20t%5C%7D%7DZ_%7BR_c%7D]%5Cquad%20(%7B%5Crm%20by%5C%20the%5C%20monotone%5C%20convergence%5C%20thoerem%7D)%5C%5C%0A%26%3D%26%5Clim_%7Bt%5Cto%5Cinfty%7DE[1_%7B%5C%7BR_c%5Cleq%20t%5C%7D%7DZ_%7BR_c%5Cwedge%20t%7D]%5C%5C%0A%26%3D%26%5Clim_%7Bt%5Cto%5Cinfty%7DE[1_%7B%5C%7BR_c%5Cleq%20t%5C%7D%7DE[Z_t%7C%5Cmathcal%7BF%7D_%7Bt%5Cwedge%20R_c%7D]]%5Cquad%20(%7B%5Crm%20by%5C%20the%5C%20optional%5C%20sampling%5C%20theorem%7D)%5C%5C%0A%26%3D%26%5Clim_%7Bt%5Cto%5Cinfty%7DE[E[1_%7B%5C%7BR_c%5Cleq%20t%5C%7D%7DZ_t%7C%5Cmathcal%7BF%7D_%7Bt%5Cwedge%20R_c%7D]]%5C%5C%0A%26%3D%26%5Clim_%7Bt%5Cto%5Cinfty%7DE[1_%7B%5C%7BR_c%5Cleq%20t%5C%7D%7DZ_t]%5C%5C%0A%26%3D%26%5Clim_%7Bt%5Cto%5Cinfty%7D(EZ_t-E[1_%7B%5C%7BR_c%3E%20t%5C%7D%7DZ_t])%5C%5C%0A%26%3D%261%5Cquad%20(%7B%5Crm%20by%5C%20the%5C%20Lebesgue%5C%20convergence%5C%20theorem%7D).png)
よって 
. また
より
.png)
・・・(1)
あとは %7DR_c%3C%5Cinfty.png)
であることを示せば十分である．なぜならばこれが成立すればWaldの恒等式から
%7DW_%7BR_c%7D%3DE%5E%7B(%5Cnu)%7D%5Ctilde%7BW%7D_%7BR_c%7D+%5Cnu%20E%5E%7B(%5Cnu)%7DR_c%3D%5Cnu%20E%5E%7B(%5Cnu)%7DR_c.png)
となり，(1)の両辺の期待値をとって整理すれば
%7DR_c%3D%5Cfrac%7B2%5Clog%20c%7D%7B%5Cnu%5E2%7D.png)
を得るからである．実際計算してみると
%7DR_c%26%3D%26%5Cint_0%5E%7B%5Cinfty%7DP%5E%7B(%5Cnu)%7D[R_c%5Cgeq%20t]dt%5C%5C%0A%26%5Cleq%26%5Cint_0%5E%7B(4%5Clog%20c)/%5Cnu%5E2%7DP%5E%7B(%5Cnu)%7D[R_c%5Cgeq%20t]dt+%5Cint_%7B(4%5Clog%20c)/%5Cnu%5E2%7D%5E%7B%5Cinfty%7DP%5E%7B(%5Cnu)%7D[R_c%5Cgeq%20t]dt%5C%5C%0A%26%5Cleq%20%26(4%5Clog%20c)/%5Cnu%5E2+%5Cint_%7B(4%5Clog%20c)/%5Cnu%5E2%7D%5E%7B%5Cinfty%7DP%5E%7B(%5Cnu)%7D[%5Ctilde%7BW%7D_t%5Cleq%20-%5Cnu%20t/4]dt%5C%5C%0A%26%5Cleq%20%26(4%5Clog%20c)/%5Cnu%5E2%20+%20%5Cint_0%5E%7B%5Cinfty%7D%5Cbigg(%5Cint_%7B-%5Cinfty%7D%5E%7B-%5Cnu%20t/4%7D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%20t%7D%7D%5Cexp(-%5Cfrac%7Bx%5E2%7D%7B2t%7D)%5Cbigg)dt%5C%5C%0A%26%3D%26(4%5Clog%20c)/%5Cnu%5E2%20+%20%5Cint_0%5E%7B%5Cinfty%7D%5Cbigg(%5Cint_%7B%5Cnu%20%5Csqrt%7Bt%7D/4%7D%5E%7B%5Cinfty%7D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%7D%5Cexp(-%5Cfrac%7Bu%5E2%7D%7B2%7D)%5Cbigg)dt%5C%5C%0A%26%5Cleq%20%26(4%5Clog%20c)/%5Cnu%5E2%20+%20%5Cint_0%5E%7B%5Cinfty%7D%5Cfrac%7B4%7D%7B%5Cnu%5Csqrt%7Bt%7D%7D%5Cexp(-%5Cnu%5E2t/32)dt%5C%5C%0A%26%3D%26(4%5Clog%20c)/%5Cnu%5E2%20+%5Cfrac%7B4%7D%7B%5Cnu%7D%5CGamma(3/2)(%5Cnu%5E2/32)%5E%7B-3/2%7D%3C%5Cinfty.png)
となり，証明が終わる．[証明終]
最後のR_cの可積分性を示すのに苦労しました．
