View source for Stochastic indistinguishability

A property of two random processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901202.png" /> which states that the random set

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901203.png" /></td> </tr></table>

can be disregarded, i.e. that the probability of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901204.png" /> is equal to zero. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901206.png" /> are stochastically indistinguishable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901207.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901208.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s0901209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090120/s09012010.png" /> are stochastically equivalent (cf. [[Stochastic equivalence|Stochastic equivalence]]). The opposite, generally speaking, is not true, but for processes that are continuous from the right (left), stochastic indistinguishability follows from stochastic equivalence.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Dellacherie,   "Capacités et processus stochastiques" , Springer  (1972)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Dellacherie,   P.A. Meyer,   "Probabilities and potential" , '''A''' , North-Holland  (1978)  (Translated from French)</TD></TR></table>