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 Assume a r.v.  >0 in a probability space ( , , ) with  (*) and survivalfunction /  ( )(*).Let  (*) be the tail quantile function of  , show that for

 Assume a r.v.  >0 in a probability space ( , , ) with  (*) and survivalfunction /  ( )(*).Let  (*) be the tail quantile function of  , show that for any pin(0,1), wehave /  ( )( (1 ))≤ .Show that the first two definitions of the SOC (secondary order condition),i.e. (**1) and (**2) are equivalent.Given  →0, prove the following with given fact that  . (1(/  ( ))( (1 ))

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