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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