# Convex Analysis and Optimization by Dimitri Bertsekas, Angelia Nedic, Asuman Ozdaglar By Dimitri Bertsekas, Angelia Nedic, Asuman Ozdaglar

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Example text

Z/ ! h; x/ ! x/; n ! h; x/ ! h; x/ ! 1; x/ ! z/d z in probability as n ! 1 for -almost all x. It remains to prove the uniform integrability. 33) 24 E. Elharfaoui et al. 1 is proved. 3 below. j / j. This norm is equivalent to the Euclidian norm and easy to work with here. 3 below see also . 3. 35) i 1 Then n 1 n X Vi ! 0 with probability 1, as n ! 1: i D1 Proof. For > 0, 1X Vi n n P ! j / ˇ Vi ˇ max ˇ ˇ 1Äj Äd ˇ n ! j / ˇ Vi ˇ ˇ ˇn ˇ ! j / ˇ Vi ˇ ˇ ˇ ˇn i D1 ! 1 Ä r rE n ˇr ! 2 of , one has that ˇ n ˇr !

U and where 0 log 0 D 0 by assumption. t. P or mean information per observation of P for discriminating of Q from P . s. P jQ/ is not a metric: it violates the symmetry and the triangle rules. x/; Sp 1 where 0 log 0 D 0 is assumed. P / measures the uncertainty inherent in P or in f . P / measures the expected amount of information gained on obtaining a direction from P , based on the principle that the rarer an event, the more informative its occurrence. P /. x/ D ˛r ; where i1 Ä : : : Ä ik 2 f1; 2; : : :g, mr 2 S p 1 , ˛r 2 R and r D 1; : : : ; k.

This norm is equivalent to the Euclidian norm and easy to work with here. 3 below see also . 3. 35) i 1 Then n 1 n X Vi ! 0 with probability 1, as n ! 1: i D1 Proof. For > 0, 1X Vi n n P ! j / ˇ Vi ˇ max ˇ ˇ 1Äj Äd ˇ n ! j / ˇ Vi ˇ ˇ ˇn ˇ ! j / ˇ Vi ˇ ˇ ˇ ˇn i D1 ! 1 Ä r rE n ˇr ! 2 of , one has that ˇ n ˇr ! 38) i D1 From the above two inequalities, one deduces that ˇr ! j / ˇ Vi ˇ ˇ ˇ ˇn ! <1 i D1 n 1 which, in turn, implies that X n 1 1X Vi n n P ! 3 then follows by Borel–Cantelli theorem.