Given two vectors - u = ( 1 n , 1 n , . . . , 1 n ) of the uniform distribution on { 1 , , . . . , n}, and a stochastic vector p ∈ R n , I need to show that: ‖ p − u ‖ 2 ≤ 1I started with the following step: ‖ p − u ‖ 2 = ( p 1 − 1 n ) 2 + ( p 2 − 1 n ) 2 + ( p 3 − 1 n ) 2 + . . . + ( p n − 1 n ) 2 ≤ ( 1 − 1 n ) 2 + ( 1 n ) 2 + ( 1 n ) 2 + . . . + 1 n ) 2 Is this step correct? If not, what am I missing?

Question

Given two vectors -   u  =  (            1      n        ,            1      n        ,  .  .  .  ,            1      n        ) of the uniform distribution on {  1  ,  ,  .  .  .  ,  n}, and a stochastic vector   p  ∈            R              n      , I need to show that:  ‖  p  −  u      ‖    2    ≤  1I started with the following step:  ‖  p  −  u      ‖    2    =      (          p      1        −                  1        n                    )              2              +    (          p      2        −                  1        n                    )              2              +    (          p      3        −                  1        n                    )              2              +    .    .    .    +    (          p      n        −                  1        n                    )              2              ≤      (    1    −                  1        n                    )              2              +    (                  1        n                    )              2              +    (                  1        n                    )              2              +    .    .    .    +                  1        n                    )              2            Is this step correct? If not, what am I missing?

Izabella Henson · Accepted Answer

If &quot;p is stochastic&quot; is to be interpreted as       p    i    ≥  0 and       ∑          i      =      1        n        p    i    =  1, then using       p    i    ≤  1 one gets  ‖  p  −  u      ‖    2    2    =  ‖  p      ‖    2    2    −  2  ⟨  p  ,  u  ⟩  +  ‖  u      ‖    2    2    =      ∑          i      =      1        n        p    i    2    −      1    n    ≤      ∑          i      =      1        n        p    i    −      1    n    =  1  −      1    n    &amp;lt;  1  ,so that indeed  ‖  p  −  u      ‖    2    ≤      1    −          1      n        &amp;lt;  1  −      1          2      n        &amp;lt;  1

vedentst9i · Accepted Answer

The function   f  :  p  ↦  ‖  p  −  u      ‖    2   is convex, and the set of &quot;stochastic vectors&quot; is closed and convex. You can write   p  =  ∑      p    i        e    i   where the       p    i   are the entries in p and the       e    i   are standard unit vectors; then   f  (  p  )  ≤      ∑    i        p    i    f  (      e    i    ). Or you could argue that f attains its maximum at an extreme point, and that the extreme points are exactly the       e    i

Given two vectors - u=((1)/(n),1/n,...,1/n) of the uniform distribution on {1,,...,n}, and a stochastic vector p in bbbR^n, I need to show that: norm(p−u)_2 <= 1

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