Let R 1 , . . . , R n be independent continuous uniform over [0, 1] random variables.The geometric mean of R 1 , . . . , R n is defined by G n = 􏰠 R 1 × . . . × R n n = ( R 1 × . . . × R n ) 1 n .Show that G n converges in probability as well as with probability 1 (i.e. almost surely) to a constant, and identify the limit. Make sure you state any theorem you use.

Question

Let       R    1    ,  .  .  .  ,      R    n   be independent continuous uniform over [0, 1] random variables.The geometric mean of       R    1    ,  .  .  .  ,      R    n   is defined by      G    n    =      􏰠                      R        1            ×      .      .      .      ×              R        n              n    =  (      R    1    ×  .  .  .  ×      R    n        )          1      n        .Show that       G    n   converges in probability as well as with probability 1 (i.e. almost surely) to a constant, and identify the limit. Make sure you state any theorem you use.

Carleigh Tate · Accepted Answer

Step 1Consider       F    n    =  ln  ⁡  (      G    n    ). Then by logarithm rules       F    n    =      1    n        ∑          i      =      1        n    ln  ⁡  (      R    i    ). This is the arithmetic mean of the iid random variables   ln  ⁡  (      R    i    ). Now prove that these have finite mean   μ. Then the strong law says that       F    n   converges almost surely to   μ.Step 2Now       G    n    =  exp  ⁡  (      F    n    ). Note that exp is continuous. So if       F    n    (  ω  ) converges to   μ then       G    n    (  ω  ) converges to   exp  ⁡  (  μ  ). Hence       G    n   converges almost surely to   exp  ⁡  (  μ  ).

Open question

Answer & Explanation

New Questions in High school geometry