I am trying to find a limit for this expression lim_(k->oo)(2^k)/(gamma)log bbbE[e^(−γX/2^k)]

Brooklyn Farrell

Brooklyn Farrell

Open question

2022-08-14

lim k 2 k γ log E [ e γ X 2 k ]
I am trying to find a limit for this expression
lim k 2 k γ log E [ e γ X 2 k ]
I have so far found these bounds:
1 γ log ( E [ e X ] ) γ lim k 2 k γ log E [ e γ X 2 k ] 1 γ log E [ e γ X ]
I was wondering if there are better bounds in particular a better lower bound estimate. γ ( 0 , ) and X L ( Ω , F , P )

Answer & Explanation

Jazmyn Bean

Jazmyn Bean

Beginner2022-08-15Added 18 answers

lim s 0 + 1 s log E ( e s X ) = E ( X )
allucinemsj

allucinemsj

Beginner2022-08-16Added 5 answers

Since l o g is concave
2 k γ log E [ e γ X 2 k ] 2 k γ E [ log e γ X 2 k ] = 2 k γ E [ γ X 2 k ] = E [ X ]

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