How does this come ?: int_(I_n) phi(x)e^(n(h(x)−h(xi)))dx =(1+o(1))phi(xi) int_(I_n)e^(n(h(x)−h(xi)))dx as n -> oo

skilmarka8j

skilmarka8j

Answered question

2022-09-22

ϕ ( x ) is bounded on I n = [ ξ ϵ n , ξ + ϵ n ] and ϕ ( ξ ) 0 ,where ϕ is continue at ξ, and ϵ n = o ( 1 ) . h ( x ) C 2 is continous ,and ξ is the only maximum point of h(x).
How does this come ?:
I n ϕ ( x ) e n ( h ( x ) h ( ξ ) ) d x
= ( 1 + o ( 1 ) ) ϕ ( ξ ) I n e n ( h ( x ) h ( ξ ) ) d x
as n

Answer & Explanation

Mackenzie Lutz

Mackenzie Lutz

Beginner2022-09-23Added 13 answers

Let M n = max I n ϕ ( x ) and m n = min I n ϕ ( x ). Then for the integral
m n I n e n ( h ( x ) h ( ξ ) ) d x I n ϕ ( x ) e n ( h ( x ) h ( ξ ) d x M n I n e n ( h ( x ) h ( ξ ) ) d x
and
m n I n e n ( h ( x ) h ( ξ ) ) d x ϕ ( ξ ) I n e n ( h ( x ) h ( ξ ) d x M n I n e n ( h ( x ) h ( ξ ) ) d x
but since ( M n m n ) = ϕ ( ξ ) + o ( 1 ) due to the continuity of ϕ, we get it.

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