Let f be a bounded probability density function. How do I show that ∫ R σ exp ⁡ ( − σ 2 ξ 2 2 ) f ( ξ ) d ξ → c &lt; ∞ as σ → ∞?I was thinking of doing dominated convergence theorem, but this doesn't work here. Could anyone give me any direction?

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Let   f be a bounded probability density function. How do I show that       ∫          R        σ  exp  ⁡  (  −                    σ        2                    ξ        2              2    )  f  (  ξ  )  d  ξ  →  c  &amp;lt;  ∞ as   σ  →  ∞?I was thinking of doing dominated convergence theorem, but this doesn&#039;t work here. Could anyone give me any direction?

gaiageoucm5p · Accepted Answer

f  (  ξ  ) is bounded and all terms are positive in the integral. Can you find an upper bound for the integrand?Let us assume   f  (  ξ  )  ≤            c              2        π            .      ∫          −      ∞              ∞        σ  exp  ⁡      [    −                                        σ            2                                ξ            2                          2              ]    f  (  ξ  )  d  ξ  ≤      ∫          −      ∞              ∞        σ  exp  ⁡      [    −                                        σ            2                                ξ            2                          2              ]              c              2        π              d  ξ  =  σ            c              2        π                  ∫          −      ∞              ∞        exp  ⁡      [    −                                        σ            2                                ξ            2                          2              ]    d  ξNote, that the upper bound of the integral is gaussian integral. And we have      ∫          −      ∞              ∞        exp  ⁡      [    −                                        σ            2                                ξ            2                          2              ]    d  ξ  =                    2        π                              |                σ                  |                      .Hence, we obtain      ∫          −      ∞              ∞        σ  exp  ⁡      [    −                                        σ            2                                ξ            2                          2              ]    f  (  ξ  )  d  ξ  ≤  σ            c              2        π                                2        π                              |                σ                  |                      =  c  .

Armeninilu · Accepted Answer

With the change of variables   u  =  σ  ξ we can rewrite your integral as      ∫                  R                  e          −              u        2                    /            2        f  (  u      /    σ  )    d  u  .This integrand is dominated by       e          −              u        2                    /            2        (      sup          x      ∈              R              f  (  x  )  ) which is integrable. If you show that the pointwise limit (as   σ  →  ∞) of the integrand is zero almost everywhere (except at   u  =  0 where the limit is   f  (  0  )) then you can conclude by applying the dominated convergence theorem.

Let f be a bounded probability density function. How do I show that ∫<!-- ∫ -

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