Taylor approximation (probability)

[Pere Callahan]

I have the following problem: Assume g is a (smooth enough) function, X a random variable and $\varepsilon^h$ a sequence of random variables, whose moments converge to 0 as h goes to zero.

I would then like to prove that
$$\mathbb{E}\left|g(X+\varepsilon^h)-g(X)\right|$$

converges to zero as well as h goes to 0, if possible at the same rate as the first moment of $\varepsilon^h$.

I tried using Taylor's theorem which states that
$$g(X+\varepsilon^h)-g(X) = g'(X)\varepsilon^h + R(X,\varepsilon^h),$$

where the absolute value of the remainder satisfies $\left|R(X,\varepsilon^h)\right|\leq C(X)|\varepsilon^h|^2$. Using this and the Cauchy-Schwarz inequality I could show that
$$\mathbb{E}\left|g(X+\varepsilon^h)-g(X)\right|\leq\sqrt{\mathbb{E}(g'(X)^2)}\sqrt{\mathbb{E}(\varepsilon^h)^2} + \sqrt{\mathbb{E}(C(X)^2)}\sqrt{\mathbb{E}(\varepsilon^h)^4}.$$

For some reason, however, I'm not quite sure if that's correct In particular I don't know how to argue that C(X) should have finite variance. Maybe i must just assume that.

Anyway, I'd appreciate any input on how to bound the expected value of the increment in term of the moments of $\varepsilon^h$. Thanks,

PereEDIT:

I think what I did is actually not correct because C(X) might also depend on on the value of $\varepsilon^h$...Intuitively, however, I find it quite plausible that $\mathbb{E}\left|g(X+\varepsilon^h)-g(X)\right|$ should not go to zero more slowly than $\mathbb{E}|\varepsilon^h}$ .. Thanks again

 

[mmwave]

for any input!Dear Pere,

Thank you for sharing your problem on the forum. Your approach using Taylor's theorem is a good start, but as you mentioned, it may not be entirely correct. In order to prove that \mathbb{E}\left|g(X+\varepsilon^h)-g(X)\right| converges to zero, we need to show that the expected value of the increment can be bounded by the moments of \varepsilon^h.

One approach that may work is to use the dominated convergence theorem. This theorem states that if we have a sequence of random variables that converge in probability to a constant, and we can find a dominating function that bounds the absolute value of the sequence, then the expected value of the sequence also converges to the same constant.

In your case, we can consider the sequence g(X+\varepsilon^h)-g(X) as a sequence of random variables that converge in probability to zero. This is because the moments of \varepsilon^h converge to zero, and as h goes to zero, the values of \varepsilon^h also get closer to zero.

To apply the dominated convergence theorem, we need to find a dominating function that bounds the absolute value of g(X+\varepsilon^h)-g(X). One way to do this is to use the Lipschitz continuity of g. This means that there exists a constant L such that for all x and y, we have |g(x)-g(y)|\leq L|x-y|. Using this, we can rewrite the absolute value of the increment as:

|g(X+\varepsilon^h)-g(X)| = |g(X+\varepsilon^h)-g(X)|\leq L|\varepsilon^h|.

Now, we can use the Cauchy-Schwarz inequality to bound the expected value of the increment as:

\mathbb{E}\left|g(X+\varepsilon^h)-g(X)\right|\leq\sqrt{\mathbb{E}(L^2)}\sqrt{\mathbb{E}(\varepsilon^h)^2}.

Since L is a constant, it has finite variance and the expected value of the increment is bounded by the moments of \varepsilon^h. This means that as h goes to zero, the expected value of the increment also goes to zero at the same rate as the first