# pr.probability – Which version of the central limit theorem should we apply here?

Let $$f in C ^ 3 ( mathbb R)$$ with $$f> 0$$ and $$g: = ln f$$. assume $$g$$ Is Lipschitz continuous? Let $$d in mathbb N$$ and $$X$$ be a $$mathbb R ^ d$$random variable evaluated with density $$mathbb R d ni x mapsto p_d (x): = prod_ {i = 1} ^ df (x_i)$$ regarding the Lebesgue measure $$lambda ^ d$$ sure $$mathcal B ( mathbb R ^ d)$$. In addition, leave $$ell> 0$$, $$sigma_d: = ell d ^ {- 1/2}$$ and $$Q_d (x, ; cdot ;): = mathcal N_d (x, sigma_d ^ 2I_d).$$ Now let $$Y$$ be a $$mathbb R ^ d$$random variable evaluated according to the composition $$(p_d lambda ^ d) Q_d$$ of $$p_d lambda ^ d$$ and $$Q_d$$.

How can we show that $$sigma_d sum_ {i = 1} 0, underbrace { int f | g & # 39; ^ 2 : { rm d} lambda ^ 1} _ {=: : M} right) tag1$$ in distribution?

For fixed $$d in mathbb N$$, $$W_1, ldots, W_d$$ should be mutually independent and equally distributed (it should be noted that $$(X_i, Y_i-X_i)$$ to the distribution $$(f lambda ^ 1) otimes mathcal N_1 (0, sigma_d ^ 2)$$ which does not depend on $$i$$), mean $$0$$ and the variance $$sigma_d ^ 2M$$.

What worries me is that $$d$$ varies, distribution and variance change … Is there a version of CLT that is still applicable?