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?