Let $X = X(t)$, $t \in [0,1]$, be a stochastic process such that $\int_0^1 |X(t)|^p dt < \infty$ almost surely, where $1 \leq p < \infty$. Thus $X$ takes values in $L^p = L^p([0,1], dt)$. Assume in addition (for simplicity) that $E(X(t)) = 0$ for every $t$. Let now $X_1, X_2, \ldots$ be a sequence of independent copies of $X$. Under which assumption on the law of $X$ is there a central limit theorem, that is the sequence $\frac {1}{\sqrt n} (X_1 + \cdots + X_n)$ is tight in $L^p$ and converges towards some Gaussian process (in $L^p$)? Are there necessary and sufficient conditions on the law of $X$?
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$\begingroup$ I'm irritated. Does there exist any non trivial (support in finite dimensional subspace) Gaussian measure on $L^p$? $\endgroup$– Dieter KadelkaCommented Nov 30, 2020 at 15:33
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1$\begingroup$ @DieterKadelka : E.g., let $X:=\sum_{j=1}^\infty G_jx_j$, where the $G_j$'s are iid standard normal random variables and the $x_j$'s are linearly independent vectors in $L^p$ such that $\sum_{j=1}^\infty \|x_j\|_p<\infty$. Then (say, by Kolmogorov's three-series theorem) $X$ will be a Gaussian random vector in $L^p$ with an infinite-dimensional support. $\endgroup$– Iosif PinelisCommented Nov 30, 2020 at 17:00
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$\begingroup$ @Iosif Pinelis: Thank you! $\endgroup$– Dieter KadelkaCommented Nov 30, 2020 at 19:19
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The following are necessary and sufficient conditions for the central limit theorem you want:
(i) for $p\in[1,2]$: $$\int_0^1(EX(t)^2)^{p/2}\,dt<\infty\tag{1}$$ (see e.g. Giné, p. 147);
(ii) for $p\in(2,\infty)$: (1) & $u^2P(\|X\|>u)\to0$ as $u\to\infty$ (see e.g. Pisier and Zinn).
answered Nov 30, 2020 at 16:46