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In some lecture notes here, pages 7, 8 and 9, a proof is given. The direction 2. $\Rightarrow $ 1. rests on extraction of a sub-sequence such that $\left(X_{n_k}\mathbf{1}_{\lvert X_{n_k}\vert \leqsla …

Convergence holds in any $L^p$ for $p\geqslant 1$. By Theorem III.4.3 page 106 in Garsia, Adriano M. Martingale inequalities: Seminar notes on recent progress. Math. Lecture Note Ser. W. A. Benjamin, …

Let us give a name to the partial sums $$ S_{P,Q}(f)=\frac 1{PQ}\sum_{i=1}^P\sum_{j=1}^Q f(X_i,Y_j). $$ and define the functions $$ f_1\colon x\mapsto \mathbb E\left[f(x,Y_1)\right]-\mu, \quad, f_2\co …

Consider an independent sequence of events $\left(A_i\right)_{i\geqslant 1}$ such that if $2^N+1\leqslant i\leqslant 2^{N+1}$, $\mathbb P(A_i)=2^{-N}$. Define for $2^N+1\leqslant i\leqslant 2^{N+1}$ t …

In the Math Stack Exchange post, I gave a proof based on Lemma 2 in Bai and Yin (1993). I will give an alternative proof. Expressing $\sum_{1\leqslant i\neq i'\leqslant n}X_{i,j}X_{i',j}$ as $\left(\s …

Let $D_n:=S_n-S_{n-1}$ for $n\geqslant 2$ and $D_1=S_1$. It is temping to want to work with martingales in order to study the properties of $\sum_{n=1}^ND_n$ and one can have the feeling that we are n …

As a immediate corollary of the real-valued case, a necessary condition is that for all $f\in H$, $\langle X,f\rangle$ should be centered and have a finite moment of order two. For $n\geqslant 3$, den …

We can construct a strictly stationary sequence $\left(X_k\right)_{k\in\mathbb Z}$ having the following properties: $X_0$ has finite moments of any order. $\beta(k)\leqslant Ck^{-1+\delta}$ for some …

Assume that $X_t$ have independent and centered increments, but not necessarily identically distributed. Let $D_i=X_i-X_{i-1}$ for $i\geqslant a+1$ and $D_a=X_a$. Let $ S_t=X_t\left(X_T-X_t\right). …

Let $U$ and $V$ be two i.i.d. random variables having finite expectation. Let $X_{3k}=X_{3k+1}:=U$, $X_{3k+2}:=V$ and $f\left(\left(x_i\right)_{i\in\mathbb Z}\right)=x_0x_1$. Then $\mathbb E\left[f\le …

It is not exactly the mentioned result, but in Ouchti, Lahcen On the rate of convergence in the central limit theorem for martingale difference sequences, Ann. Inst. H. Poincaré Probab. Statist. 41 …

First, by a self-similarity argument, it suffices to consider the case $T=1$. We can use the equivalence of the usual Hölder norm with the sequence norm, defined by $$ \lVert x\rVert_\alpha:=\sup_{j\ …

Let $A_N$ be the event defined by $$ A_N:=\bigcup_{n=2^{N-1}+1}^{2^N}\left\{\max_{1\le j\le q}I_{n,Z}(\omega_j)\neq \max_{1\le j\le q}I_{n,\tilde Z^{(n)}}(\omega_j)\right\}. $$ Then the following inc …

Here are some ideas, which may be too long for a comment. Denote for fixed $n$ and $k$: $$ Z_{n,k}:=\sum_{i=1}^n\left(X_i\sum_{j=1}^k Y_{i+j-1 \mod n}-\frac 1k\right). $$ Let $$ A_{n,k}:=\sum_{i=1}^{ …

Let us state Corollary 2.1 of these notes. Let $p>1$, $X\in\mathbb L^p$ and let $\left(\mathcal F_n\right)_{n\geqslant 1}$ be a filtration. Denote by $\mathcal F$ the $\sigma$-algebra generated by …