Rosenthal like inequality for weak $\mathbb L^p$-norms

Question

Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if $(X_i)_{i=1}^n$ is a real valued martingale difference with respect to the filtration $(\mathcal F_i)_{i=1}^n$ (that is, $(S_j:=\sum_{i=1}^jX_i)_{j=1}^n$ is a martingale with respect to this filtration), then $$\frac 1{C_p}\mathbb E\left(\sum_{i=1}^nX_i^2\right)^{p/2}\leqslant \mathbb E\left|\sum_{i=1}^nX_i\right|^p\leqslant C_p\mathbb E\left(\sum_{i=1}^nX_i^2\right)^{p/2}.$$ Hence the $\mathbb L^p$ norm of the partial sum is controlled by those of the quadratic variation.

Now define for a real valued random variable $X$: $$\lVert X\rVert_{p,\infty}:=\left(\sup_{t\geqslant 0}t^p\mu\{|X|\geqslant t\}\right)^{1/p}.$$ This is equivalent to a norm (namely $N(X):=\sup_{\mu(A)>0}\mu(A)^{-1+1/p}\int_A|X|\mathrm d\mu$).

I would like to know whether there is a similar inequality to Rosenthal's one, that is, a control of $N(S_n)$ in terms of those of $N\left(\sqrt{\sum_{i=1}^nX_i^2}\right)$ plus maybe an other term. This seems to be a natural question which has probably been investigated, but I didn't manage to find a reference.

There are weak-$\mathbb L^p$ versions of Rosenthal's inequality for independent random variables, but I would like to see a reference dealing with an extension to martingale differences, namely:

Let $(\Omega,\mathcal F,\mu)$ be a probability space $p\gt 2$. Is there a constant $C_p$ such that if $n$ is an integer and $(X_j)_{1\leqslant j\leqslant n}$ is a martingale difference with respect to the filtration $(\mathcal F_j)_{1\leqslant j\leqslant n}$ with $\lVert X_j\rVert_{p,\infty}\lt\infty$, then $$C_p^{-1}\left\lVert \sqrt{\sum_{j=1}^nX_j^2}\right\rVert_{p,\infty}\leqslant \left\lVert \sum_{j=1}^nX_j\right\rVert_{p,\infty}\leqslant C_p\left\lVert \sqrt{\sum_{j=1}^nX_j^2}\right\rVert_{p,\infty}~? $$

Answer 1

The result you want is mentioned in Remark 6 in

Johnson, W. B.(1-TXAM); Schechtman, G.(IL-WEIZ) Martingale inequalities in rearrangement invariant function spaces. Israel J. Math. 64 (1988), no. 3, 267–275 (1989).

Answer 2

In any Banach lattice, a sufficient condition for the equivalence of the norm of a sum of elements and the norm of the square function is that the sequence be unconditional and the lattice has non trivial cotype. This is proved in many places, in particular in the introductory book of Albiac and Kalton and in volume 2 of Lindenstrauss and Tzafriri. In a rearrangement invariant function space, martingale difference sequences are unconditional provided the upper and lower Boyd indices are in the range $(0,\infty)$ because you can then do an interpolation argument (vol. 2 of [LT]). IIRC, for martingale difference sequences you don't also need non trivial cotype because you get an upper estimate for a linear combination of martingale differences in terms of the square function both in the space and in the dual space. You have some duality because the filtration the martingale determines gives you commuting contractive projections that converge to the identity operator.

This is either an outline of a proof or utter nonsense.

Answer 3

I think I came out with a proof of the existence of a constant $C_p$ such that if $(S_n,\mathcal F_n)$ is a martingale with differences $X_i$, then $$C_p^{-1}\left\lVert\left(\sum_{i=1}^nX_i^2\right)^{1/2}\right\rVert_{p,\infty} \leqslant \lVert S_n\rVert_{p,\infty}\leqslant C_p\left\lVert\left(\sum_{i=1}^nX_i^2\right)^{1/2}\right\rVert_{p,\infty} +C_p\left\lVert\left(\sum_{i=1}^n\mathbb E[X_i^2\mid\mathcal F_{i-1}]\right)^{1/2}\right\rVert_{p,\infty}.$$

The proof is quite technical and relies on the proof of classical Rosenthal's inequality for the $p$-th moment. Some intermediate steps gives estimates on the tails and then are converted into bounds on the moments.

The following lemma is a part of Lemma 2.3. in Hall and Heyde's book.

Lemma. Let $(S_i,\mathcal F_i)_{i\geqslant 1}$ be a non-negative sub-martingale and for $\theta>0$, define $$Y:=\max\left\{\theta\left(\sum_{i=1}^nX_i^2\right)^{1/2}, \max_{1\leqslant i\leqslant n}S_i\right\}.$$ Then for each $t>0$, $$t\mu\{Y\gt \sqrt{1+2\theta^2}t\}\leqslant 3\mathbb E[S_n\chi_{Y>t}].$$

The proof relies on the use of the non-negative submartingale $(T_i,\mathcal F_i)$ with $$T_i:=S_i\chi\left\{\theta\left(\sum_{j=1}^iX_j^2\right)^{1/2}\gt t\right\}$$ and the stopping time $\tau:=\min\{n,\inf\{i\mid\theta\left(\sum_{j=1}^iX_j^2\right)^{1/2}\gt t \}\}$.

Choosing $\theta:=2$, noticing that $2\left(\sum_{i=1}^nX_i^2\right)^{1/2}\leqslant Y$ and the link between $\lVert X\rVert_{p,\infty}$ and $\sup t^p\mu\{X\geqslant t\}$, we obtain the inequality $$\left\lVert\left(\sum_{i=1}^nX_i^2\right)^{1/2}\right\rVert_{p,\infty} \leqslant \frac{3^{p+1}}2\lVert S_n\rVert_{p,\infty}.$$ Using the martingales $(\mathbb E[S_n^+\mid\mathcal F_i])_i$ and $(\mathbb E[S_n^-\mid\mathcal F_i])_i$ we can extend to any martingale $(S_n)$, namely $$\left\lVert\left(\sum_{i=1}^nX_i^2\right)^{1/2}\right\rVert_{p,\infty} \leqslant 3^{p+1}\lVert S_n\rVert_{p,\infty}.$$

Here is the weak moment version of Lemma 2.3 of Hall and Heyde

Lemma. If $X$ and $Y$ are two non-negative random variables, $\beta>1$, $\delta\gt 0$ and $\varepsilon\gt 0$ are such that for each positive $t$, $$\mu\{X\geqslant\beta t,Y\leqslant \delta t\}\leqslant \varepsilon\mu\{X\geqslant t\}$$ and $\beta^p\lt 1$, then $$\lVert X\rVert_{p,\infty}\leqslant \frac{\beta}{(1-\beta^p\varepsilon)^{1/p}}\delta^{-1}\lVert Y\rVert_{p,\infty}.$$

To conclude the proof, we need to apply this lemma to $X:=\max_{1\leqslant i\leqslant n}|S_i|$ and $Y:=\max\left\{\left(\sum_{i=1}^n\mathbb E[X_i^2\mid\mathcal F_{i-1}]\right)^{1/2}, \max_{1\leqslant i\leqslant n}|X_i|\right\}$ with $\varepsilon:=\frac{\delta^2}{(\beta-\delta-1)^2}$, $\beta >1$ and $\delta$ small enough. For a fixed $\lambda$, define $$E_k:=\left\{\lambda<\max_{1\leqslant i \leqslant k-1}|S_i|\leqslant \beta\lambda;\max_{1\leqslant i\leqslant k-1}|X_i|\leqslant \delta\lambda;\sum_{i=1}^k \mathbb E[X_i^2\mid\mathcal F_{i-1}]\leqslant \delta^2\lambda^2\right\}$$ and $T_i:=\sum_{k=1}^i\chi_{E_k}X_k$. From the inclusion $$\{X>\beta\lambda;Y\leqslant \delta\lambda\}\subset \{\max_{1\leqslant i\leqslant n} |T_i|>(\beta-\delta-1)\lambda\},$$ the fact that $(T_n,\mathcal F_n)$ is a martingale and Doob's maximal inequality $$\mu\{X>\beta\lambda;Y\leqslant \delta\lambda\}\leqslant (\beta-\delta-1)^{-2}\lambda^{-2} \mathbb E(T_n^2).$$ We conclude by the estimate $$\mathbb E(T_n^2)\leqslant \delta^2\lambda^2\mu\left\{\max_{1\leqslant i\leqslant n}|S_i|\geqslant \lambda\right\}.$$