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There are several treatments of Hairer's theory, including lecture notes of his that try to give the "big picture". Brief answers to some of your questions: 1) Those are the solutions to the underl …

First, an upper bound that beats your second bound is the following: use the equality $$\max(a,b)=(a+b+|a-b|)/2.$$ Then $$E\max(X,Y)=E|X-Y|/2\leq \frac{1}{2} (E|X|+E|Y|)=E|X|=\sqrt{2/\pi}\sim 0.798$$ …

An early occurence of such bounds is in the theorem of Theorem of vonBahr and Eseen vonBahr, B., Esseen C.-G.: Inequalities for the rth absolute moment of a sum of random variables, $1\leq r \leq 2$. …

My initial answer was wrong. Instead of completely deleting it, I left it at the bottom of the post. I use notation now that slightly differ from my original post. As before, I give only a partial a …

Tilting refers to a change of measure of the form $e^{\lambda \cdot x}/C(\lambda)$ where $C(\lambda)=E(e^{\lambda\cdot X})$. (The setup is that the measure for which you are proving large deviations i …

The following is a partial answer. It is useful to rescale. So think of a Poisson process in the plane and consider the box of side $n$. There will be roughly $n^2$ points there. Let $A_{[a,b]}$ be …

I assume your random $M_i$s are from $O(3)$, not $SO(3)$. In fact, I suspect that the answer depends on whether $M_1M_2$ has eigenvalue $1$ or eigenvalue $-1$. If I did not make a mistake, when you e …

The object you look at is called the Gaussian Free Field (on your graph, with zero boundary conditions) in dimension $2$. There is a lot known about it. For some pointers see the Wiki page http://en.w …

Divide the interval $[|x|,1]$ into $-\log_2 |x|$ intervals $[r_i,r_{i+1}]$. Radially, the BM performs a (simple) random walk between the circles of these radii, and there will be at least $-\log_2 |x| …

A good way to measure the set of maxima is the Hausdorff dimension of the set of records, which for BM is a.s. 1/2. Because of time/scale invariance, the dimension is the same for $\alpha B_\cdot$, an …

There are several versions of this type: 1) K. Marton has results for dependent variables. Maybe closest to what you ask (for convex functions $f$) is the paper of Samson: Samson paper 2) For a mart …

This has been treated in the literature: http://arxiv.org/pdf/0801.0059v3.pdf also see http://arxiv.org/pdf/1201.3261.pdf In particular, the upper bound goes to 0, but only polynomialy in $n$.

Of course not. Let $X,Y$ be negatively correlated centered standard Gaussian (say with correlation -1/2). Let $Z=X+Y$ and let ${\cal A}=\sigma(X)$, ${\cal B}=\sigma(Y)$. Then $E(Z|{\cal A})=X+E(Y|{\ca …

For $p>1$, the random variables you discuss do not possess exponential moments; You are in the regime of large deviations with stretched exponential tails. See for example the following recent paper b …

There are links, for example in the theory of singular integrals, see section 6.2 in Stroock's book on probability theory. Also, googling "BMO and martingales" will give you information in the directi …