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One way to do it in a computational way is to use a representation as time changed Brownian motion $\{W_t\}$ and known results concerning Brownian motion. Write $X_t=xe^{-t}+e^{-t}W_{u(t)}$ where $u( …

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 $n$.

Look up Varadhan's lemma in any large deviations text. Specifically, Deuschel-Stroock, Exercise 2.1.24

Define $u(x)=E^x\int_0^\tau X_s ds$, then $u$ satisfies $\sigma^2 u_{xx}/2-\mu u_x=-x$ with boundary condition $u(b)=0$ and $u(\infty)=\infty$. (This is missing a boundary condition, but a good way to …

For point 1, search for ``CLT for mixing sequences'' you will be drowned by the number of hits. Also, there are CLT's for stationary sequences in many textbooks - Hall and Heyde's book has a section …

You should quantify the range of $x$ that you are interested in. For example, for $1<<x<<\sqrt{n}$ you should be in the moderate deviations regime, and you could give lower bounds by performing the a …

Background: this question is motivated by the OP's paper http://arxiv.org/pdf/1111.2328.pdf. Amir Dembo and I had answered this question privately to Aryeh a long time ago. I paste the answer here, ad …

Here is one way to formalize "concentrated": for large time, the density of the conditioned process converges to the top eigenfunction of the laplacian with Dirichlet boundary conditions, while the re …

The correct statement is the following: $$E(Y(t)X(s))=F(t) E(X(t)X(s))$$ for some function $F(t)$ which is periodic. To see that, follow verbatim the proof in the appendix of http://www3.alcatel-lucen …

Define $S_i$, $i=1,\ldots,2n$ to be the number of $1$s in the interval $[0,i]$, minus $i/2$. Then, with your notation, $Y_i:=X_{i+1}-n/2=S_{n+i}-S_i$. You are interested in $M^*_n:=\max_{i=1}^n |Y_i| …

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 …

Not so much in terms of high dimensional probability (though this is definitely a high dimensional probability question). I suspect however that you meant a slightly different question, which I answer …

Consider the binary tree of ancestry (thus, the vertices at depth $n$ are the particles that reach location $n$). Write on each edge $e$ between level $n-1$ to level $n$ the time it takes to generate …

I assume all your moments are finite. In that case, a simple proof is to approximate your random variable by truncation, use the additivity for the approximated cummulants, and then pass to the limit …

It depends on more than the variance of the increments. If they have exponential tails of high enough order, the variance is of order $1$ (look up "branching random walks", or the lecture notes of Shi …