5

Of course the $X_k$ are not independent as random variables. So I assume you are referring to some notion of asymptotic independence, and it would help if you state your conjecture more precisely. One natural guess is equidistribution (see [1]) of r-tuples $(X_k,\ldots,X_{k+r-1})$. However, The triples $(X_k, X_{k+1},X_{k+2})$ will lie on a bounded ...

3

Assuming you mean $f(X_1,\ldots,X_n) = |\sum_{i=1}^N X_i|^{-1}$, the case $n\geq 2$ is actually easier than $n=1$. You could use that there exists a constant $C>0$ such that for any $x\in\mathbb{R}^n$ the expectation $\mathbb{E}[ 1 / |X_1 + x| ] < C$. Then automatically also $\mathbb{E}[ 1 / |X_1 + \ldots + X_n| ] < C$ by taking $x$ to have the ...

3

The probability distribution $P(R)$ of $R=n|E(X)|$ was calculated by Kluyver (1906), it is given by
$$P(R)=\frac{1}{2\pi}\int_0^\infty [J_0(x)]^n J_0(rx)x\,dx.$$
For $n\gg 1$ one has a Rayleigh distribution (here is derivation including higher order corrections):
$$P(R)=\frac{2R}{n}e^{-R^2/n}.$$
The desired integral then becomes
$$I=\int_0^{\infty}\frac{n}{R}...

2

It is known that:
if $b\log \alpha+ c\log \beta>0$, then $X_t\to\infty$ almost surely,
if $b\log \alpha+ c\log \beta<0$, then $X_t$ is recurrent and has a unique stationary measure, with (if $c>0$) tail $\mu([t,+\infty))\sim c t^\kappa$ for some $c>0$, $\kappa<0$. One can also derive a Central Limit Theorem, and many other limit theorems.
...

2

Let $f\in D(0,T)$. Let $\mathsf{Disc}_n(f)=\{t\colon |f(t^+)-f(t^-)|\ge \frac 1n\}$. I claim this set is discrete. If not, there is a sequence of distinct points $(t_k)\in\mathsf{Disc}_n(f)$ converging to some $t_0$. There is then a subsequence converging to $t_0$ consisting entirely of points on the left or entirely of points on the right. This contradicts ...

2

If you have a second order elliptic operator L on a smooth noncompact connected manifold then you can always find a smooth function f>0 such that Lf > 0 . See the paper by Napier and myself in L'Enseignment Mathematique vol 50 2004 pages 367-390 .

2

In view of your Note 1, it appears that the definition of $H_w Q(s,a)$ has to corrected as follows:
$$H_w Q(s,a):=\gamma w\left(r(s,a)+\sum_{s'\in S}p(s'|s,a)\max_{a\in A}Q(s',a)\right)
+(1-w)\max_{a\in A}Q(s,a).$$
(Your definition is missing $\gamma$.)
Now it is clear that $H_w Q$ is not Lipschitz in $(w,Q)$, because $H_w Q$ is "of degree $2$" (...

1

The truly minimal condition on $X$ that guarantees that the function $(t,x)\mapsto p_t(x):=\chi_t(x)$ is continuous is tautological: $p_t(x)$ is continuous in $(t,x)$ if and only $p_t(x)$ is continuous in $(t,x)$. As far as the minimality is concerned, I don't think you can do much better than this.
However, one can rather easily see that the sample ...

1

I do not think sample path continuity suffices. Here is my alleged counterexample. The densities are 1 + .5*sin(x/(1-t)), 0 < t < 1 . As t -> 1 this converges to the uniform by Riemann-Lebesgue, but, of course, it isn't continuous on [0,1]x[0,1]. To get a stochastic process whose densities these are, let F_t be the cumulant and simple take $X_t(...

1

This Theorem is a direct consequence of the Skorohod representation of Martingales. You can find it, along with many variants, in
Hall, Peter, and Christopher C. Heyde. Martingale limit theory and its application. Academic press, 2014.

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