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You have too much notation here - which is quite confusing. For instance, what is your "combined probability space" $\Omega$ (it seems to be $X\times Y$ as this is where your Radon-Nikodym derivatives …

Yes - take $Y=-X$ if $X<0$ and $Y$ be "negative gaussian" independent of the value of $X$ if $X>0$. Then $X+2Y=-X>0$ for $X<0$ (i.e., $X+2Y$ is positive at least with probability 1/2); on the other ha …

It can be done in pretty much the same way as for a single vector by using the fact that if you fix $x$, then the conditional distribution of $y-x$ is uniform on the sphere of radius $\sqrt{1-\gamma^2 …

Theorem 2.1 in the quoted paper of Diaconis and Zabell actually states that boundedness of the ratios $Q(\omega)/P(\omega)$ is a necessary and sufficient condition for obtaining $Q$ from $P$ by condit …

Yes. I presume that your "1-Wasserstein" distance is what is otherwise called the transportation metric. Let $M$ be any measure with marginals $\mu_1$ and $\mu_2$, so that $\mu_1=\int \delta_x\,dM(x …

In the first place the answer depends on the nature of your data (e.g., numerical continuous, numerical discrete, nominal etc.). In each of these cases the empirical measures on the range of your data …

First, the definition of the KL divergence has nothing to do with compactness as it is defined entirely in terms of the density of one measure with respect to the other one. Second, the KL divergence …

Such a family doesn't exist. If you fix the maximal atom (say, $p$) of a distribution $\mu$ supported by $n$ points, then its entropy is maximal when all the remaining atoms have the same weight $(1-p …

I will begin with a reformulation of your question which makes it not only more symmetric, by also (at least for me) more natural and interesting. I will pass from your variables $(W,H,Q)$ to new vari …

I presume that the state space $X$ is compact. Then your set is precisely the set of functions $f\in C(X)$, for which the Cesaro averages $$ C_n f(x) = \frac1{n+1} \Bigl( f(x) + f(Tx) + \dots + f(T^n …

Since you are talking about finite (and presumable connected) graphs, a.e. sample path (no matter where it starts) visits all vertices, so that the answer to your second question (about probability of …

Let $f$ be a piecewise constant function which takes value $f_i$ on an interval of length $p_i$, and let $q_i=f_i p_i$. Then $$ H(f) = - \int f \log f\,dx = - \sum q_i \log\frac{q_i}{p_i} = - D_{KL} ( …

To begin with, endowing the set of integers with the upper density is quite far from making it a probability space. Nonetheless, the question you ask still makes sense. Namely, you consider the $G$-va …

The example one should have in mind is the Fubini theorem for the unit square (endowed with the Lebesgue measure) projected onto the horizontal base. The conditional measures are then just the Lebesgu …

The point is that the nature of these two distributions is completely different. The degree distribution is local: in order to find it one one just has to know how the 1-neighbourhoods of vertices loo …