4
votes
Wasserstein distance between product measures
$\newcommand{\de}{\delta}\renewcommand{\S}{\mathcal S}\newcommand{\T}{\mathcal T}$The answer to your question is negative if $p<2$.
Indeed, let $\nu_i=\de_0$ for all $i$, where $\de_a$ is the Dirac ...
- 106k
3
votes
Accepted
Kolmogorov-Smirnov distance and expectation
$\newcommand\ep\epsilon\newcommand\R{\mathbb R}$No such bound exists.
Indeed, take any real $\ep>0$.
Take any natural $n\ge1/(2\ep)$. Let
$$p(x):=
\sum_{j=0}^{2n-1}1\Big(\frac{2j}{2n}<x<\frac{...
- 106k
3
votes
Wasserstein distance between product measures
From the previous answer, it follows that the inequality is true when $p\ge 2$. Let $X_i, Y_i$ be the optimal choice of random variables for which
$$\|X_i-Y_i\|_p=W_p(\mu_i,\nu_i) \ \forall i=1,\ldots,...
- 249
2
votes
Stochastic volatility model question
The statement you provided suggests that a certain conditional expectation involving the logarithm of the ratio of two increments of the process, $(\ln(S_{t+\delta}/S_t))$, has two properties:
It is ...
- 2,238
2
votes
Accepted
Does this inequality hold for the cumulant generating function?
This is not true in general. Indeed, let $X$ be a zero-mean random variable (r.v.) such that $Ee^{tX}<\infty$ for $t\in[0,\tau)$ but $Ee^{\tau X}=\infty$. Then for all $t\in(0,\tau)$ the left-hand ...
- 106k
2
votes
Accepted
Does this KL divergence inequality hold?
The answer is no. E.g., if $p_1=1/2$, $p_2=1/2$, $q_1=1/100$, $q_2=99/100$, and $\beta=1/10$, then the ratio of the left-hand side of the conjectured inequality to its right-hand is $0.00877\ldots<...
- 106k
1
vote
Accepted
Concentration inequality for square roots
Let $X:=X_n$. For your probability
$$p_{a,t}:=P(|\sqrt X-\sqrt a|>t)$$
to make sense, we need to assume that $X\ge0$ and $a\ge0$. Also, if $t<0$, then $p_{a,t}=1$. So, without loss of generality ...
- 106k
1
vote
Kolmogorov-Smirnov distance and expectation
You probably can’t get the sort of bound you want, take any continuous cdf on R, and approximate it stepwise as closely as you like. The stepwise approximation puts mass only at the steps. Take a ...
- 901
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