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 ...

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{...

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,...

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
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 ...

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<...

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 ...

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 ...

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