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 ...
Iosif Pinelis's user avatar
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{...
Iosif Pinelis's user avatar
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,...
Ribhu's user avatar
  • 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 ...
zeraoulia rafik's user avatar
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 ...
Iosif Pinelis's user avatar
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<...
Iosif Pinelis's user avatar
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 ...
Iosif Pinelis's user avatar
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 ...
mike's user avatar
  • 901

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