New answers tagged pr.probability
3
votes
Accepted
Surjectivity of pushforward on image
It is true; the trick is to use the von Neumann-Jankov measurable selection theorem to construct a right inverse of $\Phi$ on $\Phi(\mathcal X)$.
The result is essentially Lemma 2.2. of [Varadarajan, ...
0
votes
Accepted
Proving bound on expectation of likelihood ratio involving mixtures
I am now trying to show that $H(\mu) \leq 1$ for $|\mu| \geq c$. As $H$ is an even function, it is sufficient to show that $H(\mu) \leq 1$ for $\mu \geq c$.
This is not true. E.g., suppose that $c=1$ ...
- 128k
2
votes
How can I randomly draw an ensemble of unit vectors that sum to zero?
This is long past the time the question was asked, but I thought I'd add some references here in case people still come across this post. First, there's an $O(n^2)$ algorithm for generating closed ...
- 747
1
vote
Accepted
Median of cardinality of set union
It turns out there is a (somewhat absurd) counterexample.
Consider $U=\{1,2,3,4,5,6\}$, $S_1 = \{1\}, S_2 = \{2\}, S_3 = \{3,4,5,6\}$. Then $f_{\mathbf{S}}(1) = \text{median}\{1,1,4\}=1$, yet $f_{\...
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2
votes
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What (continuous) stochastic processes have path measures that are absolutely continuous w.r.t. Wiener measure?
Let $\mu_0$ be the law of a Brownian motion $B$. Let $\mu$ be any measure equivalent to $\mu_0$. Then by a converse version of Girsanov there exists a progressively measurable $F$ whose sample paths ...
- 1,904
0
votes
Intensity and compensator for a jump process
You can refer to the Theorem 2.3 in this paper https://arxiv.org/pdf/2407.21651. Hope this can help you find the solution.
1
vote
Variance of bins for N balls into M bins
I think you are asking for the variance of a multinomial distribution.
Let $I_i\in[M]$ be selected independently and uniformly at random for each $i$ and let $e_{I_i}\in \mathbb{R}^M$ be a standard ...
- 111
4
votes
Accepted
Lower bound in the singularity of random Bernoulli matrices
On the singularity of random Bernoulli matrices - novel integer partitions and lower bound expansions
derives the lower bound (theorem 1)
$$\mathbb{P}\{ \text{det}(A_n) = 0\} \geq 2^{2-n} \binom{n}...
- 188k
2
votes
Hermite–Fourier expansion for the median
$\newcommand{\si}{\sigma}\newcommand{\R}{\mathbb R}$As I understand the problem, it is to compute
$$e_{k_1,\dots,k_n}:=EM_nH_{k_1}(X_1)\cdots H_{k_n}(X_n),$$
where $n\ge1$ is an odd integer, the $H_k$'...
- 128k
8
votes
Accepted
Convergence of random functions
No. Take $f^n$ all independent to be $0$ with probability $1/\sqrt{n}$ and consisting of a bump of height $1$ and width $1/n$ at a uniformly distributed location otherwise. Clearly $f^n \to 0$ in law (...
- 10.3k
3
votes
Hermite–Fourier expansion for the median
I presume you want the coefficients
$$a_k=\int_{-\infty}^\infty p(M)\psi_k(M)\,dM,$$
with
$$\psi_k(x)=\frac{1}{\pi^{1/4}\sqrt{ 2^k k!}} e^{-x^2/2} H_k(x)$$
the normalized Hermite function of order $k$ ...
- 188k
1
vote
Accepted
On the stationarity of Gaussian processes
Suppose that $(X_t)_{t\in\Bbb R}$ is a Gaussian process stationary in the wide sense, so that $m(t):=EX_t=m$ and $Cov\,(X_s,X_t)=g(s-t)$ for some real $m$, some real-valued function $g$, and all real $...
- 128k
5
votes
Accepted
Interpretation of an asymptotic result in probability
$\newcommand\ka\kappa$Intuition for this result is as follows.
The condition on $h'$ implies that $h(y)=(1+o(1))y$ (as $y\to\infty$). So, for the tail function $T$ given by $T(y):=P(Y\ge y)$ we have
$$...
- 128k
0
votes
Upper and lower bounds for a Rademacher-type expectation
You can get one of the bounds using the recently proved Tomaszewski's Conjecture, which can be reformulated as that for
$$
X= \sum_{i=1}^{n} a_{i} \epsilon_{i}
$$
we have
$$
\mathbb P\left(|X|\le \...
- 258
1
vote
Accepted
Does convergence in probability of iid samples imply convergence in measure of the sampled functions?
Counterexample: $g_i(x)=x-1/2$ for all $i$ and all $x\in[0,1]$.
- 128k
12
votes
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Moments of a random variable related to uniform distribution on sphere
Note that the random vector $u=(u_1,u_2,\ldots u_n)$, uniformly distributed on the unit sphere, can be replaced by the ratio $u=y/|y|$, with $y=(y_1,y_2,\ldots y_n)\sim N(0,I_n)$ a multivariate ...
- 188k
2
votes
Unique coupling
For two measurable sets $A,B$, let $p=\mu(A)$ and $q=\nu(B)$. Consider any coupling of a Bernoulli$(p)$ and a Bernoulli$(q)$, say, $C:\{0,1\}^2 \to [0,1]$. Then we can find a coupling $(X,Y)$ of $\mu$...
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2
votes
Accepted
Unique coupling
The only way this can happen is the situation that you described, where at least one of the measures gives full measure to a single point. In fact, the proof below does not require the two Polish ...
- 23.2k
2
votes
The cars problem, again
Here's a sketch for a linear upper bound, but considering how similar it is to first-passage percolation on the oriented grid I suspect determining the exact constant is hard.
I'll use the convention ...
1
vote
Accepted
Lower Bound on the Probability for the Sum of IID Random Variables
This conjecture is not true.
E.g., let $P(X_i=q)=p=1-P(X_i=-p)$, where $p\in(1/2,1)$ and $q:=1-p$, so that $P(X_i>0)>1/2$, $EX_i=0$, and $Var\,X_i=pq$ for each $i$. Then for any real $c>0$ ...
- 128k
3
votes
Identities and inequalities in analysis and probability
I found the following elementary identity quite useful in proving some non-trivial inequalities:
$$
\min\{u,v\}
=
\sqrt{uv}\exp\left(-\frac12\left|\log\frac{u}{v}\right|\right),
\qquad
u,v>0.
$$
...
1
vote
Accepted
On the behaviour of individual random walks of a Markov Chain
Indeed, there exist well-established results that provide quantitative bounds on the probability that the empirical distribution of states along a finite path deviates from the stationary distribution....
- 232
1
vote
Accepted
Reconstruction of law of diffusion process from call option values
For any random variable $X$ with $E\max(X,0)<\infty$, you can determine the distribution of $X$ if you know the values of
$$g(c):=E\max(X,c)$$
for all real $c$.
Indeed, take any real $c$ and any ...
- 128k
1
vote
Accepted
What happens to an SDE conditional on the underlying Brownian motion being close to $f \in C[0, T]$?
It is not the case that the support is a singleton in general, and in fact I believe the support will be generically full when the dimension is high enough. I think the full picture is a bit subtle, ...
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0
votes
Nash equilibria of a "minority game"
You may want to consult the following paper:
https://pure.uvt.nl/ws/portalfiles/portal/854659/dp2007-61.pdf
- 745
1
vote
Accepted
How do the total variation distances of the marginals relate to the total variation distance of the joint under independence?
There is, but it is not tight, e.g. the upper and lower bounds qualitatively differ.
What you're asking about is typically referred to as "tensorization" of the total variation distance.
It ...
- 2,183
0
votes
Upper bounds on quotients of binomial coefficients
After cancellations and a bit of algebra, for $t:=\gamma>1$, we get
$$\begin{aligned}
f(n_0)&=\prod_{j=0}^{n_0-1}\Big(1-\frac{m}{n-j}\Big) \\
&\le\exp\Big(-m\sum_{j=0}^{n_0-1}\frac1{n-j}\...
- 128k
0
votes
Accepted
Upper bounds on quotients of binomial coefficients
Your formula yields $$f(n_0)=\prod_{j=0}^{m-1}\frac{n-n_0-j}{n-j}\leqslant \left(\frac{n-n_0}{n}\right)^m=
\left(1-\frac{n_0}{n}\right)^m\leqslant e^{-mn_0/n},$$
that is about $e^{-1/\gamma}$.
- 109k
5
votes
Accepted
Sub-Gaussian concentration without the sub-Gaussian norm
$\newcommand\si\sigma$The answer is no.
E.g., suppose that $P(X_i=1)=2/e=1-(X_i=0)$ for $i=0,1$.
Then $X_0$ and $X_1$ are sub-Gaussian with parameter $\si=1/\sqrt2$, so that we can take $\si_0=\si_1=1/...
- 128k
2
votes
Problem in Probability Theory and Functional Analysis
This statement is false in general. E.g., suppose that $C=I=C[0,1]$.
Then the function $1_{[0,1/2]}$ is bounded and $\sigma(C)$-measurable but not in $I$.
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