Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
5,175 questions
0
votes
1answer
72 views
Estimating expectation of a slightly strange sum
Let $X$ be a random variable with support on the positive integers (you can assume $\mathbb{E}[X^2] <\infty$ if needed, or even higher moments if needed), and let $S(i)=\mathbb{P}(X\geq I)$. ...
-4
votes
0answers
37 views
Mathematics grade 12 probability [on hold]
Stella has 12 coins in her hands, six of these coins are Looney. She accidentally (at random) drops three of these points. Determine the probability that exactly two of the drop coins are loonies.
4
votes
1answer
51 views
The minimum of the reciprocals of some Poisson random variables
Let $X_1,\dots,X_k$ denote a collection of independent samples of a Poisson random variable whose mean also happens to be equal to $k$. Does the quantity $$k\boldsymbol{E}\min\left\{ \frac{1}{1+X_{1}}...
-3
votes
0answers
36 views
Calculating the probability of obtaining exactly four distinct values when a die is rolled six times [on hold]
What is the probability of getting
a) 4 distinct numbers (no order in the outcome e.g. 1,2,3,4 or 4,5,6,2 etc)
b) 5 distinct from rolling a die 6 times
So far I was able to calculate the ...
1
vote
0answers
22 views
Asymptotics for a random set cover problem
Suppose you are given a positive integer $k$ and a probability distribution $f$ on the positive reals. I am interested in the limiting behavior of the following process as $n\to\infty$:
Create an ...
0
votes
0answers
46 views
Ladder times of a Brownian motion with drift
Let $(B_s)_{s_\geq 0}$ be a standard Brownian motion and fix $t>0$. For $u>0$, set $T_u=\inf\{s>0, B_s+s t>u\}$. Now consider $x>0$ such that $\sup_{0 \leq s \leq x} (B_s+st)=B_x+xt$ ...
-2
votes
0answers
57 views
Birthday Calendar Gaps [on hold]
I work at a company that posts a birthday calendar. I noticed that there was a string of four consecutive days with no birthdays. What is the probability of that happening?
Problem Statement
Given n ...
0
votes
1answer
45 views
Product of independent random variables and tail deconvolution
Suppose $X, Y$ are two independent non-negative random variables. The conditions
(i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$
(ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...
2
votes
0answers
69 views
Markovian Control in a stochastic control problem
I have a very simple control problem.
\begin{align*}
V(0,X) &= \sup_{(C_t)_t} E_0 \left[ |X_t| 1_{C_t = 0} + C_t \right] \\
\text{ s.t. } & d X_t = C_t d B_t, \quad X_t \in [-1,1], C_t \ge 0,...
1
vote
1answer
95 views
Existence of certain event
Suppose that $X$ is an unbounded random variable such that $\operatorname EX=0$ and $\operatorname E|X|^q=1$ with some $q>2$. Only the distribution of $X$ matters, so the probability space can be ...
0
votes
1answer
42 views
How to calculate the probability of 2 events happening in time series under only cdf information?
In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...
7
votes
2answers
809 views
Differentiating an integral that grows like log asymptotically
Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...
1
vote
1answer
137 views
+50
Showing $o(1)$ convergence for ratio of successive binomial tail probabilities
For a Binomial$(n,p)$ random variable $X$, I'm interested in showing that
$$
\frac{P(X>c)}{P(X>c-1)}=1-o(1)
$$
uniformly in $c\in\mathcal{R}$, where $\mathcal{R}$ is the range of interest (Note ...
5
votes
1answer
140 views
Shannon entropy of $p(x)(1-p(x))$ is no less than entropy of $p(x)$
If $p(x)$ is a discrete probabilistic density function, one could construct another discrete probabilistic density function proportional to $p(x)[1-p(x)]$ with a corresponding partition function to ...
-5
votes
0answers
34 views
Is the effect of Tr on Y identifiable conditioning on G? [closed]
enter image description here
I wonder if Tr and Y are independent conditioning on G? I am working on a causal inference problem, and I wonder whether and why the effect of Tr is identifiable if G is ...
0
votes
0answers
26 views
p-Variation distance defines semi-martingales
Question
When, does the process $\tilde{X}_t$, defined path-wise by
$$
\tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right),
$$
define a ...
8
votes
2answers
503 views
Random Walks on high dimensional spaces
I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in arbitrary directions (uniformely on the unit sphere $S^1$, not left-right-up-down), ...
1
vote
1answer
38 views
Jump size of process in proof of martingale CLT bounded
I'm currently trying to understand the proof of the final theorem of (Helland, I. S. (1982). Central limit theorems for martingales with discrete or continuous time) where I need the following:
For $\{...
5
votes
1answer
165 views
Relation between the two possible KL divergences of two distributions
Given that I know $$D\left(P\parallel Q\right)<\alpha,$$ can I say anything about $D\left(Q\parallel P\right)$ in terms of an upper bound on it?
Also, given this upper bound on $D\left(P\parallel ...
1
vote
1answer
117 views
+50
Time discretization in the Feynman-Kac formula with boundary conditions
I am applying the Feynman-Kac theory for solving a PDE with boundary conditions.
For the SDE simulation I use the Euler-approximation, which introduces a time-step $h$ for the Brownian Motion, and ...
2
votes
0answers
110 views
Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)
In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
3
votes
1answer
67 views
Powers of Frobenius norm of sum of random matrices
For $i= 1, \ldots, n$, let $A_i \in \mathbb{R}^{d \times d}$ be random i.i.d. matrices with $E [A_i] =0$.
Can we relate (upper bound) $E[\|\sum_{i=1}^n A_i \|_F^4]$ to $E[\|A_i\|^4_F]$ ?
6
votes
1answer
171 views
Approximating $\mathbb{E}[1/X]$
I am well aware (as for instance discussed here https://math.stackexchange.com/questions/910846/is-it-true-in-general-that-e1-x-1-ex) that for an arbitrary random variable $X$ it does not hold that $\...
4
votes
1answer
105 views
Is there a counterexample to the Thin Shell Conjecture for sub-exponential distributions?
The thin shell conjecture states that there exist universal constants $C,c>0$ such that every logconcave isotropic random vector $X$ in every Euclidean space $\mathbb{R}^n$ satisfies
$$\mathbb{P}\...
3
votes
0answers
72 views
Upper bounding the start of a distribution's CDF, given bounds on first moments
Take nonnegative random variables $X$ whose first $K$ moments have bounds:
$\mu^k\leq E[X^k]\leq c\mu^k$ for each $k=1,\dots,K$.
In this case what is an upper bound for $P(X\leq O(\mu))$?
I am ...
4
votes
0answers
79 views
On a much weaker version of the Normal conjecture
I would like to ask you about the following question. It is conjectured that every algebraic irrational number is normal (absolutely normal). I know the result by Bugeaud and Adamczewski about the non-...
3
votes
1answer
109 views
Expected size of the smallest preimage set
Let $f$ a function from $\{0, 1 \}^{2n}$ to $\{0, 1 \}^{n}$ uniformly picked at random. I would like to have an estimation of the expected size of the smallest premiage set of $f$, more formally $\...
0
votes
1answer
43 views
Anti-concentration: upper bound for $P(\sup_{a \in \mathbb S_{n-1}}\sum_{i=1}^na_i^2Z_i^2 \ge \epsilon)$
Let $\mathbb S_{n-1}$ be the unit sphere in $\mathbb R^n$ and $z_1,\ldots,z_n$ be a i.i.d sample from $\mathcal N(0, 1)$.
Question
Given $\epsilon > 0$ (may be assumed to be very small), what is ...
2
votes
0answers
79 views
The Kleisli Category of the Monad of Measures of Finite Support and its composition formula
In this post, I was introduced to the monad of finitely supported measures.
$HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad. Let's call this ...
-1
votes
0answers
32 views
A limit for two correlated variables [on hold]
Suppose we have two correlated Normal variables $X_A$ and $X_B$, with respective standard deviation $A$ and $B$, and correlation $\rho$. The variable $X_A + X_B$ has a standard deviation in excess (...
2
votes
1answer
100 views
bp continuity of Markov operators / semigroups
Let $B_b(E)$ be the space of bounded measurable functions on some Polish space $E$ endowed with the supremum norm. It seems quite classical that Markov semigroups $P_t:B_b(E)\to B_b(E)$ are in one to ...
6
votes
0answers
66 views
mean distance between subspaces
Consider the Haar measure $\mu$ on the Grassmannian $G(n, k)$ of $k$-dimensional subspaces in $\mathbb{R}^n.$ Now, pick pairs of subspaces uniformly at random with respect to $\mu,$ and compute their ...
2
votes
1answer
99 views
How to uniformly sample a square (0,1)-matrix whose trace is 0 and whose row sums and column sums are the same?
Happy New Year!
Suppose I would like to sample a $n \times n$ (0,1)-matrix whose trace is 0, and whose row sums and column sums are all $m$ with $1 \le m \le n-1.$ How can I sample this matrix ...
4
votes
1answer
66 views
Sampling uniformly from the vertices of a polytope
I'm looking for a reference on how to sample uniformly (and preferably efficiently, elegantly, etc.) from the vertices of a polytope. I gather that enumerating vertices is hard. I also note the MO ...
0
votes
0answers
56 views
Does the linear combination of the quantile $\alpha F^{-1}(\tau)+\beta G^{-1}(\tau)$ still a quantile
$F(x)$ and $G(y)$ are distribution functions.
Define the $\tau$th quantile for cdf $F(x)$, $G(y)$ as
$$\xi_\tau\equiv F^{-1}(\tau)=\inf\{x:F(x)\ge \tau\}$$
and
$$\eta_\tau\equiv G^{-1}(\tau)=\inf\{y:...
1
vote
0answers
61 views
Is there solution to a backward stochastic differential equation with $yz$ in the generator?
Please consider the following backward stochastic differential equation:
$$ Y(s)=\xi+\int_{s}^{T}a(u)Y(u)+b(u)Y(u)Z(u)du-\int_{s}^{T}Z(u)dW(u)$$
Here $a(s)$, $b(s)$ are square-integrable stochastic ...
0
votes
0answers
72 views
What are the open problems in rough volatility models?
The 2014 paper Volatility is Rough argues that empirically, fractional Brownian motion with $H=0.1$ is a good description of volatility that comes out of high frequency trading.
Since then there has ...
-1
votes
1answer
116 views
What's the probability of two independent events in time domain?
Suppose there are two independent events A and B. The probability that A or ...
2
votes
1answer
135 views
Fourier transform of a simple random walk
Consider the usual simple random walk on $\mathbb{Z}$, taking steps of +1 or -1 with equal probability. Of course, each trajectory corresponds uniquely to an element of $\{-1,1\}^\infty$. Now, there ...
0
votes
0answers
41 views
Limiting a sequence of moment generating functions [migrated]
I was trying to solve the following problem:
Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables with the probability mass function $P\{X_n = \pm1 \} = \frac{1}{2}$, $n \in \...
0
votes
0answers
56 views
Binomial CDF - Wiki vs Reality [migrated]
In https://en.wikipedia.org/wiki/Binomial_distribution the binomial CDF is defined as
$F_{Bin(n,p)}(k) = I_{1-p}(n-k, k+1)$,
Where $I$ is the regularized incomplete beta function.
There is a proof ...
3
votes
0answers
118 views
Are there any conditions on the moments that make a measure a probability measure?
For a positive Borel measure $\mu$ on the real line interval $[-1, 1]$, let $\displaystyle{m_n = \int_{-\infty}^\infty x^n d\mu(x)}$, i.e. the $n$th moments of the measure. Are there any conditions ...
5
votes
1answer
80 views
What is the pdf of Laplace distribution conditioned on a plane? How can I sample from it?
Our goal is to sample from the Laplace distribution conditioned on a linear subspace. Here are the details of this problem.
Let
$$p(x) \propto \exp(-\|x\|_1/\sigma)$$
be the pdf of the Laplace ...
8
votes
1answer
153 views
On the existence of a particular type of finite measure on $\mathbb N$
Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: ...
6
votes
0answers
64 views
Distributions of “sequential” binomials
I have come across the following stochastic process which seems very elementary, although I do not know any appropriate terminology for it; I greatly appreciate any suggestions!
Suppose I am given ...
1
vote
1answer
64 views
Sharp tail bounds for the maximum of an iid sample of a random variable supported on $[0, 1]$
Let $X_1,\ldots,X_n$ be an iid sample from a distribution supported on $[0, 1]$.
Question
What are some sharp concentration inequalities (i.e tail bounds) empirical statistic defined by $Z_n := \max(...
2
votes
1answer
109 views
About a pattern of hitting times for a simple random walk
Let $\omega_1, \omega_2, \ldots$ be uniform iid on $\{-1,1\}$, and let $X_n = \sum_{i=0}^n \omega_i$ be the corresponding simple random walk. Fix some integer $N$, and let $h^+_N$ be the first time $...
5
votes
1answer
58 views
Estimating the size of the remainder in a random partition
Pick a sequence of real numbers $x_i$ as follows. Put $x_0=1$. If $x_i$ is chosen, then pick $x_{i+1}\in[0, x_i]$ according to the uniform distribution. Obviously we have $x_i\rightarrow 0$ with ...
1
vote
0answers
51 views
Almost orthogonality of independent random vectors [closed]
If $X_1$ and $X_2$ are two independent isotropic random vectors in $\mathbb{R}^n$,
then $\mathbb{E}\|X_i\|_{2}^{2}=n$, $\mathbb{E}\langle X_1,X_2\rangle^{2}=n$.
How can I show from the above result ...
0
votes
1answer
32 views
Minimizer for Mean-Variance Portfolio Optimization [closed]
Let $\lambda \in (0,\infty).$ Does there exists a minimizer for the set
$$
\{ -\text{E}[X] + \lambda \text{Var}[X],\; X \in L^2(\Omega,\mathcal{F},P) \} ?
$$
