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.
9,022 questions
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Minimum conditional expectation of complement of event given conditional expectation of event?
Suppose $X$ is a pdf over $[0,m]$ and $Y$ is a binary experiment on $X$ such that $P(Y=1|X)$ is continuous, and we have that $\mathbb{E}[X|Y=1] = \mu_y$ and $\mathbb{E}[X] < \mu_y$. Is it always ...
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530
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Giving a general term of a recursive function, and upper bound for it
Let a constant $B \ge 1$, and let $l_1 = 0$, $b_1 = 0$ be the values of $l$ and $b$ (respectively) at time $t = 1$.
Let $l_{t+1} = l_t + 1$ if $b_i < B$, and $l_{t+1} = l_t$ otherwise
Let $b_{t+1}...
- 123
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1
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1k
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Expected value with a kronecker product and Gaussian distributional assumption
What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $? The kronecker product is where the confusion ...
- 101
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1
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816
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Two different definitions of Erdos-Rényi random graph
There are two or more ways to define an Erdos-Rényi random graph. Let consider the following two:
1) $G_n=(V_n,E_n)$ with vertex set $V_n=(1,\dots,n)$ and edge set $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \...
- 293
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1
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217
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Concerning Jump process (Lévy process)
Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is
$$ \nu \left( dx\right) = A \sum_{...
- 99
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258
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Convergence of Dirichlet Forms
If a sequence of Dirichlet forms convergence to 0, then what about the diffusion processes associated with these Dirichlet forms? Do the finite dimensional distributions of them converges weakly? and ...
- 431
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752
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transform a polynomial into another one upto a constant
I have a polynomial $p(x)=a_Nx^N+a_{N-1}x^{N-1}+\dots+a_0$. I want to convert this into another polynomial of same order, say $b_Ny^N+b_{N-1}y^{N-1}+\dots+b_0$. Is it possible to find a transformation ...
- 169
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1
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320
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Simple markov chain problem
I know this is an easy problem, but I can't figure it out.
A particle takes discrete steps $σ_1,σ_2,σ_3,…,σ_n$ which take on values +1 or −1. However, $P(σ_i=+1)=p$ and $P(σ_i=−1)$ will be $1-p$.
...
- 39
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333
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Limit of the stochastic process at time 0
This is not a homework question so please be kind not to remove it right away. I am working on some research but have to justify the following argument: Assume $S_t$ is a continuous stochastic process,...
- 153
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123
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Enumeration of quadrangulations with a boundary and simple faces.
I wish to enumerate all quadrangulations of a $2p$ gon with $n$ internal vertices. Quadrangles are required to have simple faces. Simple face means all four vertices of each quadrangle are distinct.
...
- 141
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1
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262
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Conditional Density of Random Variables
Hi all,
I read recently that for any three continuous random variables, X,Y and Z, the conditional densities are related by the following formula:
$p(x|y) = \int g(x| z) h(z | y ) dz $
where $p(x|...
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514
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Lower Bound on $E[X Y]$
(Cross-post from math.stackexchange.com Q#166689)
I would like to lower-bound $E[X Y]$ where $X, Y$ are two random variables such that:
$X \in [x_0, 1], Y \in [y_0, 1]$
$E[X] = x, E[Y] = y$
$X \geq ...
- 3,475
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2
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225
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Estimating joint and conditional probabilities with incomplete information
I'm working on an application for which it would be great to have the following functionality:
Say that you have a collection $C$ of $n$ events, for now let's set $n = 3$ and call the events $a, b,$ ...
- 21
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289
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Dither in Leech lattice quantization!
Can you please help me how to generate a dither signal $\mathbf{U}$, where $\mathbf{U}$ is a random vector of length 24 that is uniformly distributed over the Voronoi region of the Leech lattice.
Best,...
- 197
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442
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Calculate $\mathbb{E}[\int_o^T N_{t-}dS_t]$ - what went wrong?
First note, I had asked a similar question here, but the thread seems to have died, so I'll revive it here with more details. As a simplification of my real problem, I want to compute
$\mathbb{E}[\...
- 278
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1
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156
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Expected number of trials to cover certain probability mass for a probability density function?
Suppose we have a univariate random variable $X\sim\mathcal{P}$ with probability density function $f(x):\mathbb{R}\to\mathbb{R}$, $\int_{-\infty}^{\infty}f(x) dx = 1$, we then draw $n$ samples $x_1,...
- 103
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666
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A Cauchy–Schwarz Type Inequality Involving Scaled Distributions
I have stumbled upon a rather intriguing inequality involving the product of the scaled distribution and the scaled density of a random variable. The inequality has a very attractive form, and it ...
- 197
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577
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Expectation of little o in probablity [closed]
If I have $Z=o_p(1)$ where $o_p$ is the little-o in probability. I'm interested in find some properties about $E(Z)$.
My first idea was
$E(Z)=E(Z (1_{Z>\varepsilon} + 1_{Z\leq\varepsilon}) ) \...
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303
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Integrated colored Gaussian noise
Assume we have a colored Gaussian process $z_t$, with an autocorrelation function $cov(z_t,z_s)$ given by an analytical function $\alpha(t,s)$ (if it helps, one can assume that $\alpha(t,s) = \kappa e^...
- 221
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285
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Is there a monotone coupling of Dirichlet random variables?
Let $X=(X_1,X_2,X_3)\sim \text{Dirichlet}(a_1,a_2,a_3)$ and $Y=(Y_1,Y_2,Y_3)\sim \text{Dirichlet}(a_1+b_1,a_2+b_2,a_3)$, where all $a_i$ and $b_i$ are positive. Is there a natural coupling between $X$ ...
- 13
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2
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327
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Copulas and time series
Please, can anybody give a reference(s) to some good recent review papers about copulas and time series?
- 2,600
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1
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3k
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Conditional expectation of a product
I have an expression: $E[(b+X)^2|Y]$ where $X$ and $Y$ are normally distributed random variables, being two components of a final unknown outcome $Z$ ($Y$ is known, $X$ is the noise component):
$Y$ =...
- 13
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1
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389
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Radius of random walk on Z
I'm trying to find a set of uniform measure 1/2 over $ \{ -1,1 \} ^n \times \{-1,1\}^n$ such that the inner product of $(x,y)\in\{ -1,1 \} ^n \times \{-1,1\}^n$ will hold $|\langle x,y\rangle|< \...
user17253
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4
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386
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Recovering a function from a set of approximations
We assume that we have a finite set of agents with approximate knowledge about a certain function, and from this collection of approximations we want to recover the actual value of the function.
More ...
- 1,228
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435
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Heavy Tailed Network
In his paper Kronecker Graphs: An approach to modeling Networks Jure et Al, mention that an important property of networks are that they are heavy tailed.
I'm trying to get an insight on what this ...
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1
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137
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Mean of an experiment
Suppose we have a bag of n different balls, and each time m (m<n) balls are taken out for checking from the bag and put back. ...
- 9
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1
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275
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Conditional distribution of the modulus of the output of AWGN channel given the modulus of the input
Hi everyone,
I will be too happy if anybody help me find a solution for the following problem.
In fact, I have a big problem that I could not solve it for weeks.
Assume that we have we have two ...
- 197
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774
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A question on independence
For each natural number $n \geq 2$, define the set $A_n$ to be the set of points $p/n$ with $0 < p < n, \gcd(p,n) = 1$. Now define a sequence of independent random variables $X_1, X_2, \cdots$, ...
- 26.9k
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370
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Analytical expression for variance of nested binomials?
Hi all,
I want to compute the variance of a variable that is defined at each step as a recursion of binomials in the following way:
A=1
B=Bin(1,A)*Bin(1,p)
C=Bin(1,B)*Bin(1,p)
D=Bin(1,C)*Bin(1,p),...
- 1
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2
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200
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Good probability measues on $S^1$ reprented by a kernel
I was looking for some good references for properties/theorems/characterizations of 'good/important' probability measures on the unit circle $S^1$ ( and/or on spheres $S^n$ ).In particular, I want ...
- 1,471
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292
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Probability of preserving connectivity between pair of vertices in weighted graph
Let $G=(V,E)$ be an undirected graph and $p \colon E \mapsto (0,1]$ defines weights of its edges.
Let's fix two connected vertices $v_1, v_2 \in V$.
Random graph $G'=(V,E')$ is obtained from $G$ by ...
- 3
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1
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938
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Convergence of sets
Let $E$ be a compact subset of $\mathbb{R}^n$. Let the density function $\phi(x,y)$ be Lipschitz continuous and such that
$$
\int\limits_E \phi(x,y)dy=1
$$
for all $x\in E$. Let us consider the non-...
- 1,655
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1
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801
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Information criteria for ridge regression
Hi -- is there any analogue or adjustment of, say, Schwartz Bayesian (or other) information criterion that would be applicable to model selection in ridge regression with a given ridge parameter $\eta$...
- 177
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2
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339
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Efficient Method for Calculating the Probability of a Set of Outcomes?
Let's say I'm playing N different independent "games". For each game, I know the probability of winning, the probability of tying, and the probability of losing.
From these values, I've also ...
- 41
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3
answers
848
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What the the probability distribution of a mean?
There is an unknown set of values of unknown size, from which a known subset of N values is drawn at random.
Based on the known random subset, what is the probability distribution of the mean of the ...
- 269
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1
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241
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Constraints for different probability measures to have the same expectation.
Take different $D_i \in \mathbb{R} \rightarrow \mathbb{R}$ functions $f_1, f_2$ (i.e. $\exists x : f_1(x) \neq f_2(x)$). We have
$E[f_1(x)] = E[f_2(x)]$
Are there conditions that $f_1, f_2$ must ...
- 151
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1
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1k
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Kernel width in Kernel density estimation
Hi,
I am doing some Kernel density estimation, with a weighted points set (ie., each sample has a weight which is not necessary one), in N dimensions.
Also, these samples are just in a metric space (...
- 481
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1
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284
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The density of x_1^n+x_2^n where x_i are Gaussian
We define a process $\chi_k^n=\sum _{i=1}^k x_i^n$ where x_i are iid gaussian processes.
I try to find the distribution of $\chi_k^n$. If k=1 then we get $f(x^n=y)=\frac1n y^{\frac{1-n}{n}}\exp(-y^{2/...
- 1
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1
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665
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Expectation maximum i.i.d rv´s
If I have a fixed positive integer $N$ and $N$ i.i.d rv´s. $X_1,X_2,...,X_N$, and parameters $a_i$ such that $\displaystyle\sum_{i=1}^N{a_i}=1$, it is well known that there is a global maximum of
$...
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2
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429
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E[log(Z_t^2)], proof of convergence with Law of Large Numbers
Hi all,
question:
Let $Z_t$ be an iid sequence with $$\mathbb{E}\log(Z_t^2)<0 $$
Show that $$\sum_{j=0}^\infty Z_t^2 Z_{t-1}^2 ... Z_{t-j}^2 < \infty$$ almost surely
I am supposed to use LLN ...
- 109
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1
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2k
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Conditional Covariance
It is well known that for two increasing functions $f$,$g$ and for any random variable $X$ then $cov(f(X),g(X))\geq{}0$. Now assume $f,g$ have the same domain $D$ and let $A\subset{}D$. What can I say ...
- 1
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2
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595
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univariate prior corresponding to weighted sum of L1 and L2 penalties?
Is there a univariate probability distribution $p_{\lambda,\alpha}(\beta)$ over the reals, parameterized by $\lambda > 0$ and $1 >= \alpha >= 0$, such that $p_{\lambda,\alpha} \propto \exp(-\...
- 103
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1
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207
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Correlation of Statistical Tests
Suppose I have a sequence $\{x_i\}_{i=1}^\infty$ of zeros and ones. I want to test if they are randomly generated according to a conjectured scheme (the example to keep in mind is that they are ...
- 1,588
0
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2
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144
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Random values and their probability of reoccuring [closed]
I have a web application that prompts users to answer a question when the computer they are using is not recognized. A user complained today saying she is always prompted for the same question. I ...
- 113
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1
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189
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Difference Equations & Possible Limits
The answer to this may well be in some elementary textbook - a reference might be more useful than a short answer here.
If we look at the behaviour of a point in R n under matrix multiplication, we ...
0
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0
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24
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Characterisation of a family of continuous martingales
I look for a full characterisation of the continuous martingales $X=(X_t)_{0\leq t\leq T}$ (defined on some filtered probability space as nice as possible) such that
$$X_0=0\quad \mbox{ and } \quad\...
- 1,399
0
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0
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38
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Bounding the error of a truncated moment problem
Let $\{x_{i}\}_{i=1}^{\infty}$ be a non-increasing sequence of non-negative real numbers, and let $\{y_{j}\}_{j=1}^{B}$ be a non-increasing sequence of non-negative real numbers, where $B$ is a finite ...
- 433
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0
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31
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Looking for a citation for this simple generalization of the Markov bound to non-negative super-martingales
Does anybody know a reference for the following theorem?
Theorem 1. Let $(X_t)_{t=0}^\infty$ be a non-negative supermartingale.
Then, for any constant $c > 0$, the event $(\exists
> t)\, X_t \...
- 183
0
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0
answers
36
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Contribution of Fisher information near jump points in convolved probability distributions
I am trying to compute the contribution to the Fisher information from jump points $b_i(\theta)$ in the convolved function $f(x; \theta)$ with respect to the parameter $\theta$. I am unsure whether it ...
- 99
0
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2
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116
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Upper bounds on quotients of binomial coefficients
Let $\gamma>1$ be a real number and let $n\in \mathbb{N}$.
Define $f\colon\mathbb{N}\to[0,1]$
$$
f(n_0) = \frac{\binom{n-n_0}{m}}{\binom{n}{m}},
$$
where
$$
m = \Big\lfloor{\frac{n}{\lceil\gamma ...
