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.
8,632
questions
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Uniform boundedness of integral?
I have perhaps a very simple question where I lack some inutition at the moment: Is the expression
$$\sup_{\alpha < 0, \lambda \in \mathbb N}\int_{-\infty}^{\alpha} e^{-\lambda t^4} \ dt \int_{\...
4
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3
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1k
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Converse to Girsanov's theorem?
Roughly speaking, Girsanov's theorem says that if we have a Brownian motion $W$ on $[0,T]$, we can construct a new process with a modified drift that has an equivalent law to $W$ (subject to ...
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0
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99
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Dislocations and Random Matrix Theory
Does anyone have a good reference book that works as a good starting point for an analyst to learn about the connection between Random Matrix Theory and Dislocations? Thank you for your help.
By ...
4
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1
answer
289
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Variance of random variable decreasing in parameter
I did quite a few numerical computations and think the following is true, but I cannot prove it:
Let $\varphi(x):=\sum_{i=1}^n \varphi_i(x_i)$ where $x=(x_1,...,x_n) \in \mathbb{R}^n$ and $\varphi_i \...
1
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0
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Given a large random matrix, how to prove that every large submatrix whose range contains a large ball?
Context. Studing a problem in machine-learning, I'm led to consider the following problem in RMT...
Definition. Given positive integers $m$ and $n$ and positive real numbers $c_1$ and $c_2$, let's ...
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1
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75
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Definition of a system of recurrent events
[I asked a version of this question on MSE a few weeks ago and didn't get any useful feedback. Apologies if I am just being stupid.]
I am reading the paper A note on the Borel-Cantelli lemma by Kochen ...
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0
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84
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Regularity with respect to the Lebesgue measure through dimensions
Let us consider two probability measures $\mu \in \mathcal{P}(\mathbb{R}^{p})$ and $\nu \in \mathcal{P}(\mathbb{R}^{q})$ with $p,q \in \mathbb{N}^{*}$. We note $\#$ the push forward operator i.e for $...
3
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Canonical forms for block-positive-definite matrices
Suppose we are given a block $2\times 2$ matrix that is positive-definite, and let's suppose for simplicity that the blocks along the main diagonal are the identity. So
$$
\begin{bmatrix} I & X \\\...
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0
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54
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Convergence of empirical measure to Mc-Kean Vlasov equation for mean-field model with jumps
I am interested in the following mean-field model introduced in the reference below:
There are $N$ particles. At each instant of time, a particle's state is a particular value taken from the finite ...
2
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0
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154
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Representing a continuous time-inhomogeneous Markov chain by a stochastic integral
I am interested in the following mean-field model introduced in the reference below:
There are $N$ particles. At each instant of time, a particle's state is a particular value taken from the finite ...
1
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0
answers
96
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Comparison of two Fourier transforms
I am looking for $\delta>0$, such that
$$
\delta \int_{-\infty}^{\infty} \exp(its)
{ \Gamma\{2(it+1)/3\}\over \Gamma\{(it+1)/2\} }dt \le \\
\int_{-\infty}^{\infty} \exp(its)
{ \Gamma (it+1)\over \...
3
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0
answers
152
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Minimizing an f-divergence and Jeffrey's Rule
My question is about f-divergences and Richard Jeffrey's (1965) rule for updating probabilities in the light of partial information.
The set-up:
Let $p: \mathcal{F} \rightarrow [0,1]$ be a ...
3
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0
answers
78
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Random walk in a switching scenery
For each $x \in \mathbf{Z}$ let $(\eta_t(x))_{t\geq0}$ denote independent copies of a process $(\eta_t(0))_{t\geq0}$ defined as follows. The process $\eta_t(0)$ takes values in $\{-1,1\}$, where $-1$ ...
2
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1
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English translation of "Une inégalité pour martingales à indices multiples et ses applications"
Does anyone know of a English translation of "Une inégalité pour martingales à indices multiples
et ses applications" by Renzo Cairoli. Or could translate the statement of the martingale ...
2
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0
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125
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Exit time for Brownian motion with stochastic barriers
I am interested in the expected exit time of a one-dimensional Brownian particle from a stochastically evolving interval as follows.
Context:
If $L_t$ and $R_t$ denote the distance to the left and ...
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0
answers
136
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How to calculate possible arrangements of hexagons?
I was wondering someone could help. I've developed a board game which is made up of six, large hexagonal board pieces, which can be arranged in any order, and with any rotation/arrangement of sides ...
2
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0
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On the difference between Malliavin derivative and Gross-Sobolev derivative
Let $W=C_0([0,1],\mathbb R^d)$ be the classical Wiener space equipped with $\mu$ the Wiener measure.
If $F:W\to\mathbb R$ is a cylindrical function of the form
\begin{align*}
F(w)=f(W_{t_1}(w),\cdots,...
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2
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Finding the expectation of $a \mathcal{Q} \left( \sqrt{b } \gamma \right) $, where $\gamma$ is a Gamma r.v
I'm trying to analytically find the following expectation
$$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right],$$
where $a$ and $b$ are constant values, $\mathcal{Q}$ is the ...
3
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1
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On representing a continuous time Markov chain by a stochastic integral of a Poisson random measure
Let $Q=(q_{ij})$ be the transition rate matrix of a continuous time Markov chain $\{ X_t \}$ with countable state space $M$. Let $q_i = -q_{ii}=\sum_{j \neq i}q_{ij}$, and let $\Gamma_{ij}$ be defined ...
6
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Positivity of $ \int_{-\infty}^{\infty} \left\{{2^{1/\beta-1/2} \over v}\right\}^{it} { \Gamma\{(it+1)/\beta\}\over \Gamma\{(it+1)/2\} }dt$
I have the following function
$$
\int_{-\infty}^{\infty} \left\{{2^{1/\beta-1/2} \over v}\right\}^{it}
{ \Gamma\{(it+1)/\beta\}\over \Gamma\{(it+1)/2\} }dt
$$
where $1<\beta<2$, $v>0$. Need ...
1
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0
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142
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Maximizing variance of bounded random variable through convex optimization
I am interested in maximizing the variance of a random variable $X$ supported on $[0,1]$. Formally,
$$\max_{P_X: X \in [0,1]} {\rm Var}(X),$$
where $P_X$ is a distribution of $X$. This question is ...
11
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1
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354
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Graph with path of length $\geq n$ along grid diagonals - a known result in graph theory?
Is the following lemma a well known result in graph theory?
I am studying a basic existence result that appears to be simple yet powerful. I have not seen it stated as an important result in graph ...
16
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2
answers
2k
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On mathematical aspects of the most recent Nobel Prize in economics winners' work
Can somebody briefly introduce the mathematical aspects, in particular, those related to mathematical finance, of the three economists who were just awarded this year's Nobel Memorial Prize in ...
2
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1
answer
265
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On the speed of divergence of the converse of the Strong law of large numbers
By the converse of the strong law of large numbers, we know that, given a sequence of i.i.d random variables $X_1,X_2,\dots$ such that $\mathbb{P}(X_1 \ge 0)=1$ and $\mathbb{E}X_1= \infty$,
then I ...
4
votes
1
answer
184
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Asymptotics of a quotient related to a simple random walk
Let $\lambda_0 < \lambda_1$ and $\lambda_0 \lambda_1 > 1$ (i.e. at least $\lambda_1 > 1$). Further, let $S_n$ denote a simple random walk with increment distribution $$ P(X = 0)= P(X= 1) = 1/...
2
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1
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255
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Information theory for uncountably infinite-dimensional continuous random variable
I'm exploring the possibility to apply information theory on an uncountably infinite-dimensional scenario. I found the concept of generalized entropy for continuous random variables defined on finite-...
0
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0
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220
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Question regarding Ito representation theorem
Let $H$ be a Gaussian Hilbert space and $H^{:n:}$ be the homogeneous chaos of order $n$.
and let $D_n:=\{(t_1,\cdots,t_n):t_1<t_2<\cdots <t_n\}$.
For each $n\geq 0$ there exists an isometry
\...
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0
answers
256
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Projecting a vector onto a random subspace
Let $A\in\mathbb{R}^{k\times d}$ be matrix with i.i.d. $\mathcal{N}(0,1/k)$ entries with $k<d$, and let $B=A^{\top}A$. I would like to compute the distribution of $Bx$ where $x\in\mathbb{R}^{d}$ is ...
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3
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An $L^0$ Khintchine inequality
Suppose that $\epsilon_1,\epsilon_2,\ldots$ are IID random variables with the Bernoulli distribution $\mathbb{P}(\epsilon_n=\pm1)=1/2$, and $a_1,a_2,\ldots$ is a real sequence with $\sum_na_n^2=1$. ...
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Coin pusher game
While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins).
Essentially, one has a distribution of ...
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0
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55
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Quantitative bounds on convergence of Bayesian posterior
Let $Y$ be a random variable in $[0,1]$, and let $X_1, X_2, \ldots$ be a sequence of random variables in $[0,1]$. Suppose that the $X_i$'s are conditionally i.i.d given $Y$ ; in other words, I'd like ...
3
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668
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About the metrizability of the space of Probability measures $\mathcal{P}(S)$
It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the ...
4
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1
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294
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Categorified probability and statistics?
To put it simply, what if the sample space underlying our probability space is a category instead of a mere set. Has a theory or probability and statistics been developed for such a situations in ...
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0
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126
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Almost sure stochastic equicontinuity
Suppose $\mathcal{G}$ is a normed closed class of functions with finite entropy and envelope with a finite second moment (details below), and $g_0$ is a function in the interior of that class. Let $...
4
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0
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267
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Examples of measures that satisfy FKG, but not the FKG lattice condition
Let a percolation measure be a measure on $\{0,1\}^n$. We have a natural partial order on $\{0,1\}^n$ given by comparing all coordinates. An event $A$ is called increasing if for all $ \omega \in A $ ...
2
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1
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Kalman filter distribution of observation process
Let $(X_t,Y_t)$ be a pair of stochastic processes such that
$$
\begin{aligned}
dX_t =& A_t X_t dt + C_t dW_t,\\
dY_t = & H_t X_t dt + K_tdB_t
\end{aligned}
$$
for some non-random matrix-valued ...
5
votes
1
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775
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A set of questions on continuous Gaussian Free Fields (GFF)
As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...
9
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1
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Probability that random high dimensional vectors are all on the convex hull
Say I pick $n$ i.i.d. random standard normal points in $\mathbb{R}^d$. Roughly, as long as $n$ is much smaller than exponential in $d$, with high probability all points will be on the convex hull. ...
1
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1
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377
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Curious inversion formula in additive combinatorics
Let $S$ be an infinite set of positive integers, and $T=S+S=\{x+y, \mbox{ with } x,y\in S\}$.We definte the following functions:
$N_S(z)$ is asymptotic continuous version of the function counting the ...
1
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1
answer
162
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Asymptotic for the probability that a number has $k$ prime factors less than $Q$
If we let $\omega_Q(n)$ denote the number of distinct prime factors of $n$ less than a bound $Q$, then what asymptotic formulas exist for $\Pr_{n\in\mathbb{N}}[\omega_Q(n)=k]$ as $Q\to\infty$ if $k$ ...
1
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2
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485
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Moments of complex random variables
My question is that how much information we can get form integer moments of a complex random variable?
Let $\mathcal{Z}$ be a complex value random variable, and assume that we can compute $$\int \...
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1
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55
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Expected size of binomial coefficient with Poisson arrivals?
I have a Poisson process where new elements arrive to a set with Poisson intensity $\lambda$. Initially, there are $N_0$ elements in the set. The probability that there are $N_0 + M$ elements in the ...
1
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1
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199
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Dice roll expectation question [closed]
Suppose you roll a dice 100 times, How many times would you expect the most common number to show up.
I.e. roll a dice 100 times and document the frequency of each value, then repeat this process ...
6
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1
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587
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Reformulation - Construction of thermodynamic limit for GFF
I've posted a question about the thermodynamic limit for Gaussian Free Fields (GFF) a couple days ago and I haven't got any answers yet but I kept thinking about it and I thought it would be better to ...
2
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1
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451
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Stochastic integral with respect to a random field
I came across a generalized Black-Scholes equation formulation in this paper.
Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion ...
11
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3
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241
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How many operad structures are there on the symmetric sequence of simplices / finitely-supported probability measures?
Consider the symmetric sequence $P_n = \Delta^{n-1}$ of probability measures on finite sets, with coordinatewise $\Sigma_n$-action. There is a natural topological operad structure on $P$ given by ...
0
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1
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76
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Decaying probabilities
A coin $C$ is tossed $n$ times. The coin $C$ is known to have the following properties :
Let $p_i$ denote the probability of showing heads in the $i$-th toss, and $q_i$ denote the probability of ...
4
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0
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130
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Independent inner functions on the unit disk
This problem cropped up in a paper that I am writing and I have thought about it for too long to no avail: let $\mathbb{D}$ be the open unit disk in the complex plane and suppose $\varphi:\mathbb{D}\...
-1
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1
answer
343
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Convergence of Radon-Nikodým derivative
Imagine we have a sequence of finite measures $\nu_n << \mu_n$ (on the torus $\mathbb{T}^2\subseteq \mathbb{R}^2$) converging weakly to some measures $\nu << \mu$. Do we automatically have ...
2
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0
answers
97
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Replacing the usual mean by the weak one
The mean of a non-negative random variable is just its $L_1$ norm. As such, it has a "weak" version:
$$ ||X||_{1,w} = \sup_{t>0} tP(X\ge t).$$
An immediate consequence of Markov's ...