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### Weak continuous convergence of operators [migrated]

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$.
Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$.
Let $x_n \rightharpoonup x$ in ...

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### Question about the characteristics of semimartingales

Let $D=D([0,1,R)$ be the space of cadlag (right-continuous with left limits) functions defined on [0,1] and $X:=(X_t)_{t\in [0,1]}$ be the canonical process on $D$, i.e. $X_t(x)=x(t)$ for all $x\in ...

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### Does this infinite sum arising from separation of variables converge?

This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible.
Let $a_k >0$ be an increasing sequence ...

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### Variant of Skorokhod's theorem

Consider the following situation:
$S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful).
There is a a random variable $\zeta: \Omega \to S$.
$f_n(\zeta) \to^d \eta$, ...

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65 views

### Local Uniform Convergence

Suppose $f(x)$ is a positive continuous function on $[0,\infty)$ and that $f(x+u)-f(x)\to 0$ as $x\to\infty$ for every given $u\in[0,\infty)$. Prove that, given any $a>0$, $f(x+u)-f(x)\to 0$, as ...

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83 views

### Rate of convergence of Bayesian posterior

Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let ...

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100 views

### Weak convergence in $W^{1,p}_0$

Note from the answerer : this question stems from this article.
I ask this question in http://math.stackexchange.com/questions/1206617
I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so ...

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82 views

### On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$):
$$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n ...

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87 views

### Convergence proof for fictitious play!

In "Fictitious play property for games with identical interests" by D. Monderer and L.S. Shapley, the convergence of fictitious play to a Nash equilibrium is proved for a potential game with players ...

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70 views

### Division methods for divergent continued fractions

I hadn't even noticed before entering the subject above the parellel with the title of an earlier question posted here titled Summation methods for divergent series. And it's just mutatis mutandis as ...

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35 views

### Likelihood convexification

I am doing constrained vector optimization using a non-convex non-linear likelihood function. My problem is of the following form:
$$\begin{align*}\hat Q &= \underset{\vec Q}{\arg\min} -\log ...

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39 views

### Error for the convergence by distribution

A sequence of random variables $X_n$ converges in distribution to $X$, if there is pointwise convergence of its characteristic functions, i.e. $\lim_{n\rightarrow\infty}\phi_{X_n}(\lambda) = ...

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243 views

### Convergence of random variables with hypergeometric distribution

This is a very interesting conjecture of large scale property of hypergeometric distribution.
Let $a>1$ be a integer constant, $N\in\mathbb{N_+}$, for any $x<N-1$, consider $N+(a-1)x$ balls in ...

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### A curious example envolving moment's convergence

Let $\{X_n\}$ be a random variable sequence and $X\sim N(0,\sigma)$. In general, the convergence $E(X_n^k) \stackrel{n}{\longrightarrow}E(X^k)$ doesn't implie that $E(X_n^{k+1}) ...

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173 views

### Book on Convergence Concepts in Probability without Measure Theory [closed]

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...

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54 views

### convergence in distribution and convergence of moments

Suppose that the sequence of r.v $\{X_{n}\}_{n\geq 1}$ has all the moments, and $X_{n}\stackrel{D}{\longrightarrow}X\sim N(0,\sigma)$. Assume that $E\left\{(X_{n})^{K}\right\} ...

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### Weak convergence of SDE

Let $(X_t,Y_t)$ be the solution to the SDE
\begin{equation}
\begin{split}
dX_t &= f(X_t,Y_t)dt + \sigma_1 X_t dW^1_t\\
dY_t &= g(X_t,Y_t)dt + \sigma_2 Y_t dW^2_t
\end{split}
\end{equation}
...

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182 views

### Abscissa of absolute convergence of the product of two Dirichlet series

I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...

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### Perturbation method of a boundary value problem

Let $u(x, \epsilon, \theta)$ be the solution of $$u''+(\epsilon \cos(x)+\theta-u)u=0$$ with boundary conditions $u'(0)=0$ and $u'(\pi)=0$. Here $\theta\in [0, 1]$.
I tried to put the solution in ...

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105 views

### Speed of Approach to Invariant Measure

Let $X_t$ represent a continuous-time Markov process on $\mathbb{R}^d$, say a diffusion with locally Lipschitz coefficients. Suppose that there exists a unique invariant measure $\mu$ on the space, ...

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86 views

### Uniform approximation of increasing function in $C^{\infty}$

I have an increasing continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$ which is not differentiable everywhere, and I would like to approximate it with an infinitely differentiable function ...

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75 views

### Summability of ratios of moments a weight

Recently, I encounter the following problem:
Let $w$ be a probability density on $[0,1]$. Let mk be the $k$-th moment, i.e.,
$$m_k=\int_0^1t^kw(t)dt.$$
Under what condition can we have
...

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35 views

### Convergence in distribution and ODE

Assuming we have an ODE $y'_n(x) = f_n(x) y_n(x)$
with $f_n$ be Gauß-densities with mean value 0 and variance $\frac{1}{n}$, then we have that they converge in distribution to a delta peak $δ(x)$. ...

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153 views

### On the domain of convergence of usual definition of Riemann zeta function [closed]

If we define Riemann zeta function as it is usually defined so that $$\zeta (s)=\sum_{n=1}^{\infty}n^{-s}=\sum_{n=1}^{\infty}e^{-s\ln n}$$ then we can rewrite it as:
$$\zeta ...

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### Where the following interpolation method converges?

In this question about discrete-analytic functions (that is functions, who equal to their Newton series) I asked for a solution for the following problem:
Is there a method to extend the notion ...

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### Finite element convergence rates for mixed problems [closed]

I've coded up a Stokes Flow problem using finite elements and am in the process of verifying that it works. I'm just not sure what convergence rate I should be expecting as I globally refine the mesh.
...

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235 views

### Using a probability measure, P, defined on uncountable sets to construct a probability measure, P' on singleton P-null sets

Let $\Omega$ be an uncountable set and $(\Omega, \mathcal{F},P)$ be a probability space built on $\Omega$.
Let $S \subset \{A \in \mathcal{F}: P(A)=0,\;|A|=1\}:|S|<\infty$ be a finite subset of ...

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### Pointwise convergence of ergodic averages of unconventional conditional expectations

Let $(X_i,Y_i)_{i\in\mathbb{Z}}$ be a finite-valued stationary process whose $\sigma$-algebra of tail events is trivial. Let $\mathcal{F}_n^m$ be the $\sigma$-algebra generated by $X_n,\dots,X_m$ ...

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### Exchange limit and sum in certain conditions

Let
$\sum_{i=1}^{\infty} f_i(s)$ be a series of analytic functions and suppose it converges on some neighbourhood $V$ of $s=0$ and it converges uniformly on $V\cap\{s \;|\; |s|> \varepsilon \}$ for ...

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108 views

### Growth of the truncation of the integral multiples of an irrational number

Let $[a]$ denote the integral part of a real number $a$.
Let $a$ be an irrational number and $b$ a real number greater than $1$.
Consider the sequence $(b^n(na-[na]))$ with $n$ running on the ...

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192 views

### Central limit theorem and convergence of means [closed]

If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = P(Z_i=+1) = \frac 12,$ then we have by the Central Limit Theorem that $\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\stackrel{d}{\to} \mathcal{N}(0,1),$ so ...

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### What good is (strong) mixing in dynamical systems?

For measure-preserving dynamical systems, there exist several notions of mixing. The most basic ones are strong mixing, weak mixing and ergodicity (see the wikipedia page, for instance), asserting ...

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### Eigenvalues of matrix products [closed]

Hi my problem is with row stochastic matrices. Its known that if we keep multiplying this row stochastic matrices, we will get a rank one row stochastic matrix. Rank one means that all the rows has ...

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### Linear or quadratic combinations of i.i.d. random variables [closed]

I already posted this question here http://math.stackexchange.com/questions/769920/law-of-large-numbers-for-linear-quadratic-combinations-of-i-i-d-random-variab but I received no answers.
Let ...

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243 views

### A convergence issue [Edited]

Let $\{x_n\}_{n=1}^\infty$ be a sequence of vectors in a Hilbert space $$l^2_{k^{-2}}:=\{z=\{z(k)\}_{k=1}^\infty:\sum\limits_{k=1}^\infty z(k)^2k^{-2}<\infty\}.$$ It is known that for some $x\in ...

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63 views

### Weak convergence of 4-th degrees

Good day!
We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, ...

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128 views

### Is $\int_0^\infty \sin(Kx)f_K(x)\,dx$ of larger order than $\int_0^\infty \cos(Kx)f_K(x)\,dx$?

Suppose we have a function $f$, such that $f$ is of some smoothness degree $m$, and $f,f^{(k)} \in L_1[0,\infty)$ $k=1,...,m$. Now if $f^{(k)}(0) = \lim_{x\rightarrow\infty}f^{(k)}(x) = 0$ for ...

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### Preserving Predimension Functions under Functional Convergences

Definition 1. If $\mathcal{L}$ is a countable relational language, a predimension class $C$ is a class of $\mathcal{L}$-structures with the following properties:
C1: ...

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### Epi-convergence to indicator function

Let the indicator function be defined as $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and $I_{\nu}(x) \in [0,1]$ be a continuous approximation of ...

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### Convergence rate of the central limit theorem near the center of the distribution

I'm looking for fast convergence rates for the central limit theorem - when we are not near the tails of the distribution.
Specifically, from the general convergence rates stated in the Berry–Esseen ...

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### Convergence of Symmetric Iterative Proportional Fitting

Let $A$ denote a symmetric matrix of non-negative entries, whose rows (and columns) sum up to the all-positive vector $d := A\mathbf{1}$ with $\mathbf{1}$ the all-one-vector. Denote $D := diag(d)$ by ...

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265 views

### Convergence of a Trigonometric Series

After working with a Fourier series for a while, I realized that it would be of great help to me if I could prove that the following limit is zero:
$$\lim_{N\to\infty}_{N\in ...

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249 views

### Estimate on gaussian distribution

Let X be an $\mathbb R^d$-valued random variable with distribution $N_d(0,\Sigma)$. I'm looking for a function $f$ such that
$$P(|X_1|\leq M, |X_2|\leq M,\dots, |X_d|\leq M)\geq f(M),$$
and such that ...

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### Some convergence similar to weak-$\ast$ convergence on the space of finite measures

I have a question:
Let $D$ be the space of cadlag functions defined on $[0,1]$ and $V$ be its subspace consisting of $x$ with finite variation and $x(0)=0$.
Define $TV(x)$ as the total variation ...

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615 views

### Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series
$$f_{\sigma}(z) = ...

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### Berry-Esseen result for triangular arrays

Let $\{\{X_{n1},\ldots,X_{nn}\},n=1,2,\ldots\}$ be a triangular array of independent random variables where each row contains identically distributed random variables. Let $E[X_{n1}]=\mu_n<\infty$ ...

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### Quantile convergence

Let $X^1,\dots,X^n$ be a sample of (not necessarily iid) random variables and denote
$$F^n(x)=\frac{1}{n}\sum_{i=1}^n \mathbf 1_{X^i\leq x}$$
the empirical distribution function. Suppose that we know ...

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363 views

### Convergence of the empirical distribution function

Let $\alpha\in\mathbb R^d$, with $\alpha\neq 0$. Take a sequence of iid random variables $X^{1},\dots,X^{n}$ with values in $\mathbb R^d$ and denote $R$ the cdf of $\alpha X^1$ (where the product is a ...

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### Greedy interpolation of functions

Let $f:[-1,1]\rightarrow \mathbb{R}$ be a continuous function. Consider the following greedy algorithm for interpolation:
Set $r_0 = f$.
for $k = 0,1,\ldots,$
Find the location of the global ...

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### Convergence rate of the convolution of almost uniform measures on $\mathbb{Z}_p$

Statement Given a finite abelian group $G$ and two independent random variables $X,Y$ taking values in $G$ and satisfying $d_{TV}(X,U_G)\leqslant \delta$ and $d_{TV}(Y,U_G)\leqslant \delta$ (where ...