The convergence tag has no usage guidance.

**40**

votes

**1**answer

792 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) = ...

**3**

votes

**1**answer

92 views

### Convergence of Fixed-Point Iteration of a dependent map

Suppose that we have two mappings $T_1(\cdot): Y \mapsto Y$ and $T_2(\cdot,\cdot) : X \times Y \mapsto X$ where both $X$ and $Y$ are compact and convex subsets of the same Euclidean space. Furthermore ...

**3**

votes

**1**answer

287 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 ...

**4**

votes

**0**answers

189 views

### Well-founded families of sets and topological convergence

Background/Motivation
A space is scattered if every non-empty subset has an isolated point. A space is pseudoradial if every non-closed set contains a transfinite sequence (a well-ordered net) ...

**3**

votes

**0**answers

165 views

### 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 ...

**2**

votes

**0**answers

77 views

### Convergence rate of Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...

**2**

votes

**0**answers

82 views

### Swapping sums and integration for a kernel in Fourier space (the non absolutely convergent case)

Under what conditions on $c_{r}^{m}$
does
$$\int_0^{2\pi} k(p,q)\exp(-inq)dq=\sum_{r=0}^{\infty}c_{r}^{m}\exp\left(-imp\right) \text{ in } L^2_{per}$$
hold for
...

**2**

votes

**0**answers

58 views

### Decomposition of the space according to the Ergodic Theorem

Given a space $(X, T)$, it is well known that for every $T$-invariant ergodic measure $\mu$, there exists a set $E_\mu$ of $\mu$-measure $1$ s.t. for every "nice" function $f$
$$ ...

**2**

votes

**0**answers

99 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 ...

**2**

votes

**0**answers

112 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, ...

**2**

votes

**0**answers

99 views

### 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 $X_n,\dots,X_m$ ...

**2**

votes

**0**answers

73 views

### 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 ...

**2**

votes

**0**answers

84 views

### 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 ...

**2**

votes

**0**answers

159 views

### An inequality for Lp-functions

I am interested in the following inequality:
\begin{equation}
\int\limits_\Omega \left\vert f_1 - f_2 \right\vert^p ~ d\mu \le C_p \left[ \int\limits_\Omega \left\vert f_1 \right\vert^p ~ d\mu + ...

**1**

vote

**0**answers

39 views

### Convergence of the solution of Volterra integral equation with convergent kernel (reposted, need help!)

Consider the following Volterra integral equation
$g(t)=∫_0^tK_n(t,s)w_n(s)ds$
where $g(t)$ and $K_n(t,s)$ are continuous and $K_n(t,s)≥K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ converges to ...

**1**

vote

**0**answers

92 views

### Does strong convergence in $W_p^1$ imply strong convergence of derivatives of absolute values in $L_p$?

Let $G$ be a Lipshitz domain and $u_n \to u$ in $W_p^1(G)$. Is it correct, that $\frac{\partial |u_n|}{\partial x_m} \to \frac{\partial |u|}{\partial x_m}$ in $L_p(G)$?
I know, that $\frac{\partial ...

**1**

vote

**0**answers

97 views

### Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then
$$
f(x)=\sum_{n\geq 0}a_n x^n
$$
converges absolutely for all $x$. Under ...

**1**

vote

**0**answers

38 views

### 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 ...

**1**

vote

**0**answers

104 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 ...

**1**

vote

**0**answers

48 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 ...

**1**

vote

**0**answers

41 views

### 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}) ...

**1**

vote

**0**answers

94 views

### 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 ...

**1**

vote

**0**answers

61 views

### 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 ...

**1**

vote

**0**answers

116 views

### 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 $E[X_{n1}]=\mu_n<\infty$ ...

**1**

vote

**0**answers

109 views

### Interior point optimisation using big M for L1 norm on linear system using Dikin's Affine method

I am a 4th year undergrad surveying student studying computations, specifically $L_{1}$ norm minimisation of residuals in large data sets. To start with (and probably to finish with) I'm using a set ...

**1**

vote

**0**answers

127 views

### Bounding a recursively defined sequence

I have a sequence $\lambda_0,\lambda_1,\ldots,$ which is defined recursively as
$$ \lambda_0 = \frac{1}{2},$$
and
$$\lambda_{k+1} = \max_{\lambda\in [1,b]} \left(\frac{1}{2\lambda}\prod_{0\leq ...

**1**

vote

**0**answers

111 views

### Exponential Ergodicity for Reflected Brownian Motion in a Bounded Domain

Assume we have a reflected Brownian motion in a smooth bounded domain $D \subseteq \mathbb R^d$. It can have nonzero (but constant) drift, non-identity (but constant) covariance matrix, and oblique ...

**1**

vote

**0**answers

174 views

### convergence of sets and limit of an integral

Let $X\subset\mathbb{R}$ and $Y\subset\mathbb{R}$ be compact sets.
Let $f:X\times Y\rightarrow\mathbb{R}$ be a $C^{1}$ function.
Let $s:Y\rightarrow X$ be a function (not necessarily continuous).
...

**1**

vote

**0**answers

501 views

### Definition and Convergence of Iteratively Reweighted Least Squares

I've been using iteratively reweighted least squares (IRLS) to minimize functions of the following form,
$J(m) = \sum_{i=1}^{N} \rho \left(\left| x_i - m \right|\right)$
where $N$ is the number of ...

**0**

votes

**0**answers

11 views

### Question on the convergence proof of SGD in Leon Bottou's paper

In Leon Bottou's paper "online learning and stochastic approximations" http://leon.bottou.org/publications/pdf/online-1998.pdf, the convergence
$(w_{t} - w^{*})\nabla_{w}C(w_{t})\rightarrow 0$ as ...

**0**

votes

**0**answers

64 views

### Entropy, convergence and invariant measures

Could you give conditions that a sequence of shift invariant measures $\eta_{n}$ has to satisfy in order to happen this convergence in terms of entropies $h(\eta_{n})$:
$h(\eta_{n}) \rightarrow ...

**0**

votes

**0**answers

47 views

### $A_n \not \rightharpoonup A$ in $L_1[-\pi; \pi] $ ( $A_n$ is partial fourier sum )

Let
\begin{equation*}
(A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k cos(kt) + b_k sin(kt), \\
a_k = \frac{1}{\sqrt{\pi}} \int_{-\pi}^{\pi} x(t) cos(kt) dt, \\
b_k = \frac{1}{\sqrt{\pi}} ...

**0**

votes

**0**answers

48 views

### 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}
...

**0**

votes

**0**answers

46 views

### 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 ...

**0**

votes

**0**answers

72 views

### 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 ...

**0**

votes

**0**answers

66 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.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, ...

**0**

votes

**0**answers

318 views

### Series with iterated log's: does it converge?

This came up in our office today. Let
$$f(x) = \begin{cases} x & \mbox{if } x\leq 1 \cr x\cdot f(\ln(x)) & \mbox{otherwise}\end{cases}$$
Does this series converge?
$$ \sum_{n=1}^\infty ...

**0**

votes

**0**answers

233 views

### Convergence of a function in a metric space to its metric.

Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ with a metric $d$ being the Euclidean metric:
If $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ ...