Questions tagged [limits-and-convergence]

Convergence of series, sequences and functions and different modes of convergence.

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7 votes
1 answer
256 views

Does the limit of $x_n$, defined by $x_{n+1}=1/(m+1-nx_n)$ exist?

Let $m$ be positive integer, and consider the recursion $$x_{n+1}=\frac{1}{m+1-nx_n}.$$ Does the limit of $x_n$ exist? I'm guessing the limit doesn't exists for any $m$.
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1 vote
0 answers
74 views

Limit of alternating sum of factorial moments which diverge

Consider the non-negative, integer valued random variable $X$, and its $i^{\text{th}}$ factorial moment $E_{i}[X]$. Then we have that $$ P(X=0) = \sum _{i=0}^{\infty} \frac{(-1)^i E_{r}[X]}{ i!} $$ ...
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0 votes
0 answers
37 views

Convergence of stopped stochastic processes

Let $(E,d)$ be a locally compact separable metric space. Let $\mathcal{D}=\mathcal{D}([0,\infty),E)$ denote the space of right continuous functions on $[0,\infty)$ having left limits and taking ...
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2 votes
1 answer
61 views

On the distance to the stationary distribution

A Markov Chain $M$ has only one stationary distribution $q$. For distribution $p$, with $D_{TV}(p,Mp)=x$, can we bound $D_{TV}(p,q)$? Clearly, $x=0$ implies $D_{TV}(p,q)=0$. Does general bound hold? ...
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5 votes
0 answers
161 views

Very slow continued fraction convergence

Let $a(0)+b(0)/(a(1)+b(1)/(a(2)+b(2)/(a(3)+\dots)))$ be a continued fraction, and $p(n)/q(n)$ its $n$-th convergent. If it converges (i.e., $p(n)/q(n)$ tends to some limit $S$ as $n\to\infty$), then $...
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3 votes
0 answers
172 views

Some pending questions about $\sum_{p\leq\sqrt{n}}p=\pi(n)$

Here it was showed that $S(n)\sim \pi(n)$, where $S(n)=\sum_{p\leq\sqrt{n}}p$, $p$ refers to prime numbers, and $\pi(n)$ is the prime counting function. Here it was proved that $S(n)=\pi(n)$ for ...
2 votes
1 answer
89 views

Convergence in probability of a supremum

Let $A>0$ be fixed and consider $X_1,\ldots$ i.i.d. nonnegative random variables such that $E[1/X_1]<\infty$. Is is true that $$\sup_{a\in \big (0,\frac A{\sqrt n} \big]} \sum_{i=1}^n 1_{X_i>...
4 votes
2 answers
127 views

Does the average of correlated Gaussian random variables with mean zero and different variances converge in probability to their mean?

Let $X_i\sim N(0,\sigma_i^2)$ and $\operatorname{Corr}(X_i,X_j)>0$. Is it possible to show that $$\frac{1}{N} \sum_{i=1}^N X_i \overset{p}\rightarrow E[X_i]=0.$$ Do you have a reference to a law of ...
0 votes
0 answers
52 views

Relevance of the deduction of similar theorems than Maier's theorem for other constellations of primes

A year ago I asked this question on Mathematics Stack Exchange with identifier 4245823 and same title Relevance of the deduction of similar theorems than Maier's theorem for other constellations of ...
0 votes
2 answers
108 views

Show that the set of strictly stationary, mean zero and finite variance stochastic processes is closed (or not)

Let $\mathcal{P}$ be the set of real-valued and strictly stationary processes with expectation zero and finite variance, i.e.: \begin{equation} \mathcal{P}:=\left\{ X = (X_t)_{t \in \mathbb{Z}} \, ...
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3 votes
1 answer
279 views

Limit of the average of telescopic products

I am trying to show that $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\prod_{j=0}^{i}\frac{kn-j-k}{kn-j}=\frac{k^{k+1}-(k-1)^{k+1}}{(k+1)k^{k}}$$ for all $k\in\mathbb{N}$, $k\geq 4$. I could verify the ...
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2 votes
1 answer
74 views

Approximation of a stationary process by a sequence of ergodic and stationary sequence of stochastic processes

Let $X = [X_t : t \in \mathbb{Z}] \sim P$ and $Y = [Y_t : t \in \mathbb{Z}]\sim Q$ be two stochastic processes. Let's define the Mallows metric. Let $\mathcal{M}_m$ be the random vectors $(X,Y)$ ...
  • 131
1 vote
0 answers
98 views

Normal numbers and law of the iterated logarithm

If I remember correctly, for the binary digits of a real number in $[0,1]$, I was told that satisfying the law of the iterated logarithm (LIL) is stronger than being normal. That is, supposedly, some ...
1 vote
1 answer
61 views

Locallic maps given by series

Maps between real numbers are often defined by convergent series. For example, to define the exponential map, we can just prove that series $$\sum_{n = 0}^{\infty} \frac{x^n}{n!}$$ converges, which ...
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5 votes
0 answers
101 views

Convergence of the best constant in the $s$-fractional $L^p$ Sobolev inequality

It is known that the fractional $L^p$ Sobolev inequality $$ \|f\|_{L^{p^*_s}(\mathbb R^n)}^p \leq \sigma_{n,p,s} (1-s) \int_{\mathbb R^n}\int_{\mathbb R^n} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} dx dy $$ ...
17 votes
2 answers
995 views

"Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?

Recently, I encountered this problem: "Given a sequence of positive number $(x_n)$ such that for all $n$, $$x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$$ Find the limit $\lim_{n \rightarrow \infty} \...
0 votes
0 answers
92 views

If the Dirichlet series $L(z,\chi)$ diverges for $\sigma< 1$, does its alternating version converge for some $\sigma_0 < 1$, and conversely?

Here $\chi$ is a completely multiplicative function with $\chi(p)\in\{-1,+1\}$ for any prime $p$, but not necessarily a character. Also $s=\sigma + it$ as usual. The series corresponding to $L(s,\chi)$...
1 vote
1 answer
88 views

Does pointwise convergence yield the convergence under Skorokhod topology?

Let $D_+$ be the set of non-increasing functions $f: [0,T]\to [0,1]$ that are right-continuous. Let $(f_n)_{n\ge 1}\subset D_+$ be a sequence of continuous functions s.t. $\lim_{n\to\infty }f_n(t)$ ...
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1 vote
1 answer
111 views

Interchange summation order in the limit of number of elements going to $\infty$

Considering the sum $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty} a_{ij}$, in general we are not allowed to interchange the summation order (i.e. pass to $\sum_{j=0}^{\infty}\sum_{i=0}^{\infty} a_{ij}$) but ...
2 votes
1 answer
63 views

Law of large numbers for triangular arrays whose moments "look independent"

Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be a triangular array of random variables with finite moments of all orders, with no assumptions on their independence. Suppose that $$ \mathbb{E}\left[\...
1 vote
0 answers
120 views

"Beautiful convergence" in Hejhal, The Selberg Trace Formula

A simple question: Hejhal in his volumes on the Selberg Trace formula (particularly volume 2) uses the expression "converges beautifully". I can't find the definition, even searching the ...
0 votes
0 answers
49 views

Dense subspace of square integrable functions on the complex disc

Denote by $L^{2}(D,(1-|z|^{2})^{a-1}|z|^{2b-2}dx dy)$ the set of square integrable functions on the complex disc $D= \lbrace z \in C, \; |z| <1 \rbrace$ with respect to the measure $(1-|z|^{2})^{a-...
1 vote
0 answers
38 views

Taming families of rate functions

$\newcommand\R{\mathbb R}$Let us say that a function $r\colon\R_+\to\R_+$ is a rate function if $r$ is nondecreasing and $r(x)\to\infty$ as $x\to\infty$. Let us say that a family $(r_j)_{j\in J}$ of ...
8 votes
2 answers
268 views

Rearrangement, conditional convergence, and "placid" permutations

This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. It ...
0 votes
1 answer
83 views

Fokker-Planck: uniqueness and convergence to stationary distribution

Consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of ...
0 votes
0 answers
41 views

Equivalent formulation of uniform convergence in measure

If $(X,m)$ is a measure space, and $f_n, f : X \to \mathbb C$ are measurable functions, then it is known that $f_n \to f$ in measure if and only if every sub-sequence $(f_{n_i}) _{i \ge 0}$ contains a ...
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3 votes
0 answers
73 views

On tangential approach regions for general power series converging on the unit disk

Notation and premises. Here it is a list of notations more or less explicitly used in the question: If $z\in\Bbb C$ then $z = re(t)$ where $r\in \Bbb R_{\ge 0}$, $t\in [0,1]$ and $e(t)\triangleq \exp(...
2 votes
0 answers
71 views

Relations between different "propagation of chaos" type results?

My questions come from the paper Logarithmic Sobolev inequalities for some nonlinear PDE’s written by F. Malrieu (May 2001). The basic set-up is that we have a $N$-particle system $(X^{i,N}_t)_{1\leq ...
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-2 votes
1 answer
195 views

Tricky (for me) limit

I've been trying to compute the following limit for a few hours. Let $f(\gamma, \beta)$ be defined as follows: $$f(\gamma, \beta)=\lim_{x \rightarrow \infty} (1-\gamma^{1/x})(\log(x))^{\beta}.$$ I am ...
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0 votes
0 answers
62 views

Does non-parametric density estimation converge when cross-validation is used for model selection?

Suppose you have an infinite sequence of parametric probability density models, $\phi_i(\theta)$, with monotonically increasing parameter counts as $i$ increases, and a training sample of size $N$. ...
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0 votes
0 answers
47 views

What's the convergence condition for the generating function formula of Legendre polynomials?

What is the convergence condition of the next infinite series about the Legendre polynomials $P_n(x)$? $$ \frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^\infty P_n(x)t^n $$ I know it is convergent at least ...
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2 votes
1 answer
100 views

Mikusiński's approach to Bochner integrals; replace absolute by unconditional?

In the book The Bochner Integral, Mikusiński described an approach to Lebesgue and Bochner integrals via absolutely convergent series corresponding to step functions: Defn. Let $X$ be a Banach space. ...
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1 vote
3 answers
228 views

Squeezing more convergence from the convergence in all $L^p$ spaces

Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...
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4 votes
1 answer
76 views

Distance between trunctated random walk and its normal form

I have $$X_i \sim N(0,1), \quad S_n=X_1+\cdots+X_n,$$ $$ \mathscr{S}_n (t, \omega) := \frac{1}{ \sqrt{n} } \sum_{i=1}^{n} \left[ S_{i-1} (\omega ) + n \left( t - \frac{i-1}{n} \right) X_{i}(\omega) \...
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2 votes
1 answer
232 views

One series converges iff the other converges

In Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges it is said that this sequence of partial sums converges $$ \begin{split} \sum_{1<n\leq N}\frac{a_{n}}{\...
0 votes
0 answers
27 views

Demonstrate a certain inequality related to the accompanying law theorem

Suppose I have a triangular array of r.v. $(X_{nj})_{1\leq j \leq k_n}$- $X_{nj} \sim \mu_{nj}$, $k_n \uparrow\infty$ - which are independent in each row and satisfies the uniformly asymptotically ...
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3 votes
1 answer
92 views

For a random sequence $X_0, X_1, X_2, \ldots$ and $F_n$ the empirical CDF, does $F_n(X_0)$ converge to a uniform random variable?

Let $X_0, X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with continuous CDF $F(x)$ and define a sequence of ...
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2 votes
1 answer
148 views

Are Chebyshev polynomials a Schauder basis of $\mathrm{Lip}[-1,1]$?

It is known that every Lipschitz function $f \colon [-1,1] \to \mathbb R$ can be expressed as a series in the Chebyshev polynomials $$f = \sum_{n = 0}^\infty a_n T_n $$ which is absolutely convergent ...
-1 votes
1 answer
146 views

Convergence to a constant or not? Reference request [closed]

Consider the function $$f(n) = \log n /(n\ \log\theta(p_n)),$$ where $\theta$ is the first Chebyshev function and $p_n$ is the $n$-th prime. Does $f$ converge to a constant as $n$ grows to infinity, ...
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1 vote
1 answer
101 views

A problem of the limit of $\frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a}}$

Suppose that $f$ is a continuous function on $[0,1]$. For $0<a<1$, if $$ \varlimsup_{\delta \rightarrow 0} \frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a}} = \...
  • 389
0 votes
0 answers
39 views

Persistence of planar trajectory converging to a node / focus

I consider a planar system $\dot u =F(u,p)$ where p is a scalar parameter. Suppose that the flow $\phi^t(u_0; 0)$ from $u_0$ converges to a stable node / focus $x^{eq}_0$ for the parameter value $p=0$....
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3 votes
1 answer
87 views

Boundedness and convergence

If I know that $\Phi_\varepsilon$ is bounded in $L^{\infty}(\mathbb{R}^{2d})$ and that $\nabla \Phi_\varepsilon$ is bounded in $L^{\infty}(\mathbb{R}^{2d})$, is it true that $\nabla \Phi_\varepsilon \...
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1 vote
0 answers
73 views

$ \varlimsup_{x\rightarrow 0^+}\frac{f(x)}{x^a}=\varliminf_{x \rightarrow 0^+}\frac{f(x)}{x^a} $

$f(x)$ is continuous for $\forall x \geq 0$ and monotonically decrease. $f(0)=0$. $a>0$. Is it true $$ \varlimsup_{x\rightarrow 0^+}\frac{f(x)}{x^a}=\varliminf_{x \rightarrow 0^+}\frac{f(x)}{x^a} $...
  • 389
2 votes
1 answer
50 views

Compare two limits related to Hölder condition

Suppose $f$ is a continuous function on $\mathbb{R}$. $0<a<1$. $B(x,r)$ is open ball centered at $x$ with radius $r$. Is it true that $$ \varlimsup_{r\rightarrow 0} \frac{|f(x+r)-f(x)|}{|r|^\...
  • 389
2 votes
2 answers
474 views

About roots of polynomials [closed]

Let $n\in\mathbb N^*$, $P(x)=a_0+\dotsb+a_{n-1}x^{n-1}+x^n$ and $r_1,\dotsc,r_n\in\mathbb C$ the roots of $P$. Is it true $\lim\limits_{\max(\lvert a_i\rvert,i=0\dotsc n-1)\rightarrow 0} \max(\lvert ...
  • 3,185
1 vote
1 answer
281 views

Lower bound and limit of a sum with binomial coefficients

Let $$A_k = \sum_{i=1}^k i {3k-2i-1 \choose i-1} {2i-2 \choose k-i}$$ $$B_k = \sum_{i=1}^k i {3k-2i-2 \choose i-1} {2i-1 \choose k-i}$$ $$C_k = \sum_{i=1}^k (3k-2i-2) {3k-2i-3 \choose i-1} {2i\...
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5 votes
4 answers
711 views

Limit of a sum with binomial coefficients

Let $$A_k = \frac{\sum_{i=1}^ki{2k-i-1 \choose i-1}{i-1 \choose k-i}}{k{2k-1\choose k}}$$ $$B_k = \frac{\sum_{i=1}^ki{2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}$$ $$C_k = \frac{\sum_{i=1}^k(...
  • 115
2 votes
2 answers
210 views

Convergence almost everywhere of characteristic functions

Let $(\Phi_n)_n$ be the characteristic functions of probability measures $(\mu_n)_n$ and let $\Phi$ be the characteristic function of a probability measure $\mu$. Do you know an example where $\Phi_n(...
1 vote
1 answer
61 views

Convergence of the average weight of an infinite path through a weighted directed graph

Consider a directed graph $G = (V, E, w)$, where $V$ is the set of vertices, $E \subseteq V \times V$ is the set of directed edges (with self-loops allowed), and $w : E \to \mathbb{R}_+$ is a weight ...
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4 votes
2 answers
220 views

$\lim\inf_{t\to \infty} f(x(t))=0\Longrightarrow \lim\inf_{t\to\infty} \|x(t)\|=0$

I am a PhD student and during my research I was presented to the claim that For a positive definite function $f:\mathbb{R}\to \mathbb{R}$ continuous in $0$, with $0$ a stable point at $t=0$ for $x$, ...

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