Questions tagged [limits-and-convergence]
Convergence of series, sequences and functions and different modes of convergence.
774
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Convergence in probability of quadratic form with positive mean
Let $\boldsymbol{X}_n\in\mathbb{R}^n$ be a sequence of Gaussian random vectors with independent entries, such that $X_{n,i}\sim \mathcal{N}(\mu_i,\sigma^2)$ (that is, all entries of the $n$th vector ...
1
vote
0
answers
36
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Bound on a two-dimensional recursive series
For $n,k\in\mathbb{N}$, let $f(n,k)$ be defined as follows.
If $n \geq k$ and $n > 2$, then
$$
f(n,k) = \frac{k(n-k)}{n(n-1)}f(n-2,k-1) + \frac{k(k-1)}{n(n-1)}f(n-2,k-2) + \frac{n-k}{n}f(n-1,k) + \...
0
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1
answer
50
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Uniform convergence of differential quotients in $L^1$
I know that the question arose already in other contexts. However, I think this question might be different. If I have $f\in W^{1,1}$ then it is obvious that $\frac{f(x+t)-f(x)}{t}$ converges ...
2
votes
2
answers
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"Completeness" for weak convergence of unbounded closed operators on a separable Hilbert space $H$
Let $H$ be a separable Hilbert space with the inner product $\langle, \rangle$ and $\{ T_n \}$ be a sequence of unbounded closed linear operators with a common dense domain $D \subset H$ such that $...
1
vote
1
answer
71
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Mittag-Leffler expansions converging to bounded function
Is it true that
$$\lim_{N\to\infty}\left\langle\sum_{n=-N^2}^{N^2}\frac1{(Nx-n)^2}\right\rangle_N=\pi^2$$
for some suitable definition of "minima smoothing" such as $\langle f(x)\rangle_N\...
0
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0
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43
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How to find A(i, d)?
Let $s(n)$ denote the digit sum of a natural number $n$. For $i, d\in \mathbb{N}$ define $$A(i, d) = \limsup_{m\to \infty}\frac{|\{n\leq m | s(n)\equiv i\mod d\}|}{m}.$$ Compute $A(i, d)$ for all $i, ...
1
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2
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Show convergence result
Consider the following sets:
$$
A = \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n, [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\},
$$
and
$$
A_n = \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)...
2
votes
1
answer
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Lipschitz continuity of eigenprojections
This question has the same flavor of this and this questions, but asks for something stronger.
Assume that
$A$ is a symmetric $n \times n$ matrix,
$H$ is a $n \times n$ perturbation matrix.
Moreover ...
1
vote
1
answer
108
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Limit of $F_{n}(\lfloor{nx}\rfloor)$ where $ F_{n}(k)=G_{n}(k)+H_{n}(k)F_{n}(k+1) $ and $F_{n}(n)=\mu.$
The following conjecture is inspired by asymptotic results in generalizations of the secretary problem.
CONJECTURE
Consider a sequence of functions {$F_n$} with $F_{n}:[0,n]\cap
\mathbb{Z}\rightarrow\...
0
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1
answer
300
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Necessary conditions for convergence of convolution
In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
7
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1
answer
515
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A variation on the Borel–Cantelli lemma theme
Let $X,X_0,X_1,\dots$ be nonnegative independent identically distributed (i.i.d.) random variables. Let
\begin{equation*}
E:=\bigcap_{n\ge0}B_n,
\end{equation*}
where
\begin{equation*}
B_n:=\...
2
votes
2
answers
302
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Existence of the limit of periodic measures
Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
3
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2
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$\lim_{n \to \infty} \frac{2^n}{n} \left[ 1 - \sum_{k = 1}^{n-1} \frac{(1 - \lambda 2^{-n})^{2^k}}{2^k} \right]$
Find the limit
\begin{equation*}
\lim_{n \to \infty} \frac{2^n}{n} \left[
1 - \sum_{k = 1}^{n-1} \frac{(1 - \lambda 2^{-n})^{2^k}}{2^k}
\right]
\end{equation*}
where $\lambda > 0$.
My guess is that ...
2
votes
5
answers
904
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Binomial series
I am interested in the limit $\frac{\sum_{k=0}^n \sqrt{k}\cdot\binom{n}{k}}{\sqrt{n}\cdot2^n}$ as $n$ goes to infinity. Any reference or argument?
In general what do we know about the asymptotic ...
2
votes
0
answers
155
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Question about the ergodic mean
This is a repost from this MathStackExchange question, where unfortunately I was not able to resolve this question.
I've read a thesis where there is an example on ergodic mean, where however there is ...
7
votes
3
answers
858
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Using the Stone-Weierstrass theorem to solve an integral limit
The following question was posted on math stack exchange here but it got no answers
Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
5
votes
1
answer
382
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On the convergence of a martingale
Let $W$ be a standard one dimensional Brownian motion and let $A$ be the process defined by :
$$\forall \ t\geq 0: \quad A_t := \int_0^t\left(1 + e^{W_s}\right)\mathrm{d}s$$
and for $t\geq 0$, we ...
-1
votes
2
answers
81
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Limits of integral series
Suppose we have the series of functions:
\begin{equation}
F(x)=\sum_{n=1}^{\infty} f_n(x)
\end{equation}
where convergence is uniform.
Additionally, consider the partial functions of the series:
\...
27
votes
5
answers
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How to show a function converges to 1
Consider the following recurrence relation in two variables:
$$f(a, b) = \frac{a}{a+b} f(a-1,b)+ \frac{b}{a+b}f(a+1,b-1) $$
for positive integers $a$ and $b$,
with the boundary conditions $f(0,b)=0$ ...
3
votes
2
answers
237
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Definition of weak conditional convergence of random variables
I am looking for a definition of conditional convergence. Suppose that $X_1, X_2, \dots, X_n$ are $\mathbb R$-valued random variables with finite second moments, and $W_1, W_2, \dots, W_n$ are iid $\...
7
votes
1
answer
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High dimensional Fekete's subadditive lemma: does $|\vec x_{n+m}|\leq |\vec x_n+\vec x_m|$ imply the convergence of $\{\vec x_n/n\}$?
Let $d\geq 1$ be a positive integer. If $\{\vec x_n\}_{n=1}^\infty$ is a sequence of $d$-dimensional vectors satisfying $$\lvert\vec x_{n+m}\rvert\leq \lvert\vec x_n+\vec x_m\rvert\qquad \text{for all ...
10
votes
1
answer
314
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Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap converge?
Posting in MO since this questions has been unanswered in MSE for 3 months.
Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $...
0
votes
1
answer
131
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Does weak $H^1$ convergence imply $L^2$ convergence when multiplied with an exponentially decaying function?
I'm trying to see if given a sequence $\{f_n\}_n\in H^1$ which converges weakly in $H^1$ to a function $f_*$, the $L^2$ norm $\|R^2f_n\|_{L^2}^2$ converges to $\|R^2f_*\|^2_{L^2}$, where $R$ is a ...
4
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0
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Question about $n$ random points in a regular polygon, and a limiting probability
Suppose we choose $n$ uniformly random points in a disk, then draw the smallest circle that encloses all of those points. There is evidence suggesting that the probability that the enclosing circle is ...
0
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1
answer
243
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Nature of $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $ [closed]
I'm trying to determine the nature of this series $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $, but I'm not getting anywhere. I've tried using the Abel and trigonometric formulas, but I can't ...
2
votes
2
answers
198
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Weak convergence of measures on continuous function spaces
Let $S$ be the unit sphere of $C[0,1], \|\cdot\|_{\infty})$, let $(B_{t})_{t}$ the brownian motion.
I would like to show that the measure $\mu_r$ defined on $\mathbb{B}(S)$ by
$\mu_r(A):=P\Big(\frac{...
2
votes
2
answers
190
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Limit of a integral whose integrand diverges under the limit
I am trying to simplify the following limit of integral where $\mu$ is given:
$$p(y) = \lim_{\sigma \to 0} \int_{\mathbb R} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^...
0
votes
1
answer
105
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Functional CLT with an asymptotically small time change
This question was posted to MathSE but it seems like MathOverflow might be the more appropriate place for it.
Suppose I know that $(\frac{1}{\sqrt{m}}X(mt))_{0\leq t\leq 1}\xrightarrow[m\to\infty]{\...
1
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0
answers
99
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Asymptotic distribution of L infinity norm of Gaussian random vector
Let $\mathbf{X}_n = (X_{n,1}, \ldots, X_{n,n})$ be a $n$-dimensional random vector with $N_n( \mathbf{0}_n, \boldsymbol{\Sigma}_n )$ distribution. The asymptotic distribution of the $L_\infty$-norm of ...
11
votes
2
answers
921
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Twice continuously differentiable implied by existence of limit
I have the following question. Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions (vanishing at infinity) and assume that
$$
\frac{f(x+t)+f(x-t)-2f(x)}{t^2}\to g(x)
$$
for all $x\in X$ when ...
22
votes
1
answer
4k
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A challenging (for me) limit calculation
How to calculate the following limit
$$
\lim_{n\to\infty}\sqrt{n}\underbrace{{}\sin(\sin(\sin(\sin(\cdots\sin(\frac{1}{\sqrt{n}})\cdots))))}_{n \text{ sin's}} \text{?}
$$
${}{}$
3
votes
1
answer
359
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On the convergence in total variation
$\newcommand\R{\mathbb R}$For a probability measure $\mu$ over $\R^2$ and a unit vector $u\in\R^2$, let $\mu^u$ denote the pushforward of $\mu$ under the projection map $\R^2\ni x\mapsto u\cdot x\in\R$...
1
vote
1
answer
99
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How to prove this iterative convergence of trigonometric functions [closed]
Consider the set of sequences $ S=\{\{s_n\}_{n \ge 0}\mid s_n \in \{-1,+1\}\} $
For any set: $ s=\{s_n \}_{n \ge 0}\in S$, we define the sequence
$$ c_n=\sum_{k=0}^n \frac{s_0s_1 \dots s_k}{2^k}$$
Now,...
0
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1
answer
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If $a_1=1$ and $a_n=\sec (a_{n-1})$ then what does the proportion of positive terms approach, as $n\to\infty$?
Consider the sequence $a_1=1$ and $a_n=\sec (a_{n-1})$ for $n>1$.
What does the proportion of positive terms approach, as $n\to\infty$?
At first I thought the limiting proportion might be $\frac{...
2
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0
answers
56
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A lemma in the application of Lions's concentration compactness pricnciple in Hardy-Littlewood-Sobolev inequality
I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2".
The Hardy-Littlewood-Sobolev (HLS) ...
2
votes
1
answer
72
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Conditions for absorption
Let $X$ be a Markov chain with countable state space $S$ and transition kernel $P$, and let $h \colon S \to [0,1]$ be a sub-harmonic or super-harmonic function. Assume that for all $\varepsilon >0$ ...
1
vote
0
answers
76
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Asymptotics of ${2n \choose n+k} {2n \choose n}^{-1}$ when $k$ grows with $n$
The quotient $Q(n,k) := \frac{{2n \choose n+k}}{{2n \choose n}}$ clearly converges to one for $k \in \mathbb{N}$ fixed and $n \rightarrow \infty$. Simultaneously it converges to zero, if $k$ grows ...
3
votes
2
answers
252
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On convergence of convex-concave functions
Let $(f_n)$ be a sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that:
$f_n$ is strictly convex on $(-\infty,x_n)$,
$f_n$ is ...
1
vote
1
answer
221
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Convergence of concave/convex function
Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ ...
1
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1
answer
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If a Hilbert space-valued mapping is norm-decreasing, can we make sense of the limit of sums consisting of projected-values?
Let $H$ be some separable Hilbert space with a given orthonormal basis $\{ e_n \}$. Write the projection onto the subspace spanned by first $N$ basis elements to be $P_N$.\
Now,let $g(t) : [0,1] \to H$...
2
votes
1
answer
150
views
Continuity of an upper semi-continuous function over periodic points
Let $f: X \to \mathbb{R}$ be an upper semi-continuous function on $X$, which is a compact subspace of a vector space. Let sequence $x_n, n \in \mathbb{N}$, with positive elements - periodic: there ...
0
votes
1
answer
430
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A complex question related to a certain convergence of Lévy measures
Consider the sequence of stochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and:
\begin{equation}\label{I}\tag{SP}
X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...
0
votes
0
answers
50
views
Characterizing functions that are limits of integrable lower-bounded functions
Let $X$ be a separable Hausdorff topological space, endowed with a positive finite Borel regular measure. Consider those (trivially measurable) functions $f : X \to \mathbb R$ such that their ...
0
votes
0
answers
70
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Error bound for stochastic gradient descent method
To solve an optimization problem $\min_x G(x)$ using standard stochastic gradient descent method, we let $x_0$ be the initial point and $x_k$ be the $k$-th point such that
\begin{equation}
x_k = x_{k-...
4
votes
1
answer
461
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How "correct" is Knuth's fast addition $(a,b) \mapsto (a \oplus b) \oplus ((a\land b) \ll 1)$?
Donald Knuth suggested a bitwise approximation for addition on the non-negative integers that is very fast on common processors:
$(a,b)\mapsto (a\oplus b) \oplus ((a\land b) \ll 1)$,
where $a,b$ are ...
3
votes
1
answer
621
views
Sequence of $L^2$ functions converging to zero weakly s.t. $|f_n|^2$ converges to 1 weak-star?
I am trying to construct a sequence $\{f_n\} \in L^2([0,1])$ with $f_n \geq 0$ a.e. such that $f_n \to 0$ weakly in $L^2$ (meaning $\int f_n dx \to 0$ for all $f \in L^2([0,1]))$ and such that for all ...
1
vote
1
answer
79
views
Asymptotic scaling of mean and variance for the norm of random vector (non gaussian components)
The norm of a vector whoose components $X_i$ are normally distributed follows the Non-central chi distribution and it can be shown that, increasing the number of components $k$ (i.e. the dimension of ...
0
votes
0
answers
68
views
Meromorphic extension of a limit function
Suppose $f_j(z)$, $j=1,2,..$ is a sequence of meromorphic functions on the complex plane $\mathbb{C}$. With a common set of all poles given by $S = \{-i,-2i, -3i,..\}$.
Assume that each of them is ...
1
vote
0
answers
29
views
Can we modify this extended pseudometric such that its convergence is equivalent to that in measure?
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
4
votes
1
answer
199
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Show that $\frac{1}{2 \pi i} \oint_{\mathbb{S}^1} \frac{1-\hat{f}(\xi)}{1-\xi}\cdot \frac{\mathrm{d} \xi}{\xi^{n+1}} \to 0$ as $n \to \infty$
Let $f = (f_0,f_1,\ldots,f_n,\ldots) \in \mathcal{P}(\mathbb N)$ be a probability distribution on $\mathbb N$ and denote by $$\hat{f}(z) = \sum_{n\geq 0} z^n f_n$$ for its probability generating ...