Questions tagged [limits-and-convergence]
Convergence of series, sequences and functions and different modes of convergence.
787
questions
2
votes
1
answer
73
views
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 $\varepsilon >0$ ...
user514580
1
vote
0
answers
76
views
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 ...
- 1,087
3
votes
2
answers
262
views
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 ...
- 118k
1
vote
1
answer
235
views
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 $x_n\in\mathbb{R}$ ...
- 175
1
vote
1
answer
57
views
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 $g(t) : [0,1] \to H$...
- 2,749
2
votes
1
answer
152
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 ...
- 1,031
0
votes
1
answer
434
views
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-...
- 13
0
votes
0
answers
51
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 ...
- 5,282
0
votes
0
answers
72
views
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-...
- 771
4
votes
1
answer
462
views
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 ...
- 45.5k
3
votes
1
answer
705
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 ...
- 31
1
vote
1
answer
81
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 ...
- 185
0
votes
0
answers
70
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 ...
- 647
4
votes
1
answer
200
views
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 ...
- 700
1
vote
1
answer
80
views
Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers
This is a repost from MSE because I got no answers there.
I have been trying to find the supremum of this bivariate function over a specific region. However, the expressions that I get are horrible. I ...
- 573
1
vote
0
answers
48
views
Convergence of random variables based on shifts of a markov chain
Suppose we have a discrete time (not necessarily stationary) Markov chain $X=(X_0,X_1,X_2,\dots)$ on $(\Omega, F)$. We assume $X$ is Harris ergodic with an invariant distribution.
Suppose we have a ...
- 11
3
votes
0
answers
122
views
Convergence of the perimeter of level sets
I have already posted this question on Math StackExchange. Suppose you have a sequence of $C^1$ functions $\{\phi_n\}_{n\in \mathbb{N}}$ defined on $\mathbb{R}^n$ that converges in $C^{1}_{\mathrm{loc}...
3
votes
1
answer
161
views
Quantitative version of ergodic theorem in Markov chains
Consider an irreducible Markov chain $X_t$ with finite state space $E$, and unique invariant measure $\pi$. Fix a function $V:E\to\mathbb R$ such that $E_\pi[V]=0$. The ergodic theorem tells us that, ...
- 59
2
votes
2
answers
237
views
Convergence of the row sums in a triangular null array with zero mean
Let $(X_{jn})_{1\leq j \leq n}$, $n\in \mathbb N$, be a triangular array of random vectors in $\mathbb R^d$ (the $X_{jn}$ are understood to be independent in $j$ for fixed $n$.). We say that the ...
- 13
0
votes
0
answers
136
views
Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
- 781
1
vote
1
answer
174
views
Does $\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert \, dt$ converge for $L^1_\text{loc}$ $f : [0,\infty) \to \mathbb{R}$?
Let $f(t) : [0,\infty) \to \mathbb{R}$ be an $L^1_\text{loc}$ function.
Then, I wonder if the following series
\begin{equation}
\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert
\, dt
...
- 2,749
1
vote
2
answers
238
views
Reference request and clarification for Central Limit Theorem for complex random variables
I'm looking for a reference and a proof of the following version (or eventually a more general version) of the Central Limit Theorem for complex random variables.
Theorem. Let $Z_1, Z_2, \dots, Z_n$ ...
- 361
1
vote
0
answers
81
views
Central limit theorem with limit of functions
Suppose that
$$\sqrt{n}(X_n - \theta)\xrightarrow{d} X,$$
according to the delta method, we have
$$\sqrt{n}(g(X_n)-g(X))\xrightarrow{d} g'(\theta)X$$
when $g$ is differentiable.
My question is, if
$$\...
- 11
4
votes
1
answer
231
views
$\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}$ for various $x$
Let $$f(x)=\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}.$$
Compute $f(1)$ and $f(2)$.
1
vote
0
answers
57
views
Convergence of stochastic linear recurrences
Suppose that $\zeta_t$ is a univariate, stationary stochastic process ($t\in\mathbb{N}^+$).
Let $x_0\in\mathbb{R}^n$, and let $f:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$ be a continuously ...
- 183
2
votes
1
answer
161
views
A question about convergence of stochastic processes converging to a random walk
Consider the following random walk $(y_t)_{t \in \mathbb Z_+}$:
$$y_t = y_{t-1} + u_t,\quad (u_t)_{t \in \mathbb Z_+} \overset{iid}{\sim} N(0,1), \quad (t \in \mathbb Z_+)$$
where $y_0, u_1, u_2,...$ ...
- 13
1
vote
1
answer
141
views
Convergence rate of a sequence of sets to a set-theoretic limit?
Suppose $n\in\mathbb{N}$ and set $A\subseteq\mathbb{R}^{n}$.
If we define a sequence of sets $\left(F_r\right)_{r\in\mathbb{N}}$ with a set theoretic limit of $A$; how do we define the rate at which $\...
- 1
1
vote
0
answers
88
views
Numerical strategies for evaluating a modular invariant infinite sum
I'm working on a problem that involves the numerical evaluation of the following infinite sum:
$$
\sum_{m=-\infty}^{\infty} \ln \left|1\pm e^{-2\pi \tau_1 \sqrt{m^2+x^2/(4\pi^2\tau_1)}-2 \pi i \tau_0 ...
- 11
1
vote
1
answer
116
views
calculating a double limit
We have the following term:
$$ (e^{-a h}+e^{-b h})^n / 2^n$$
Now we take the limit:
$$ h\to 0, n\to \infty $$
What relation of $h$ and $n$ must be satisfied for the following limit to hold?
$$\lim_{h\...
- 105
2
votes
1
answer
254
views
Rate of convergence of Fejer kernel to the Dirac delta function
This seems like something one might find in a book so I would be grateful for any references you think may be helpful.
I am interested in the rate at which of a function integrated against the $N$th ...
- 131
0
votes
1
answer
76
views
Convergence in expectation of a discontinuous function
Consider a random variable $X\in \mathbb{R}^d$. Let ${\theta_m}$ be a sequence of real numbers that converge to $\theta$. Let $f(x,y)$ be a function that is not continuous. To be specific, fix, $x=a$, ...
- 11
0
votes
0
answers
71
views
Do finite exchangeable random sequences behave asymptotically independently?
Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ and $(Y_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be triangular arrays of row-wise exchangeable random variables, that is for any $n$ and permutation $\...
- 899
3
votes
0
answers
180
views
On the spectrum of Fokker–Planck with linear drift
The paper by Liberzon and Brockett, "Spectral analysis of Fokker–Planck and related operators arising from linear stochastic differential equations." SIAM Journal on Control and ...
- 116
1
vote
1
answer
217
views
convergence for series of random variables
Suppose $X_n\sim N(0,1) $ is iid, then it is easy to see that
$$\sum_{n=1}^{\infty}\frac{X_n}{n}\cos nx$$
converges a.s. for any $x$ since
$$\sum_{n=1}^{\infty}var(\frac{X_n}{n}\cos nx)<\infty$$
...
- 105
3
votes
0
answers
386
views
Surprisingly difficult limit of a sequence
Is there an easy way to prove that $|\operatorname{Re}(a_n)| \to \infty$ where $a_n=\left(\frac{1}{2}+i\frac{\sqrt{7}}{2}\right)^n$?
Of course $|a_n| \to \infty$, but we have
$$
\operatorname{Re}(a_n)=...
- 489
8
votes
2
answers
578
views
On Zagier's missing continued fraction with multiple limits?
I. Zagier's continued fraction
As pointed out by Gorodetsky in his answer, Zagier evaluated the continued fractions associated with his six sporadic sequences excepting the one for $(-9,-3,-27)$. Let $...
- 12.1k
2
votes
2
answers
356
views
Finding a new level-$7$ pi formula using the relation $j_{7A}(\tau) = \left(\sqrt{j_{7B}(\tau)}+\frac{7}{\sqrt{j_{7B}(\tau)}}\right)^2$
(Note: This third method continues from this post.)
There are level-$7$ pi formulas based on the McKay-Thompson series $T_{7A}$ and Cooper's $s_7$ sequence in this paper. This third method, among ...
- 12.1k
3
votes
0
answers
194
views
Ramanujan's pi formulas with a twist (nine years later)
(Note: The second method described here continues this post.)
About nine years ago, I made an MO post "Ramanujan's pi formulas with a twist". An answer was informative, but not completely ...
- 12.1k
4
votes
1
answer
189
views
Transformations of Ramanujan's 1/pi formulas $\sum_{n=0}^{\infty} s(n)\frac{An+ B}{C^n}$ and Monster moonshine functions
Someone with many papers on Ramanujan's work asked me how I managed to find the closed-forms for the binomial sums of level $10$ in a recent MO post. (A colleague of his wasn't able to find them.) I ...
- 12.1k
6
votes
2
answers
526
views
Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped
I have a weird question which probably seems out of place here but it has proven more difficult than anticipated. I am going to describe the game without showing work toward a solution. Numerically, ...
user504691
1
vote
0
answers
45
views
Finding a distance so that this function is a contraction mapping
Let $f(x,y)=(y,\frac{2}{x+y})$ defined on $(0,\infty)\times (0,\infty)$. Is there a distance $d$ on $(0,\infty)\times (0,\infty)$ such that $f$ is a contraction of the metric space $((0,\infty)\times (...
- 489
0
votes
0
answers
138
views
Convergence of Solutions of Integral Equations with Weakly Converging Forcing Terms
Let $\Omega$ be a bounded interval of $\mathbb{R}$ and let $y\in L^\infty(\Omega \times (0,T))$ be a mild solution of the integral equation
$$
y(\cdot,t)=S(t) y_0+\int_0^t S(t-s) \left[u(\cdot,s)y(...
- 55
1
vote
1
answer
112
views
Which kind of convergence can we get from Laplace transform convergence?
This question is a related question see this post Vague convergence VS Laplace transform convergence. But now we assume that
\begin{equation}
\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}...
1
vote
1
answer
225
views
Vague convergence VS Laplace transform convergence?
If we assume that $\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}\mu(dx), \forall s\geq0$, it is possible to show that $\mu_n\to\mu$ vaguely. Where $\mu_n$ is a measure. Please check here for ...
0
votes
1
answer
119
views
Bound the expectation of an average
Let $(a_n)_{n \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $\nu_l(b) \le P[a_n = b\mid a_1,\ldots,a_{n-1}] \le \nu_u(b)$ almost surely for every $n \ge 1$ and $b \in ...
- 76
2
votes
2
answers
220
views
Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$
When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a_{-1}$ in the Laurent expansion $f(x) = \sum_{n=-1}^\infty a_n x^n$ by evaluating the limit $\lim_{...
- 1,692
0
votes
0
answers
39
views
Fischer Information and Entropy, matrix case, determinant of covariance matrix going to zero
If $X \sim \mathcal{N}(\mu, \sigma^2)$, then
\begin{equation}
\mathcal{I}\left(\mu, \sigma^2\right)=\left(\begin{array}{cc}
\frac{1}{2\mathcal{H}...
6
votes
2
answers
485
views
Computing a limit on the unit sphere: Riemann Lebesgue?
Let $u\in L^1(\mathbb{S}^{d-1})$. I want to show that
\begin{align*}
\lim_{|\xi|\to \infty}
\int_{\mathbb{S}^{d-1}}(1-\cos(\xi\cdot w))u(w)d \sigma_{d-1}(w)
= \int_{\mathbb{S}^{d-1}}u(w)d \sigma_{d-1}(...
- 1,033
0
votes
1
answer
138
views
Finding weak LUR property of $C[0,1]$ with an equivalent norm
On the space $X=C[0,1]$, define a norm $||| f |||^2=\Vert f \Vert_{\infty}^2 + \Vert f \Vert_2^2$, where $\Vert \cdot \Vert_\infty$ is the sup norm on $C[0,1]$ space and $\Vert \cdot \Vert_2$ is the $...
- 75