Questions tagged [stieltjes-integral]
For questions about Stieltjes integrals. Use with other tags as needed, such as [riemann-integration], to specify Riemann–Stieltjes, Lebesgue–Stieltjes, etc.
198
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Infimum of the Upper Sums
Let $P$ be the partition of the interval $[a,b]$. If $P$ is divided into two partitions $P_1$ and $P_2$ such that $P_1$ covers the interval $[a,c]$ and $P_2$ covers the interval $[c,b]$, then $U(P, f, ...
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What do the limits represent on a Riemann–Stieltjes integral?
Say I have the integral
$$
\int_0^{10} 1\ d(x^2)
$$
Are the limits on the range of $x$ or of $x^2$ ?
Based on the wiki page I am assuming it is the limits of $x$ in which case I imagine the answer ...
- 141
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Riemann–Stieltjes integral with itself. [closed]
Suppose that $f:[a,b] \to \mathbb R$ is continuous everywhere and non-decreasing. I am wondering that whether $$\int_a^b f(x)^{n-1}df(x) = \frac{1}{n}[f(b)^n-f(a)^n]$$ for $n=1,2,\ldots$. Notice that $...
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Riemann-Stieltjes Integral Does not Add Up
Does there exist functions $f,\phi : [a,b]\rightarrow \mathbb{R}$ and $c\in [a,b]$ such that the Riemann-Stieltjes integrals $\int_a^c f\,d\phi$ and $\int_c^b f\,d\phi$ exist but $\int_a^b f\,d\phi$ ...
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Is there a modification of Laplace's method for obtaining the asymptotic behaviour of Riemann-Stieltjes integrals?
Is there a modification of Laplace's method for obtaining the asymptotic behaviour of Riemann-Stieltjes integrals?
In particular, I am interested in asymptotic behaviour of the Riemann-Stieltjes ...
- 423
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A Lebesgue Stieltjes integral where the integrator is a Lebesgue Stieltjes integral
Let $f: \mathbb{R} \to \mathbb{R}$ be a Borel measurable and non-negative function; let $g: \mathbb{R} \to \mathbb{R}$ have bounded variation but not necessarily be right-continuous. Then, the ...
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Proving that quotient of orthogonal polynomials is a Padé approximant of Stieltjes transform.
Conext and notation. In the question below, $R_n(z) / S_n(z)$ denotes the n-th convergent of the continued fraction
$$ \frac{1|}{|z-b_1} + \frac{-a_2|}{|z-b_2} + \dots + \frac{-a_n|}{|z-b_n} + \dots,$$...
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Total variation of a class in $L^1 [0, 1]$ and lower semicontinuity
Consider a function $f \in \mathcal L^1 [0, 1]$: we define the total variation of $f$ as usual by
$$
V_0^1 (f) = \sup \sum_{k = 0}^{n - 1} | f(x_{k + 1}) - f(x_k) |,
$$ where the supremum is taken ...
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Completely Monotonic Functions and Stieltjes-like integral
A function $f(x)$ is completely monotonic if $(-1)^nf^{(n)}(x) \geq 0$ for all $n \geq 0$ and all $x > 0$. These functions are characterized by Bernstein's theorem and are well studied in Widder's ...
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Riemann-Stieltjes Integral for discontinuous integrator
I was wondering if there is a good general technique for evaluating the Riemann-Stieltjes integral in cases where the integrator is discontinuous on a few points. For example: find $$\int_{0}^{2} f(x) ...
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Proper notation for integration on manifolds?
If I have an integral over a manifold $\mathcal{M}\subset\mathbb{R}^n$, and I have an invertible continuously differentiable map $\varphi:\mathcal{M}\to\mathcal{D}\subseteq \mathbb{R}^d$, and $\...
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Riemann-Stieltjes integral with respect to functions equal almost everywhere [closed]
Let $f, g$ be Lebesgue integrable, real-valued functions on $[0, 1]$ with bounded variation, and let $\phi : [0, 1] \to \mathbb R$ be continuous. Assuming that $f = g$ almost everywhere, does one have ...
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Where does the Change Of Variable of Riemann-Stieltjes integral theorem uses continuity of COV function? Isn't the assumption superfluous? Apostol 7.7
Here is theorem 7.7 from Tom Apostol, Mathematical Analysis:
Let $f\in \mathfrak R (\alpha)$ on $[a,b]$ and let $g$ be a strictly monotonic continuous function defined on an interval $S$ having ...
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Is the Darboux approach / definition compatible with a bounded integrator in Riemann–Stieltjes integrals? Why these hypothesis in Rudin PMA chapter 6?
I'm reading Rudin's Principles of Mathematical Analysis $6^{th}$ chapter and am wondering why doesn't he require the weaker condition of $\alpha$ (the integrator) being bounded on $[a,b]$ ?
Is he ...
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(Complex Stieltjes Integral) If $f$ is integrable wrt $\alpha$, is $\overline{f}$ also integrable wrt $\alpha$?
Let $f,\alpha$ be two bounded complex functions on $[0,1]$. We say that $f$ is integrable w.r.t. $\alpha$ iff the Riemann sum $$\sum f(t_i)(\alpha(x_i)-\alpha(x_{i-1}))$$ converges to a fixed number $...
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What is the difference between a line integral and Riemann-Stieltjes integral?
I'm particularly concerned with the visual representation of line integration. The following is provided by Wikipedia
while this is shown in the Wikihow page
Lastly, the Riemann-Stieltjes integral
...
- 119
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35
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Property of Darboux integrals that fails for some Darboux-Stieltjes integrals
Let $X$ be a closed interval, $I \subseteq X$ be a bounded interval, $\alpha : X \rightarrow \mathbb{R}$ be a monotone increasing function and $f : I \rightarrow \mathbb{R}$ be Darboux-Stieltjes ...
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Evaluating a double integral with distribution functions in the differential to find the expected value
Given that
variable $\tilde{p}$ follows a probability distribution function $\Omega$ with mean $\bar{p}$ lower and upper limits $a$ and $b$, repectively: $\tilde{p}\sim\Omega(a,b)$, and
variable $\...
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Under which conditions can a Lagrange multiplier of bounded variation (Stieltjes integral) be expressed by a continuous functional (Lebesgue integral)
My question conerns Example 1 in §9.3 of Luenberger's Optimization by Vector Space Methods. There, he considers the problem of finding $x\in D^n(t_o,t_1)$ (n-vector functions possessing continuous ...
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21
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Proof of formula for Lebesgue -Stieltjes integral with generalized inverse function
Let $F$ be a nondecreasing function, we cal assume $F(-\infty)=0$ $F(\infty)=1$, and $F^{-1}(t)=inf\{x:F(x)>t\}$ be its generalized inverse, how can we prove the following formula for Lebesgue ...
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Are two Riemann integrable functions, Riemann stieltjes integrable with respect ro each other?
Let f,g be real valued Riemann integrable functions on [a,b] then does it imply, f is Riemann Stieltjes integrable with respect to g on [a,b]?
If not please do provide a counter example demonstrating ...
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Does the $\Gamma\subset\mathbb{C}$ curve always has to be smooth in Cauchy's integral theorem?
In Cauchy's integral theorem (see: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem), does the $\gamma$ representation of the $\Gamma\subset\mathbb{C}$ curve always has to be smooth (i.e. at ...
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What is the correct way of translating a joint probability to Lebesgue-Stieltjes integrals
I have a set inequalities of several non-negative independent random variables and I want to compute the probabilities from these. While the cumulative distribution function of each of these random ...
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Stieltjes Integral of Product of Gaussian Processes
Let $(\Omega,\mathcal F, P)$ be a probability space, let $\mathcal G_1$ and $\mathcal G_2$ be two (not necessarily independent) centered real-valued Gaussian processes, i.e., measurable functions $\...
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LLN and rates in terms of Stieltjes integral
I am learning about Stieltjes integrals. Written in terms of a Stieltjes, the law of large numbers, $\frac1n\sum_{i=1}^n f(X_i)\to E(X)$ as $n\to\infty$, looks like:
$$
\int_{-\infty}^{\infty}f(x)dF_n(...
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question about definition of Lebesgue-Stieltjes measure
In some books it's defined with the outer measure:
$$\mu^*(E)=\inf\{\sum (f(b_n)-f(a_n)):E\subset\bigcup (a_n, b_n]\}$$
where $f: \mathbb{R}\to\mathbb{R}$ is not-decreasing and right continuous.
I'm ...
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If the Riemann-Stieltjes integral $\int_a^b{f\,dg}$ exists for every $f\in\mathcal{R}[a,b]$, then $g$ is absolutely continuous
In Remark 18.13 of Real Analysis: Foundations and Functions of One Variable, the author states that
The class of Riemann integrable functions and the class of absolutely continuous
functions are also ...
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A question on integrating a function, with respect to another function?
I've been asked to find the area described by
$$A={ (x,y), y^2 \leqslant2x; and, y\geqslant 4x-1 } $$
which is effectively the following area(If I can insert a graph using mathJax, please let me ...
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How does this theorem on continued fractions relate to the Riemann-Stieltjes integral?
I'm going to include two precursor sections here to introduce all of the content referenced in my actual question (if you're familiar with both continued fractions and the Riemann-Stieltjes integral, ...
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What is the „limit type” in the definition of Stieltjes-integral?
Let's say $f$ and $g$ are „nice enough” to $$\sum_{k}f\left(\xi_{k}\right)\left(g\left(x_{k+1}\right)-g\left(x_{k}\right)\right)\rightarrow\int_{a}^{b}f\left(x\right)dg\left(x\right),$$ where $\xi_{k}\...
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Let $\alpha$ be an increasing function on $[a,b]$. Show that $\int^a_b\alpha d \alpha = \frac{1}{2}[\alpha (b)^2 - \alpha(a)^2]$
I am wanting to try to prove the question below, but there is a step that I can't get pass. I know that the proof is worthless if I assume incorrectly, and should have stopped proving from there, but ...
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Is a real random variable's moment-generating function the Laplace-Stieltjes transform of its CDF?
Wikipedia gives that the moment-generating function for a real random variable $X$ with cumulative distribution function $F$ equals
$M_x(t) = \int \limits_{-\infty}^\infty e^{tx}\, dF(x)$, using the ...
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A special case of the Riemann–Stieltjes integral
A special case of the Riemann–Stieltjes integral is found here. The claim is that if $g(x)$ is continuously differentiable over $\mathbb{R}$, then
$$\int_{a}^b f(x)\ \mathrm{d}g(x)=\int_a^bf(x)g^{\...
user1036107
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Is $M_t:=\exp\left(-\int_0^tc(Y_s)\:{\rm d}s\right)$ differentiable at $0$?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$E$ be a topological space
$(Y_t)_{t\ge0}$ be an $E$-valued right-continuous process on $(\Omega,\mathcal A,\operatorname P)$
$c:E\to[...
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If $g$ is bounded, nondecreasing and right-continuous, how do we show $\int f(t)\:{\rm d}g(t)=\int f(t)g'(t)\:{\rm d}t$?
Let $g:\mathbb R\to\mathbb R$ be bounded, nondecreasing and right-continuous and $\mu_g$ denote the Lebesgue-Stieltjes measure on $\mathcal B([0,\infty))$ associated with $g$.
Moreover, let $\lambda$ ...
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Convergence of Riemann-Stieltjes integral of square function.
I got this question which I think I know the solution to but I am not certain I am doing it right, will gladly use some help about it.
So the question is like that, assume we have function $\alpha$ ...
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How can we prove $\int_a^bf\:{\rm d}g=\int_a^bf(s)g'(s)\:{\rm d}s$ if $g'$ is not continuous?
Let $a,b\in\mathbb R$ with $a<b$, $$\mathcal D_{[a,\:b]}:=\{(t_0,\ldots,t_k):k\in\mathbb N\text{ and }a=t_0<\cdots<t_k\}$$ and $$\mathcal T_\varsigma:=\{(\tau_1,\ldots,\tau_k):\tau_i\in[t_{i-...
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Lebesgue-Stieltjes Integration by parts formula
Let $A,B$ be two right-continuous functions of finite variation. Then the integration by parts formula states:
$$
A_{t}B_{t} = A_0B_0+\int_{0}^{t} A_{-s}dB_s \, + \int_{0}^{t} B_{-s}dA_s + \sum_{ s\...
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Sum of a locally BV right-continuous function and a locally BV left-continuous function
Let $f:\mathbb{R}\to\mathbb{R}$ be a right-continuous function that is locally of bounded variation (i.e., of bounded variation on every $[a,b]\subset\mathbb{R}$; henceforth, "LBV function"),...
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Lebesgue-Stieltjes integral and Dynkin $\pi-\lambda$ theorem
I am studying the Lebesgue-Stieltjes integral from this PDF: https://www.math.utah.edu/~li/L-S%20integral.pdf.
In Theorem 8 the authors claim to use Dynkin's theorem in a way that I do not understand. ...
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Fundamental Theorem of Calculus for Lebesgue-Stieltjes integrals?
I am wondering whether the following result is true:
Let $F: [0, m] \rightarrow [0, 1]$ be a non-decreasing right-continuous function and $q: [0, m] \rightarrow \mathbb{R}_+$ be measuable. Suppose ...
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If $M,N$ are martingales, show that $\operatorname E\left[M_tN_t\right]=\operatorname E\left[\sum_{s\in(0,\:t]}\Delta M_s\Delta N_s\right]$
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ and $M,N\in\mathcal V$ (see definition below$^2$) be càdlàg $\...
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Total variation of Lebesgue-Stieltjes integral
Setting We work on a filtered probability space. Let $A$ be a process of finite variation (FV) started from $0$ and $H$ be predictable, positive and such that $H\cdot A$ is defined $\omega$-wise as a ...
user711386
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Lebesgue–Stieltjes integration with discontinuous function
My question comes from the book Credit risk modeling: Valuation and hedging by Tomasz R. Bielecki and Marek Rutkowski, Page 137 at the bottom. It is easily described as follows:
Consider a probability ...
- 121
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How do I integrate a not necessarily differentiable multivariate (Riemann)-Stieltjes integral?
I am having a bit of trouble grasping the way of calculating a Stieltjes integral where the differential term might not be at least once differentiable with respect to every variable. I have a good ...
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Regarding Theorem 1.9, Conway, complex Analysis
I have a doubt in equation (1.11) in Chapter 4, Section 1 of J.B Conway’s, functions of one complex variable.
I know that the estimation of the integral in equation 1.10, comes from the previous ...
- 107
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Convert probability into an integral with respect to a distribution function
I'm studying a proof and have problems with a step where a probability is converted into a Lebesgue Stieltjes integral with respect to a distribution function.
Let $(X_i)$ be a sequence of independent,...
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Resources for Riemann-Stieltjes or Lebesgue-Stieltjes integrals
I'm an engineer, who works with retarded delay differential equations. So far I've encountered systems with single or multiple point-wise delays, and systems with distributed delays. Someone on a ...
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Decomposition of propagation function
In the book of Jean Prüss (Evolutionary Integral Equations and Applications, 2012, pg. 111) it says in the Proposition 4.10...
Let $dc$ be completely positive, let $w(t;\tau)$ denote the associated
...
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Let $f(x) = x^2$, and define $\alpha$ as follows, find $100\int_{-1}^{100}f\ d\alpha$.
Making edits to this question. I think I may have figured out how to do the first part of the question
Based on this website (http://www.stat.rice.edu/~dobelman/notes_papers/math/Riemann.Stiltjes.pdf)...
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