Journal of Theoretical Probability最新文献

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A Robust $$alpha $$-Stable Central Limit Theorem Under Sublinear Expectation without Integrability Condition 次线性期望下无可积条件下的稳健$$alpha $$ -稳定中心极限定理

4区数学 Q3 STATISTICS & PROBABILITY

Journal of Theoretical Probability

Pub Date : 2023-11-03 DOI: 10.1007/s10959-023-01298-x

Lianzi Jiang, Gechun Liang

Abstract This article fills a gap in the literature by relaxing the integrability condition for the robust $$alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -stable central limit theorem under sublinear expectation. Specifically, for $$alpha in (0,1]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , we prove that the normalized sums of i.i.d. non-integrable random variables $$big {n^{-frac{1}{alpha }}sum _{i=1}^{n}Z_{i}big }_{n=1}^{infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>{</mml:mo> </mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>α</mml:mi> </mml:mfrac> </mml:mrow> </mml:msup> <mml:msubsup> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msubsup> <mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> converge in law to $${tilde{zeta }}_{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>ζ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> </mml:math> , where $$({tilde{zeta }}_{t})_{tin [0,1]}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mover> <mml:mi>ζ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> </mml:math> is a multidimensional nonlinear symmetric $$alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -stable process with jump uncertainty set $${mathcal {L}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> . The limiting $$alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -stable process is further characterized by a fully nonlinear partial integro-differential equation (PIDE): $$begin{aligned} left{ begin{array}{l} displaystyle partial _{t}u(t,x)-sup limits _{F_{mu }in {mathcal {L}}}left{ int _{{mathbb {R}}^{d}}delta _{lambda }^{alpha }u(t,x)F_{mu }(dlambda )right} =0, displaystyle u(0,x)=phi (x),quad forall (t,x)in [0,1]times {mathbb {R}}^{d}, end{array} right. end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mfenced> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:ms

摘要本文通过放宽次线性期望下稳健$$alpha $$ α -稳定中心极限定理的可积性条件，填补了文献的空白。具体而言，对于$$alpha in (0,1]$$ α∈(0,1)，证明了i.i.d不可积随机变量$$big {n^{-frac{1}{alpha }}sum _{i=1}^{n}Z_{i}big }_{n=1}^{infty }$$ n- 1 α∑i = 1 n zi n = 1{∞的归一化和规律收敛于}$${tilde{zeta }}_{1}$$ ζ 1，其中$$({tilde{zeta }}_{t})_{tin [0,1]}$$ (ζ t) t∈[0,1]是一个具有跳跃不确定性集$${mathcal {L}}$$ L的多维非线性对称$$alpha $$ α稳定过程。极限$$alpha $$ α稳定过程进一步表征为一个完全非线性的偏积分微分方程(PIDE): $$begin{aligned} left{ begin{array}{l} displaystyle partial _{t}u(t,x)-sup limits _{F_{mu }in {mathcal {L}}}left{ int _{{mathbb {R}}^{d}}delta _{lambda }^{alpha }u(t,x)F_{mu }(dlambda )right} =0, displaystyle u(0,x)=phi (x),quad forall (t,x)in [0,1]times {mathbb {R}}^{d}, end{array} right. end{aligned}$$∂t u (t, x) - sup F μ∈L∫R d δ λ α u (t, x) F μ (d λ) = 0, u (0, x) = ϕ (x)，∀(t, x)∈[0,1]× R d，其中$$begin{aligned} delta _{lambda }^{alpha }u(t,x):=left{ begin{array}{ll} u(t,x+lambda )-u(t,x)-langle D_{x}u(t,x),lambda mathbbm {1}_{{|lambda |le 1}}rangle , &{}quad alpha =1, u(t,x+lambda )-u(t,x), &{}quad alpha in (0,1). end{array} right. end{aligned}$$ δ λ α u (t, x):= u (t, x + λ) - u (t, x) -⟨dx u (t, x)， λ 1 {| λ |≤1}⟩，α = 1, u (t, x + λ) - u (t, x)， α∈(0,1)。本研究中使用的方法涉及到几种工具的利用，包括弱收敛方法来获得极限过程，非线性$$alpha $$ α稳定过程的l - khintchine表示和截断技术来估计相应的$$alpha $$ α稳定l测量。此外，本文还给出了证明上述全非线性PIDE解的存在性的概率方法。

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引用次数: 0

Shannon–McMillan–Breiman Theorem Along Almost Geodesics in Negatively Curved Groups 负弯曲群中沿几乎测地线的Shannon-McMillan-Breiman定理

4区数学 Q3 STATISTICS & PROBABILITY

Journal of Theoretical Probability

Pub Date : 2023-11-02 DOI: 10.1007/s10959-023-01291-4

Amos Nevo, Felix Pogorzelski

引用次数: 0

Laws of Large Numbers for Weighted Sums of Independent Random Variables: A Game of Mass 独立随机变量加权和的大数定律:质量的博弈

4区数学 Q3 STATISTICS & PROBABILITY

Journal of Theoretical Probability

Pub Date : 2023-11-01 DOI: 10.1007/s10959-023-01296-z

Luca Avena, Conrado da Costa

Abstract We consider weighted sums of independent random variables regulated by an increment sequence and provide operative conditions that ensure a strong law of large numbers for such sums in both the centred and non-centred case. The existing criteria for the strong law are either implicit or based on restrictions on the increment sequence. In our setup we allow for an arbitrary sequence of increments, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centred case, convergence can be translated into the behaviour of a deterministic sequence and it becomes a game of mass when the expectation of the random variables is a function of the increment sizes. We identify various classes of increments and illustrate them with a variety of concrete examples.

摘要考虑由增量序列调节的独立随机变量的加权和，并提供了保证这种和在中心和非中心情况下都具有强大数律的操作条件。现有的强律准则要么是隐式的，要么是基于对增量序列的限制。在我们的设置中，我们允许任意序列的增量，可能是随机的，只要这些增量调节的随机变量满足一些温和的浓度条件。在非中心情况下，收敛可以转化为确定性序列的行为，当随机变量的期望是增量大小的函数时，它就变成了质量游戏。我们确定了各种类型的增量，并用各种具体的例子来说明它们。

引用次数: 1

On the Local Time of Anisotropic Random Walk on $$mathbb Z^2$$ 各向异性随机行走的局部时间 $$mathbb Z^2$$

4区数学 Q3 STATISTICS & PROBABILITY

Journal of Theoretical Probability

Pub Date : 2023-10-31 DOI: 10.1007/s10959-023-01297-y

Endre Csáki, Antónia Földes

引用次数: 0

A Theory of Singular Values for Finite Free Probability 有限自由概率的奇异值理论

4区数学 Q3 STATISTICS & PROBABILITY

Journal of Theoretical Probability

Pub Date : 2023-10-29 DOI: 10.1007/s10959-023-01295-0

Aurelien Gribinski

引用次数: 2

Lower Deviation for the Supremum of the Support of Super-Brownian Motion 超布朗运动支持极值的低偏差

4区数学 Q3 STATISTICS & PROBABILITY

Journal of Theoretical Probability

Pub Date : 2023-10-19 DOI: 10.1007/s10959-023-01292-3

Yan-Xia Ren, Renming Song, Rui Zhang

引用次数: 0

Some Properties of Markov chains on the Free Group $${mathbb {F}}_2$$ 自由群上马尔可夫链的一些性质 $${mathbb {F}}_2$$

4区数学 Q3 STATISTICS & PROBABILITY

Journal of Theoretical Probability

Pub Date : 2023-10-18 DOI: 10.1007/s10959-023-01294-1

Antoine Goldsborough, Stefanie Zbinden

引用次数: 1

Cutpoints of (1,2) and (2,1) Random Walks on the Lattice of Positive Half Line (1,2)和(2,1)随机漫步在正半直线格上的截点

4区数学 Q3 STATISTICS & PROBABILITY

Journal of Theoretical Probability

Pub Date : 2023-10-13 DOI: 10.1007/s10959-023-01293-2

Lanlan Tang, Hua-Ming Wang

引用次数: 0

Green Function for an Asymptotically Stable Random Walk in a Half Space 半空间中渐近稳定随机漫步的格林函数

4区数学 Q3 STATISTICS & PROBABILITY

Journal of Theoretical Probability

Pub Date : 2023-09-29 DOI: 10.1007/s10959-023-01283-4

Denis Denisov, Vitali Wachtel

Abstract We consider an asymptotically stable multidimensional random walk $$S(n)=(S_1(n),ldots , S_d(n) )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . For every vector $$x=(x_1ldots ,x_d)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with $$x_1ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , let $$tau _x:=min {n>0: x_{1}+S_1(n)le 0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>min</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:mn>0</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> be the first time the random walk $$x+S(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> leaves the upper half space. We obtain the asymptotics of $$p_n(x,y):= {textbf{P}}(x+S(n) in y+Delta , tau _x>n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mi>y</mml:mi> <mml:mo>+</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml

我们认为不合理的是，我们认为是一种复杂的多多维的稳定步行S(n)=(S_1(n)，ldots, S_d(n) $S(n)为每一个向量$ x = (x_1 ldots, x_d) $ ... 1 x = (x, x, d)和$ x_1 ge 0 $ x 1≥0,则让$知道_x: = min { {n> 0: x_ {1} + S_1 (n)的le 0 $τx: = min {n >0:×1 + S (n)≤0}成为《随机漫步第一次$ x + S (n) $ x + S (n)的树叶上半空间。asymptotics》我们得到$ p_n (x, y): = P { textbf {}} (x + S + y (n) 中三角洲,知道_x> n) $ $ P (x, y): = P (x + y + S (n)∈xΔ,τ>n)美国n tends to无限,在$ $Δ三角洲是一个固定立方体。从这一点,我们得到《绿功能(local asymptotics for $ G (x, y): sum = _n p_n (x, y) $ G (x, y): =∑n p n (x, y),美国$ | | $ | | y和y - x或x $ | | $ | | tend to无限。

引用次数: 0

General Mean Reflected Backward Stochastic Differential Equations 一般均值反映后向随机微分方程

4区数学 Q3 STATISTICS & PROBABILITY

Journal of Theoretical Probability

Pub Date : 2023-09-25 DOI: 10.1007/s10959-023-01288-z

Ying Hu, Remi Moreau, Falei Wang

引用次数: 0

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Journal of Theoretical Probability

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