arXiv subjects

Michael B. Marcus

Publications and source records attributed to Michael B. Marcus.

23 records · Page 2Linked to original sources

arXiv2009

A CLT for the $L^{2}$ moduli of continuity of local times of Levy processes

Let $X=\{X_{t},t\in R_{+}\}$ be a symmetric Lévy process with local time $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$. When the Lévy exponent $ψ(\la)$ is regularly varying at infinity with index $1<β\leq 2$ and satisfies some additional regularity conditions && \sqrt{hψ^{2}(1/h)} \lc \int (L^{x+h}_{1}- L^{x}_{1})^{2} dx- E(\int (L^{x+h}_{1}- L^{x}_{1})^{2} dx)\rc\nn && {1 in} \stackrel{\mathcal{L}}{\Longrightarrow} (8c_{β,1})^{1/2} η(\int (L_{1}^{x})^{2} dx)^{1/2} \nn, as $h\rar 0$, where $η$ is a normal random variable with mean zero and variance one that is independent of $L^{x}_{t}$, and $c_{β,1}$ is a known constant.

math.PR↗

arXiv2009

A CLT for the L^{2} modulus of continuity of Brownian local time

Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion and \[ α_{t}:=\int_{-\infty}^{\infty} (L^{x}_{t})^{2} dx . \] Let $η=N(0,1)$ be independent of $α_{t}$. For each fixed $t$ \[ {\int_{-\infty}^{\infty} (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\to}({64 \over 3})^{1/2}\sqrt{α_{t}} η, \] as $h\rar 0$. Equivalently \[ {\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- 4t\over t^{3/4}} \stackrel{\mathcal{L}}{\to}({64 \over 3} )^{1/2}\sqrt{α_{1}} η, \] as $t\rar\infty$.

math.PR↗

arXiv2008

Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in $R^{2}$ with non--zero mean

Let $G=(G_{1},G_{2})$ be a Gaussian vector in $R^{2}$ with $EG_{1}G_{2}\neq 0$. Let $c_{1},c_{2}\in R^{1}$. A necessary and sufficient condition for $G=((G_{1}+c_{1}α)^{2},(G_{2}+c_{2}α)^{2})$ to be infinitely divisible for all $α\in R^{1}$ is that \[ \Ga_{i,i}\geq \frac{c_{i}}{c_{j}}\Ga_{i,j}>0\qquad\forall 1\le i\ne j\le 2.\] In this paper we show that when this does not hold there exists an $0<α_{0}<\ff $ such that $G=((G_{1}+c_{1}α)^{2},(G_{2}+c_{2}α)^{2})$ is infinitely divisible for all $|α|\leq α_{0}$ but not for any $|\al|>\al_{0}$.

math.PR↗

arXiv2008

$L^p$ moduli of continuity of Gaussian processes and local times of symmetric Lévy processes

Let $X=\{X(t),t\in R_+\}$ be a real-valued symmetric Lévy process with continuous local times $\{L^x_t,(t,x)\in R_+\times R\}$ and characteristic function $Ee^{iλX(t)}=e^{-tψ(λ)}$. Let \[σ^2_0(x-y)=\frac{4}π\int^{\infty}_0\frac{\sin^2({λ(x- y)}/{2})}{ψ(λ)} dλ.\] If $σ^2_0(h)$ is concave, and satisfies some additional very weak regularity conditions, then for any $p\ge1$, and all $t\in R_+$, \[\lim_{h\downar row0}\int_a^b\biggl|{\frac{L^{x+h}_t-L^x_t}{σ_0(h)}}\biggr|^p dx =2^{p/2}E|η|^p\int_a^b|L^x_t|^{p/2} dx\] for all $a,b$ in the extended real line almost surely, and also in $L^m$, $m\ge1$. (Here $η$ is a normal random variable with mean zero and variance one.) This result is obtained via the Eisenbaum Isomorphism Theorem and depends on the related result for Gaussian processes with stationary increments, $\{G(x),x\in R^1\}$, for which $E(G(x)-G(y))^2=σ_0^2(x-y)$; \[\lim_{h\to0}\int_a^b\biggl|\frac{G (x+h)-G(x)}{σ_0(h)}\biggr|^p dx=E|η|^p(b-a)\] for all $a,b\in R^1$, almost surely.

math.PR↗

arXiv2006

CLT for L^{p} moduli of continuity of Gaussian processes

Let G=\{G(x),x\in R^1\} be a mean zero Gaussian processes with stationary increments and set \si ^2(|x-y|)= E(G(x)-G(y))^2. Let f be a symmetric function with Ef(η)<\ff, where η=N(0,1). When \si^2(s) is concave or when \si^2(s)=s^r$, $1<r\leq 3/2 we have \lim_{h\downarrow 0}{\int_a^b f(\frac{G(x+h)-G(x)}{\si (h)}) dx - (b-a)Ef(η)\over \sqrt{Φ(h,\si(h),f,a,b)}}= N(0,1) in law where Φ(h,\si(h),f,a,b) is the variance of the numerator. This result continues to hold when \si^2(s)=s^r, 3/2<r<2, for certain functions f, depending on the nature of the coefficients in their Hermite polynomial expansion. The asymptotic behavior of Φ(h,\si(h),f,a,b) at zero, is described in a very large number of cases.

math.PR↗