arXiv subjects

Kohei Uchiyama

Publications and source records attributed to Kohei Uchiyama.

18 recordsLinked to original sources

arXiv2021

The two-sided exit problem for a random walk on $\mathbb{Z}$ with infinite variance I

Let $S=(S_n)$ be an oscillatory random walk on the integer lattice $\mathbb{Z}$ with i.i.d. increments. Let $V_{\rm d}(x)$ be the renewal function of the strictly descending ladder height process for $S$. We obtain several sufficient conditions -- given in terms of the distribution function of the increment $S_1-S_0$ -- so that as $R\to\infty$ $$ (*) \quad P [ S\; \mbox{leaves $[0,R]$ on its upper side}\, |\, S_0=x] \, \sim\, V_{\rm d}(x)/V_{\rm d}(R)$$ uniformly for $0\leq x\leq R$. When $S$ is attracted to a stable process of index $0<α\leq 2$ and there exists $ρ= \lim P[S_n>0]$, the sufficient condition obtained are also necessary for $(*)$ and fulfilled if and only if $(α\vee 1)ρ=1$, and some asymptotic estimates of the probability on the left side of $(*)$ are given in case $(α\vee 1)ρ\neq 1$.

math.PR↗

arXiv2021

The two-sided exit problem for a random walk on $\mathbb{Z}$ and having infinite variance II

Let $F$ be a distribution function on the integer lattice $\mathbb{Z}$ and $S=(S_n)$ the random walk with step distribution $F$. Suppose $S$ is oscillatory and denote by $U_{\rm a}(x)$ and $u_{\rm a}(x)$ the renewal function and sequence, respectively, of the strictly ascending ladder height process associated with $S$. Putting $A(x) =\int_0^x [1-F(t)-F(-t)] dt$, $H(x)=1-F(x)+F(-x)$ we suppose $$A(x)/\big(xH(x)\big) \to -\infty \quad (x\to\infty).$$ Under some additional regularity condition on the positive tail of $F$, we show that $$u_{\rm a}(x) \sim U_{\rm a}(x)[1-F(x)]/|A(x)|$$ as $x\to\infty$ and uniformly for $0\leq x\leq R\in \mathbb{Z}$, as $R\to\infty$ $$P [ S\; \mbox{leaves $[0,R]$ on its upper side}\, |\, S_0=x] \, \sim\, c^{-1}A(x)u_{\rm a}(x), $$ where $c= \sum_{n=1}^\infty P[S_{n}>S_0;\, S_k < S_0$ for $0<k<n]$ and the regularity condition is satisfied at least if $S$ is recurrent, $\limsup [1-F(x)]/F(-x)<1$, and $x[1-F(x)]/L(x) $ ($x\geq 1$) is bounded away from zero and infinity for some slowly varying function $L$. We also give some asymptotic estimates of the probability that $S$ visits $R$ before entering the negative half-line for asymptotically stable walks and obtain asymptotic behaviour of the probability that $R$ is ever hit by $S$ conditioned to avoid the negative half-line forever.

math.PR↗

arXiv2020

The potential function and ladder variables of a recurrent random walk on $\mathbb{Z}$ with infinite variance

We consider a recurrent random walk of i.i.d. increments on the one-dimensional integer lattice and obtain a formula relating the hitting distribution of a half-line with the potential function, $a(x)$, of the random walk. Applying it, we derive an asymptotic estimate of $a(x)$ and thereby a criterion for $a(x)$ to be bounded on a half-line. The application is also made to estimate some hitting probabilities as well as to derive asymptotic behaviour for large times of the walk conditioned never to visit the origin.

math.PR↗

arXiv2020

Estimates of Potential functions of random walks on $Z$ with zero mean and infinite variance and their applications

Let $S_n =X_1+\cdots +X_n$ be an irreducible random walk (r.w.) on the one dimensional integer lattice with zero mean, infinite variance and i.i.d. increments $X_n$. We obtain an upper and lower bounds of the potential function, $a(x)$, of $S_n$ in the form $a(x)\asymp x/m(x)$ under a reasonable condition on the distribution of $X_n$; we especially show that as $x\to\infty$ $$a(x) \asymp \frac{x}{m_-(x)} \quad\mbox{and}\quad \frac{a(-x)}{a(x)} \to 0 \quad\;\;\mbox{if}\quad \lim_{x\to +\infty} \frac{m_+(x)}{m_-(x)} =0,$$ where $m_\pm(x) = \int_0^xdy\int_y^\infty P[\pm X_1>u]du$ and $m=m_++m_-$. Under certain conditions on the tails of the distribution of $X$ we derive precise asymptotic forms of $a(x)$ as $x\to +\infty$ or/and $-\infty$. The results are applied to derive a sufficient condition for the relative stability of the ladder height and estimates of some escape probabilities from the origin; we show among others that under the above condition on $m_+/m-$, $P[S_n>0] \to 1/α$ if and only if the probability of exiting a long interval $[-Q,R]$ through the upper boundary converges to $λ^{α-1}$ as $Q/(Q+R) \to λ$ for any $0<λ<1$.

math.PR↗

arXiv2020

A renewal theorem for relatively stable variables

Let $F\{dx\}$ be a relatively stable probability distribution on the whole real line and $S_n$ the random walk started at the origin with step distribution $F$. We obtain an exact asymptotic form of the Green measure $U\{x+dy\}= \sum_{n=0}^\infty P[S_n-x \in dy]$ as $x\to \infty$ when $S_n$ is transient and $S_n\to \infty$ in probability. If $F$ is concentrated on $[0,\infty)$, it is relatively stable if and only if $\ell(x) :=\int_0^x F\{(t,\infty)\}dt$ is slowly varying at infinity; our result entails that if $F$ is non-arithmetic and relatively stable, then $\lim_{x\to\infty}\, \ell(x)U\{[x, x+h)\} = h$ for each $h>0$. This surpasses the known result due to Erickson \cite{Ec}, the latter assuming the stronger condition that $xF\{(x,\infty)\}$ is slowly varying. An obvious analog also holds for arithmetic variables.

math.PR↗

arXiv2019

Scaling limits of random walk bridges conditioned to avoid a finite set

This paper concerns a scaling limit of a one-dimensional random walk $S^x_n$ started from $x$ on the integer lattice conditioned to avoid a non-empty finite set $A$, the random walk being assumed to be irreducible and have zero mean. Suppose the variance $σ^2$ of the increment law is finite. Given positive constants $b$, $c$ and $T$ we consider the scaled process $S^{b_N}_{[tN]}/σ\sqrt N$, $0\leq t \leq T$ started from a point $b_N \approx b\sqrt N$ conditioned to arrive at another point $\approx -c\sqrt N$ at $t=T$ and avoid $A$ in between and discuss the functional limit of it as $N\to\infty$. We show that it converges in law to a continuous process if $E[|S_1|^3; S_1<0] <\infty$. If $E[|S_1|^3; S_1<0] =\infty$ we suppose $P[S_1<u]$ to vary regularly as $u\to -\infty$ with exponent $-β$, $2\leq β\leq 3$ and show that it converges to a process which has one downward jump that clears the origin if $β<3$; in case $β=3$ there arises the same limit process as in case $E[|S_1|^3; S_1<0] <\infty$. In case $σ^2=\infty$ we consider the special case when $S_1$ belongs to the domain of attraction of a stable law of index $1<α<2$ having no negative jumps and obtain analogous results.

math.PR↗

arXiv2019

Asymptotically stable random walks of index $1<α<2$ killed on a finite set

For a random walk on the integer lattice $\mathbb{Z}$ that is attracted to a strictly stable process with index $α\in (1, 2)$ we obtain the asymptotic form of the transition probability for the walk killed when it hits a finite set. The asymptotic forms obtained are valid uniformly in a natural range of the space and time variables. The situation is relatively simple when the limit stable process has jumps in both positive and negative directions; in the other case when the jumps are one sided rather interesting matters are involved and detailed analyses are necessitated.

math.PR↗

arXiv2018

Recurrent random walks on $\mathbb{Z}$ with infinite variance: transition probabilities of them killed on a finite set

In this paper we consider an irreducible random walk on the integer lattice $\mathbb{Z}$ that is in the domain of normal attraction of a strictly stable process with index $α\in (1, 2)$ and obtain the asymptotic form of the distribution of the hitting time of the origin and that of the transition probability for the walk killed when it hits a finite set. The asymptotic forms obtained are valid uniformly in the natural domain of the space and time variables.

math.PR↗

arXiv2017

Brownian hitting distributions in space-time of bounded sets and the expected volume of Wiener sausage for Brownian bridges

The space-time distribution, $Q_A(x,dt dξ)$ say, of Brownian hitting of a bounded Borel set $A$ of the $d$-dimensional Euclidian space is studied. We derive the asymptotic form of the leading term of the time-derivative $Q_A(x, dtdξ)/dt$ for each $d =2, 3, ...$, valid uniformly with respect to the starting point $x$ of the Brownian motion, which result extends significantly the classical results for $Q_A(x, dt dξ)$ itself by Hunt ($d=2$), Joffe and Spitzer ($d= 3, 4,...$). The results are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joining the origin to a distant point.

math.PR↗

arXiv2017

Boundary behaviour of RW's on planar graphs and convergence of LERW to chordal SLE$_2$

This paper concerns a random walk on a planar graph and presents certain estimates concerning the harmonic measures for the walk in a grid domain which estimates are useful for showing the convergence of a LERW (loop-erased random walk) to an SLE (stochastic Loewner evolution). We assume that the walk started at a fixed vertex of the graph satisfies the invariance principle as in Yadin and Yehudayoff [16] in which the convergence of LERW to a radial SLE is established in this setting. Our main concern is chordal case, where a random walk is started at a boundary vertex of a simply connected grid domain and conditioned to exit it through another boundary vertex specified in advance. The primary contribution of the present paper is an estimate, which states that the excursion of the conditioned walk leaves an intrinsic neighborhood of its initial point not 'along' the boundary but through an intrinsic interior of the domain with high probability. Based on this result we give a proof for the convergence to the chordal SLE, a result that has recently been proved by Suzuki [12] under an analyticity assumption on the boundary of the domain arising in the limit.

math.PR↗

arXiv2017

The transition density of Brownian motion killed on a bounded set

We study the transition density of a standard two-dimensional Brownian motion killed when hitting a bounded Borel set $A$. We derive the asymptotic form of the density, say $p^A_t({\bf x},{\bf y})$, for large times $t$ and for ${\bf x}$ and ${\bf y}$ in the exterior of $A$ valid uniformly under the constraint $|{\bf x}|\vee |{\bf y}| =O(t)$. Within the parabolic regime $|{\bf x}|\vee |{\bf y}| = O(\sqrt t)$ in particular $p^A_t({\bf x},{\bf y})$ is shown to behave like $4e_A({\bf x})e_A({\bf y}) (\lg t)^{-2} p_t({\bf y}-{\bf x})$ for large $t$, where $p_t({\bf y}-{\bf x})$ is the transition kernel of the Brownian motion (without killing) and $e_A$ is the Green function for the \lq exterior of $A$' with a pole at infinity normalized so that $e_A({\bf x}) \sim \lg |{\bf x}|$. We also provide fairly accurate upper and lower bounds of $p^A_t({\bf x},{\bf y})$ for the case $|{\bf x}|\vee |{\bf y}|>t$ as well as corresponding results for the higher dimensions.

math.PR↗

arXiv2017

One dimensional random walk killed on a finite set

We study the transition probability, say $p_A^n(x,y)$, of a one-dimensional random walk on the integer lattice killed when entering into a non-empty finite set $A$. The random walk is assumed to be irreducible and have zero mean and a finite variance $σ^2$. We derive the asymptotic form of $p_A^n(x, y)$ for large $n$ valid uniformly in the regime characterized by the conditions $|x|\vee |y| =O(\sqrt n)$ and $|x|\wedge |y|= o(\sqrt n)$, in which $p^A_t({\bf x},{\bf y})$ behaves for large $n$ like $[g_A^{+}(x)\hat g_{A}^{\,+}(y) + g_A^-(x)\hat g_{A}^{\,-}(y)] (σ^{2}/2n) p^n(y-x)$. Here $p^n(y-x)$ is the transition kernel of the random walk (without killing); $g^\pm_A$ are the Green functions for the "exterior" of $A$ with "pole at $\pm \infty$" normalized so that $g^\pm_A(x) \sim 2|x|/σ^2$ as $x \to \pm\infty$; and $\hat g_A^{\, \pm}$ are the corresponding Green functions for the time-reversed walk.

math.PR↗

arXiv2016

Density of space-time distribution of Brownian first hitting of a disc and a ball

We compute the joint distribution of the site and the time at which a $d$-dimensional standard Brownian motion $B_t$ hits the surface of the ball $ U(a) =\{|{\bf x}|<a\}$ for the first time. The asymptotic form of its density is obtained when either the hitting time or the starting site $B_0$ becomes large. Our results entail that if Brownian motion is started at ${\bf x}$ and conditioned to hit $U(a)$ at time $t$ for the first time, the distribution of the hitting site approaches the uniform distribution or the point mass at $a{\bf x}/|{\bf x}|$ according as $|{\bf x}|/t$ tends to zero or infinity; in each case we provide a precise asymptotic estimate of the density. In the case when $|{\bf x}|/t$ tends to a positive constant we show the convergence of the density and derive an analytic expression of the limit density.

math.PR↗

arXiv2016

Asymptotic behaviour of a random walk killed on a finite set

We study asymptotic behavior, for large time $n$, of the transition probability of a two-dimensional random walk killed when entering into a non-empty finite subset $A$. We show that it behaves like $4 \tilde u_A(x) \tilde u_{-A}(-y) (\lg n)^{-2} p^n(y- x)$ for large $n$, uniformly in the parabolic regime $|x|\vee |y| =O(\sqrt n)$, where $p^n(y-x)$ is the transition kernel of the random walk (without killing) and $\tilde u_A$ is the unique harmonic function in the 'exterior of $A$' satisfying the boundary condition $\tilde u_A(x) \sim \lg |x|$ at infinity.

math.PR↗

arXiv2015

Asymptotics of the densities of the first passage time distributions for Bessel diffusions

This paper concerns the first passage times of Bessel processes to a point on the positive real line. We are interested in the case when the process starts at a position on its right and compute the densities of the distributions of the passage time to obtain the asymptotic forms of them as time tends to infinity that are valid uniformly for the starting position.

math.PR↗

arXiv2012

The expected area of the Wiener sausage swept by a disc

The expected areas of the Wiener sausages swept by a disc attached to the two-dimensional Brownian Bridge joining the origin to a point x over a time interval [0,t] are computed. It is proved that the leading term of the expectation is given by Ramanujan's function if (x,t) remains in a parabolic region. The corresponding result for unconditioned process is also obtained.

math.PR↗

arXiv2011

One dimensional lattice random walks with absorption at a point / on a half line

This paper concerns a random walk that moves on the integer lattice and has zero mean and a finite variance. We obtain first an asymptotic estimate of the transition probability of the walk absorbed at the origin, and then, using the obtained estimate, that of the walk absorbed on a half line. The latter is used to evaluate the space-time distribution for the first entrance of the walk into the half line.

math.PR↗

arXiv2010

Asymptotic estimates of the distribution of Brownian hitting time of a disc

The distribution of the first hitting time of a disc for the standard two dimensional Brownian motion is computed. By investigating the inversion integral of its Laplace transform we give fairy detailed asymptotic estimates of its density valid uniformly with respect to the point where the Brownian motion starts from.

math.PR↗