Kesten's bound for sub-exponential densities on the real line and its multi-dimensional analogues

TL;DR

This paper investigates the tail behavior of sub-exponential densities on the real line and their multi-dimensional analogues, establishing bounds for convolutions and applications to nonlocal heat equations.

Contribution

It proves Kesten's bound for sub-exponential densities and extends it to multi-dimensional functions, providing new tools for analyzing nonlocal PDEs.

Findings

n-fold convolution asymptotically equals the density times n

Kesten's bound provides uniform estimates for convolutions

applications to nonlocal heat-equation solutions

Abstract

We study the tail asymptotic of sub-exponential probability densities on the real line. Namely, we show that the n-fold convolution of a sub-exponential probability density on the real line is asymptotically equivalent to this density times n. We prove Kesten's bound, which gives a uniform in n estimate of the n-fold convolution by the tail of the density. We also introduce a class of regular sub-exponential functions and use it to find an analogue of Kesten's bound for functions on $\mathbb{R}^d$. The results are applied for the study of the fundamental solution to a nonlocal heat-equation.

Equations

$$\overline{F_1\ast F_2}(s)={\int }_{\mathbb{R}}{\overline{F}}_1(s-\tau )F_2(d\tau )={\int }_{\mathbb{R}}{\overline{F}}_2(s-\tau )F_1(d\tau ),$$

$$\underset{s\to \infty }{\mathrm{liminf}}\frac{\overline{F\ast F}(s)}{\overline{F}(s)}\ge 2.$$

$$\underset{s\to \infty }{\lim }\frac{\overline{F\ast F}(s)}{\overline{F}(s)}=2.$$

$$\underset{s\to \infty }{\lim }\frac{\overline{F}(s+t)}{\overline{F}(s)}=1\text{for each }t>0.$$

$$(f_1\ast f_2)(s):={\int }_{\mathbb{R}}{f}_1(s-t)f_2(t)dt,s\in \mathbb{R}.$$

$$\underset{s\to \infty }{\lim }\frac{(f\ast f)(s)}{f(s)}=2.$$

$$\underset{s\to \infty }{\lim }\frac{f^{\ast n}(s)}{f(s)}=n,$$

$$f^{\ast n}(s)\le c_{\delta }(1+\delta {)}^{n}f(s),s>s_{\delta },n\in \mathbb{N}.$$

$$f(s+\tau )\le Kf(s),s>\rho ,\tau >0$$

$$a^{\ast n}(x)\le c_{\alpha }(1+\delta {)}^{n}a(x{)}^{\alpha },|x|>s_{\alpha },n\in \mathbb{N}$$

$$a(x)\le p(|x|),x\in {\mathbb{R}}^d,\log p(s)\sim \log b(s),s\to \infty ,$$

$$\underset{s\to \infty }{\lim }\frac{b(s+\tau )}{b(s)}=1.$$

$$\lim_{s\to\infty}\sup_{|\tau|\leq h}\biggl{\lvert}\frac{b(s+\tau)}{b(s)}-1\biggr{\rvert}=0.$$

$$\lim_{s\to\infty}\sup_{|\tau|\leq h(s)}\biggl{\lvert}\frac{b(s+\tau)}{b(s)}-1\biggr{\rvert}=0.$$

$$\displaystyle B(s_{1})=B\bigl{(}\lambda s_{2}+(1-\lambda)(s_{1}+\tau)\bigr{)}\leq\lambda B(s_{2})+(1-\lambda)B(s_{1}+\tau),$$

$$\displaystyle B(s_{2}+\tau)=B\bigl{(}(1-\lambda)s_{2}+\lambda(s_{1}+\tau)\bigr{)}\leq(1-\lambda)B(s_{2})+\lambda B(s_{1}+\tau),$$

$$b_{+}(s):=1\!\!1_{\mathbb{R}_{+}}(s)\biggl{(}\int_{\mathbb{R}_{+}}b(\tau)d\tau\biggr{)}^{-1}b(s),\quad s\in\mathbb{R},$$

$$(b_{+}\ast b_{+})(s)={\int }_{\mathbb{R}}{b}_{+}(s-\tau )b_{+}(\tau )d\tau ={\int }_0^s{b}_{+}(s-\tau )b_{+}(\tau )d\tau \sim 2b_{+}(s).$$

$$\lim_{s\to\infty}s\,b\bigl{(}h(s)\bigr{)}=0.$$

$$\int_{0}^{s}b(s-\tau)b(\tau)\,d\tau\sim 2\biggl{(}\int_{\mathbb{R}_{+}}b(\tau)d\tau\biggr{)}b(s),\quad s\to\infty.$$

$$(b*b)(s)=\int_{\mathbb{R}}b(s-\tau)b(\tau)\,d\tau\sim 2\biggl{(}\int_{\mathbb{R}}b(\tau)d\tau\biggr{)}b(s),\quad s\to\infty.$$

$$b(s+\tau )\le Kb(s),s>\rho ,\tau >0.$$

$$\underset{s\to \infty }{\lim }\frac{b(s\pm h(s))}{b(s)}=1.$$

$$\underset{s\to \infty }{\lim }\frac{b_j(s)}{b(s)}=c_j,j=1,2.$$

$$\underset{s\to \infty }{\lim }\frac{(b_1\ast b_2)(s)}{b(s)}=c_1{\int }_{\mathbb{R}}{b}_2(\tau )d\tau +c_2{\int }_{\mathbb{R}}{b}_1(\tau )d\tau .$$

$${\int }_{h(s)}^{s-h(s)}{b}_{+}(s-\tau )b_{+}(\tau )d\tau =o(b_{+}(s)),s\to \infty ,$$

$$(b_1\ast b_2)(s)$$

$$\displaystyle\bigl{\lvert}b_{j}(s)-c_{j}b(s)\bigr{\rvert}+\sup_{|\tau|\leq h(s)}\bigl{\lvert}b(s+\tau)-b(s)\bigr{\rvert}+\int_{h(s)}^{s-h(s)}b(s-\tau)b(\tau)\,d\tau\leq\delta b(s),$$

$${\int }_{-\infty }^{-h(s)}{b}_j(\tau )d\tau +{\int }_{h(s)}^{\infty }{b}_j(\tau )d\tau \le \delta ,j=1,2.$$

$$\displaystyle\quad\int_{-\infty}^{-h(s)}\bigl{(}b_{1}(s-\tau)b_{2}(\tau)+b_{1}(\tau)b_{2}(s-\tau)\bigr{)}\,d\tau\leq\delta K(c_{1}+c_{2}+2\delta)b(s).$$

$$\displaystyle\quad\biggl{\lvert}\int_{-h(s)}^{h(s)}b_{1}(s-\tau)b_{2}(\tau)\,d\tau-c_{1}b(s)\int_{-h(s)}^{h(s)}b_{2}(\tau)\,d\tau\biggr{\rvert}\leq\delta B_{2}(1+c_{1})b(s),$$

Full text

\authornames

Dmitri Finkelshtein, Pasha Tkachov

\authorone

[Swansea University]Dmitri Finkelshtein \addressoneDepartment of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K. (d.l.finkelshtein@swansea.ac.uk) \authortwo[Gran Sasso Science Institute]Pasha Tkachov \addresstwoGran Sasso Science Institute, Viale Francesco Crispi, 7, 67100 L’Aquila AQ, Italy (pasha.tkachov@gssi.it)

\ams

60E0545M05, 62E20

1 Introduction

Let $F$ be a probability distribution on $\mathbb{R}$. Denote by \overline{F}(s):=F\bigl{(}(s,\infty)\bigr{)}, $s\in\mathbb{R}$ its tail function. For probability distributions $F_{1}$, $F_{2}$ on $\mathbb{R}$, their convolution ${F}_{1}*{F}_{2}$ has the tail function

[TABLE]

where $\overline{F}_{1},\overline{F}_{2}$ are the corresponding tail functions of $F_{1},F_{2}$.

If a probability distribution $F$ is concentrated on $\mathbb{R}_{+}:=[0,\infty)$ and $\overline{F}(s)>0$, $s\in\mathbb{R}$, then, see e.g. [Chi1964],

[TABLE]

If, additionally, $F$ is heavy-tailed, i.e. $\int_{\mathbb{R}}e^{\lambda s}F(ds)=\infty$ for all $\lambda>0$, then the equality holds in (1.1), see [FoKo2007]. An important sub-class of heavy-tailed distributions concentrated on $\mathbb{R}_{+}$ constitute sub-exponential ones, for which

[TABLE]

Any sub-exponential distribution on $\mathbb{R}_{+}$ is (right-side) long-tailed on $\mathbb{R}$, see e.g. [Chi1964], i.e. (cf. Definition 4 below)

[TABLE]

If distributions $F_{1},F_{2}$ on $\mathbb{R}$ have probability densities $f_{1}\geq 0,f_{2}\geq 0$, with $\int_{\mathbb{R}}f_{1}(s)\,ds=\int_{\mathbb{R}}f_{2}(s)\,ds=1$, then $F_{1}*F_{2}$ has the density

[TABLE]

The density $f$ of a sub-exponential distribution $F$ concentrated on $\mathbb{R}_{+}$ (i.e. $f(s)=0$ for $s<0$) is said to be sub-exponential on $\mathbb{R}_{+}$ if $f$ is long-tailed, i.e. (1.3) holds with $\overline{F}$ replaced by $f$ (see also Definition 4 below), and, cf. (1.2),

[TABLE]

It can be shown (see e.g. [AFK2003, FKZ2013, Klu1989]) that, in this case, for any $n\in\mathbb{N}$,

[TABLE]

where $f^{*n}:=f*\ldots*f$ ($n-1$ times). Note that, in general, the density of a sub-exponential distribution concentrated on $\mathbb{R}_{+}$ even being long-tailed does not need to be a sub-exponential one; the corresponding characterisation can be found in [AFK2003, FKZ2013], see (2.13) below. The property (1.5) implies, in particular, that, for each $\delta>0$, $n\in\mathbb{N}$, there exists $s_{n}>0$, such that $f^{*n}(s)\leq(n+\delta)f(s)$ for $s>s_{n}$. In many situations, it is important to have similar inequalities ‘uniformly’ in $n$, i.e. on a set independent of $n$. A possible solution is given by the so-called Kesten’s bound, see [Klu1989, AFK2003]: for a bounded sub-exponential density $f$ on $\mathbb{R}_{+}$ and for any $\delta>0$, there exist $c_{\delta},s_{\delta}>0$, such that

[TABLE]

For the corresponding results for distributions, see [Chi1964, CNW1973, AN1972, FKZ2013]. Kesten’s bounds were used to study series of convolutions of distributions on $\mathbb{R}_{+}$, $\sum_{n=1}^{\infty}\lambda_{n}F^{*n}$, and of the corresponding densities, $\sum_{n=1}^{\infty}\lambda_{n}f^{*n}$, appeared in different contexts: starting from the renewal theory (that was the motivation for the original paper [Chi1964]) to branching age dependent processes, random walks, queue theory, risk theory and ruin probabilities, compound Poisson processes, and the study of infinitely divisible laws, see e.g. [EKM1997, AFK2003, FKZ2013, Wat2008, Klu1989, BB2008] and the references therein.

If $F$ is a probability distribution on the whole $\mathbb{R}$, such that $F^{+}$, given by $F^{+}(B):=F(B\cap\mathbb{R}_{+})$ for all Borel $B\subset\mathbb{R}$, is sub-exponential on $\mathbb{R}_{+}$, then (see e.g. [FKZ2013, Lemma 3.4]) $F$ is long-tailed on $\mathbb{R}$ and (1.2) holds. The distributions on $\mathbb{R}$ and their densities were considered by several authors, see [Sgi1990, Sgi1982, RS1999, Wat2008] and others. The reference [Wat2008], in particular, gives a review of difficulties appeared in the case of the whole $\mathbb{R}$ and closes several gaps in the preceding results. However, even some basic properties of sub-exponential densities on the whole $\mathbb{R}$ remained open.

Namely, in [FKZ2013, Lemma 4.13], it was shown that if an integrable on $\mathbb{R}$ function $f$

• •

is right-side long-tail and, being restricted on $\mathbb{R}_{+}$ and normalized in $L^{1}(\mathbb{R}_{+})$, satisfies (1.4) (we will say then that $f$ is weekly sub-exponential on $\mathbb{R}$, cf. Definition 11 below), and if

• •

the condition

[TABLE]

holds, for some $K,\rho>0$ (in particular, if $f$ decays to [math] at $\infty$, cf. Definition 18),

then (1.4) holds for the original $f$ on $\mathbb{R}$ as well. We generalize this to an analogue of (1.5) with a general $n\in\mathbb{N}$. In particular, we prove in Theorem 22 below that

Theorem 1**.**

Let $f$ be an integrable weakly sub-exponential on $\mathbb{R}$ function, such that (1.7) holds. Then $f$, being normalized in $L^{1}(\mathbb{R})$, satisfies (1.5) for all $n\in\mathbb{N}$.

Moreover, in Theorem 25, we prove that then (1.6) holds as well. Namely, one has the following result.

Theorem 2**.**

Let $f$ be a bounded weakly sub-exponential probability density on $\mathbb{R}$, such that (1.7) holds. Then, for each $\delta>0$, there exist $c_{\delta},s_{\delta}>0$, such that (1.6) holds.

Note that the all ‘classical’ examples of sub-exponential functions satisfy assumptions of Theorems 1 and 2, see Subsection LABEL:subsec:examples.

The multi-dimensional version of the constructions above is much more non-trivial. Currently, there exist at least three different definitions of sub-exponential distributions on ${\mathbb{R}^{d}}$ for $d>1$, see [CR1992, Ome2006, SS2016]. The variety is mainly related to different possibilities to describe the zones in ${\mathbb{R}^{d}}$ where an analogue of the equivalence (1.2) takes place. However, any results about sub-exponential densities in ${\mathbb{R}^{d}}$, $d>1$, seem to be absent at all. Note also that if, e.g. $a$ is radially symmetric, i.e. $a(x)=b(|x|)$, $x\in{\mathbb{R}^{d}}$ (here $|x|$ denotes the Euclidean norm on ${\mathbb{R}^{d}}$) and $b$, being normalized, is a sub-exponential density on $\mathbb{R}_{+}$, then $(a*a)(x):=\int_{\mathbb{R}^{d}}a(x-y)a(y)\,dy=p(|x|)$, $x\in{\mathbb{R}^{d}}$, for some $p:\mathbb{R}_{+}\to\mathbb{R}_{+}$ (i.e. $a*a$ is also radially symmetric), however, asymptotic behaviors of $b$ and $p$ at $\infty$ are hardly to be compared. Leaving this problem as on open, we focus in this paper on an analogue of Kesten’s bound (1.6) in the multi-dimensional case.

To do this, we introduce a special class $\widetilde{\mathcal{S}}_{\mathrm{reg},d}$ of regular sub-exponential functions on $\mathbb{R}_{+}$ (see Definitions 27 and LABEL:def:super-subexp-tilde). Functions from this class are either inverse polynomials (i.e. (LABEL:ass:newA5) holds), or decay at $\infty$ faster than any polynomial (i.e. (LABEL:eq:quicklydecreasing) holds), but slower than any exponential function, with the fastest allowed asymptotic \exp\bigl{(}-s(\log s)^{-q}\bigr{)} with $q>1$, cf. Remark LABEL:rem:exception. Then, in Corollary LABEL:cor:mainRd, we show the following result.

Theorem 3**.**

Let $a=a(x)$ be a probability density on ${\mathbb{R}^{d}}$, such that $a(x)=b(|x|)$, $x\in{\mathbb{R}^{d}}$, for some $b\in\widetilde{\mathcal{S}}_{\mathrm{reg},d}$. Then, for each $\delta>0$ and for each $\alpha<1$ close enough to $1$,

[TABLE]

for some $c_{\alpha}=c_{\alpha}(\delta)>0$ and $s_{\alpha}=s_{\alpha}(\delta)>0$.

Clearly, a(x)=o\bigl{(}a(x)^{\alpha}\bigr{)}, $|x|\to\infty$, for any $\alpha\in(0,1)$, hence the inequality (1.8) is weaker than (1.6) for the case $d=1$.

The results of Corollary LABEL:cor:mainRd is based on more general Theorem LABEL:thm:mainRd, which says that if, for some $b\in\widetilde{\mathcal{S}}_{\mathrm{reg},d}$ and decreasing on $\mathbb{R}_{+}$ function $p$,

[TABLE]

then (1.8) holds with $a(x)$ replaced by $b(|x|)$ in the right hand side.

The paper is organized as follows. In Section 2, we consider properties of general sub-exponential functions on the real line and prove the results which imply Theorems 1 and 2. In Section 3, we define and study properties of regular sub-exponential functions on the real line and consider the corresponding examples. In Section LABEL:sec:KbddRd, we prove Theorem 3 and its generalizations. Finally, in Appendix, we apply the obtained results to the study of a non-local heat equation.

2 Sub-exponential functions and Kesten’s bound on the real line

Definition 4**.**

A function $b:\mathbb{R}\to\mathbb{R}_{+}$ is said to be (right-side) long-tailed if there exists $\rho\geq 0$, such that $b(s)>0$, $s\geq\rho$; and, for any $\tau\geq 0$,

[TABLE]

Remark 5**.**

By [FKZ2013, formula (2.18)], the convergence in (2.1) is equivalent to the locally uniform in $\tau$ convergence, namely, (2.1) can be replaced by the assumption that, for all $h>0$,

[TABLE]

A long-tailed function has to have a ‘heavier’ tail than any exponential function; namely, the following statement holds.

Lemma 6** (​​​[FKZ2013, Lemma 2.17]).**

Let $b:\mathbb{R}\to\mathbb{R}_{+}$ be a long-tailed function. Then, for any $k>0$, $\lim\limits_{s\to\infty}e^{ks}b(s)=\infty$.

The constant $h$ in (2.2) may be arbitrary big. It is quite natural to ask what will be if $h$ increases to $\infty$ consistently with $s$.

Lemma 7** (Cf. [FKZ2013, Lemma 2.19, Proposition 2.20]).**

Let $b:\mathbb{R}\to\mathbb{R}_{+}$ be a long-tailed function. Then there exists a function $h:(0,\infty)\to(0,\infty)$, with $h(s)<\frac{s}{2}$ and $\lim\limits_{s\to\infty}h(s)=\infty$, such that, cf. (2.2),

[TABLE]

Following [FKZ2013], we will say then that $b$ is $h$-insensitive. Of course, for a given long-tailed function $b$, the function $h$ that fulfills (2.3) is not unique, see also [FKZ2013, Proposition 2.20].

The convergence in (2.1) will not be, in general, monotone in $s$. To get this monotonicity, we consider the following class of functions.

Definition 8**.**

A function $b:\mathbb{R}\to\mathbb{R}_{+}$ is said to be (right-side) tail-log-convex, if there exists $\rho>0$, such that $b(s)>0$, $s\geq\rho$, and the function $\log b$ is convex on $[\rho,\infty)$.

Remark 9**.**

It is well-known that any function which is convex on an open interval is continuous there. Therefore, a tail-log-convex function $b=\exp(\log b)$ is continuous on $(\rho_{b},\infty)$ as well.

Lemma 10**.**

Let $b:\mathbb{R}\to\mathbb{R}_{+}$ be tail-log-convex, with $\rho=\rho_{b}$. Then, for any $\tau>0$, the function $\frac{b(s+\tau)}{b(s)}$ is non-decreasing in $s\in[\rho,\infty)$.

Proof 2.1**.**

Take any $s_{1}>s_{2}\geq\rho$. Set $B(s):=\log b(s)\leq 0$, $s\in[\rho,\infty)$. Then the desired inequality $\frac{b(s_{1}+\tau)}{b(s_{1})}\geq\frac{b(s_{2}+\tau)}{b(s_{2})}$ is equivalent to $B(s_{1}+\tau)+B(s_{2})\geq B(s_{2}+\tau)+B(s_{1})$. Since $B$ is convex, we have, for $\lambda=\frac{\tau}{s_{1}-s_{2}+\tau}\in(0,1)$,

[TABLE]

that implies the needed inequality.

Because of the terminology mentioned in the introduction, we will use the following definition.

Definition 11**.**

We will say that a function $b:\mathbb{R}\to\mathbb{R}_{+}$ is weakly (right-side) sub-exponential on $\mathbb{R}$ if $b$ is long-tailed, $b\in L^{1}(\mathbb{R}_{+})$, and the function

[TABLE]

satisfies the following asymptotic relation (as $s\to\infty$)

[TABLE]

The next statement shows that a long-tailed tail-log-convex function is weakly sub-exponential on $\mathbb{R}$ provided that it decays at $\infty$ fast enough.

Lemma 12** (cf. [FKZ2013, Theorem 4.15]).**

Let $b:\mathbb{R}\to\mathbb{R}_{+}$ be a long-tailed tail-log-convex function such that $b\in L^{1}(\mathbb{R}_{+})$. Suppose that, for a function $h:(0,\infty)\to(0,\infty)$, with $h(s)<\frac{s}{2}$ and $\lim\limits_{s\to\infty}h(s)=\infty$, the asymptotic (2.3) holds, and

[TABLE]

Then $b$ is weakly sub-exponential on $\mathbb{R}$.

Remark 13**.**

Let $b:\mathbb{R}\to\mathbb{R}_{+}$ be a weakly sub-exponential function on $\mathbb{R}$. Then, by (2.4), (2.5), we have

[TABLE]

Definition 14**.**

We will say that a function $b:\mathbb{R}\to\mathbb{R}_{+}$ is (right-side) sub-exponential on $\mathbb{R}$ if $b$ is long-tailed, $b\in L^{1}(\mathbb{R})$, and the following asymptotic relation holds, cf. (2.5), (2.7),

[TABLE]

Remark 15**.**

By [FKZ2013, Lemma 4.12], a sub-exponential function on $\mathbb{R}$ is weakly sub-exponential there. The following lemma presents a sufficient condition to get the converse.

Lemma 16** (cf. [FKZ2013, Lemma 4.13]).**

Let $b\in L^{1}(\mathbb{R}\to\mathbb{R}_{+})$ be a weakly sub-exponential function on $\mathbb{R}$. Suppose that there exists $\rho=\rho_{b}>0$ and $K=K_{b}>0$ such that

[TABLE]

Then (2.8) holds, i.e. $b$ is sub-exponential on $\mathbb{R}$.

Remark 17**.**

For $b\in L^{1}(\mathbb{R}\to\mathbb{R}_{+})$, condition (2.9) yields that $\sup_{t\geq s}b(t)\to 0$, $s\to\infty$. In particular $b(s)\to 0$, as $s\to\infty$.

An evident sufficient condition which ensures (2.9) is that $b$ is decreasing on $[\rho,\infty)$. Consider the corresponding definition.

Definition 18**.**

A function $b:\mathbb{R}\to\mathbb{R}_{+}$ is said to be (right-side) tail-decreasing if there exists a number $\rho=\rho_{b}\geq 0$ such that $b=b(s)$ is strictly decreasing on $[\rho,\infty)$ to [math]. In particular, $b(s)>0$, $s\geq\rho$.

The proof of the following useful statement is straightforward.

Proposition 19**.**

Let $b:\mathbb{R}\to\mathbb{R}_{+}$ be a tail-decreasing function. Let $h:(0,\infty)\to(0,\infty)$, with $h(s)<\frac{s}{2}$ and $\lim\limits_{s\to\infty}h(s)=\infty$. Then (2.3) holds, if and only if

[TABLE]

The next statement and its proof follow ideas of [FKZ2013, Lemma 4.13, Lemma 4.9].

Proposition 20**.**

Let $b:\mathbb{R}\to\mathbb{R}_{+}$ be a weakly sub-exponential function on $\mathbb{R}$, such that (2.9) holds. Let $b_{1},b_{2}\in L^{1}(\mathbb{R}\to\mathbb{R}_{+})$ and there exist $c_{1},c_{2}\geq 0$, such that

[TABLE]

Then

[TABLE]

Proof 2.2**.**

Since $b_{+}$, given by (2.4), is long-tailed, and (2.5) holds, we have, by [FKZ2013, Theorem 4.7], that there exists an increasing function $h:(0,\infty)\to(0,\infty)$, such that $h(s)<\frac{s}{2}$, $\lim\limits_{s\to\infty}h(s)=\infty$, and

[TABLE]

and, evidently, one can replace $b_{+}$ by $b$ in (2.13).

Next, for any $b_{1},b_{2}\in L^{1}(\mathbb{R}\to\mathbb{R}_{+})$, one can easily get that

[TABLE]

Take an arbitrary $\delta\in(0,1)$. By (2.11), (2.3), and (2.13) (the latter, with $b_{+}$ replaced by $b$), there exist $K,\rho>0$, such that (2.9) holds and, for all $s\geq h(\rho)$,

[TABLE]

Then, by (2.9), (2.14), (2.15), for $s\geq\rho>h(\rho)$,

[TABLE]

Set $B_{j}:=\int_{\mathbb{R}}b_{j}(s)\,ds$, $j=1,2$. By (2.14), for $s\geq\rho$,

[TABLE]

From the obtained estimates, it is straightforward to get that, for some $M>0$, \bigl{\lvert}(b_{1}*b_{2})(s)-(c_{1}B_{2}+c_{2}B_{1})b(s)\bigr{\rvert}\leq\delta Mb(s) for $s\geq\rho$. The latter implies (2.12).

Corollary 21**.**

The property of an integrable on $\mathbb{R}$ function to be weekly sub-exponential on $\mathbb{R}$ depends on its tail property only. Namely, for a weakly sub-exponential on $\mathbb{R}$ function $b\in L^{1}(\mathbb{R}\to\mathbb{R}_{+})$ and for any $c\in L^{1}(\mathbb{R}\to\mathbb{R}_{+})$ and $s_{0}\in\mathbb{R}$, the function $\tilde{b}(s)=1\!\!1_{(-\infty,s_{0})}(s)c(s)+1\!\!1_{[s_{0},\infty)}(s)b(s)$ is weakly sub-exponential on $\mathbb{R}$, cf. also Theorem LABEL:thm:coolthm below.

Now one gets a generalization of Lemma 16.

Theorem 22**.**

Let $b\in L^{1}(\mathbb{R}\to\mathbb{R}_{+})$ be a weakly sub-exponential on $\mathbb{R}$ function, such that (2.9) holds (for example, let $b$ be tail-decreasing). Then

[TABLE]

where $b^{*n}(s)=(\underbrace{b*\ldots*b}_{n})(s)$, $s\in\mathbb{R}$.

Proof 2.3**.**

Take in Proposition 20, $b_{1}=b_{2}=b$, i.e. $c_{1}=c_{2}=1$. Then, for $B:=\int_{\mathbb{R}}b(\tau)\,d\tau$, one gets $b^{*2}(s)\sim 2Bb(s)$, $s\to\infty$. Proving by induction, assume that $b^{*(n-1)}(s)\sim(n-1)B^{n-2}b(s)$, $s\to\infty$, $n\geq 3$. Take in Proposition 20, $b_{1}=b$, $b_{2}=b^{*(n-1)}$, $c_{1}=1$, $c_{2}=(n-1)B^{n-2}$, then, since $\int_{\mathbb{R}}b^{*(n-1)}(\tau)\,d\tau=B^{n-1}$, one gets $\lim\limits_{s\to\infty}\frac{b^{*n}(s)}{b(s)}=B^{n-1}+B(n-1)B^{(n-2)}=nB^{(n-1)}$.

Consider now some general statements in the Euclidean space ${\mathbb{R}^{d}}$, $d\in\mathbb{N}$. Fix the Borel $\sigma$-algebra $\mathcal{B}({\mathbb{R}^{d}})$ there. All functions on ${\mathbb{R}^{d}}$ in the sequel are supposed to be $\mathcal{B}({\mathbb{R}^{d}})$-measurable. Let $0\leq a\in L^{1}({\mathbb{R}^{d}})$ be a fixed probability density on ${\mathbb{R}^{d}}$, i.e.

[TABLE]

Let $f:\mathbb{R}\to\mathbb{R}$; we will say that the convolution

[TABLE]

is well-defined if the function $y\mapsto a(x-y)f(y)$ belongs to $L^{1}({\mathbb{R}^{d}})$ for a.a. $x\in{\mathbb{R}^{d}}$. In particular, this holds if $f\in L^{\infty}({\mathbb{R}^{d}})$. Next, for a function $\phi:{\mathbb{R}^{d}}\to(0,+\infty)$, we define, for any $f:{\mathbb{R}^{d}}\to\mathbb{R}$,

[TABLE]

Proposition 23**.**

Let a function $\phi:{\mathbb{R}^{d}}\to(0,+\infty)$ be such that $a*\phi$ is well-defined, $\lVert a\rVert_{\phi}<\infty$, and, for some $\gamma\in(0,\infty)$,

[TABLE]

Then $a^{*n}(x)\leq\gamma^{n-1}\lVert a\rVert_{\phi}\phi(x)$, $x\in{\mathbb{R}^{d}}$.

Proof 2.4**.**

For any $f:{\mathbb{R}^{d}}\to\mathbb{R}$, with $\lVert f\rVert_{\phi}<\infty$, we have, for $x\in{\mathbb{R}^{d}}$,

[TABLE]

In particular, since $\lVert a\rVert_{\phi}<\infty$, one gets $\lVert a*a\rVert_{\phi}\leq\gamma\lVert a\rVert_{\phi}<\infty$. Proceeding inductively, one gets $\lVert a^{*n}\rVert_{\phi}\leq\gamma\lVert a*a^{n-1}\rVert_{\phi}\leq\gamma^{n-1}\lVert a\rVert_{\phi}<\infty$, that yields the statement.

Proposition 24**.**

Let a function $\upomega:{\mathbb{R}^{d}}\to(0,+\infty)$ be such that, for any $\lambda>0$,

[TABLE]

Suppose further

[TABLE]

Then, for any $\delta\in(0,1)$, there exists $\lambda=\lambda(\delta,\upomega)\in(0,1)$, such that (2.18) holds, with

[TABLE]

and $\gamma:=\max\{1,(1+\delta)\eta\}$.

Proof 2.5**.**

By (2.21), for an arbitrary $\lambda>0$, we have $\upomega_{\lambda}(x)\leq\lambda$, $x\in{\mathbb{R}^{d}}$; then $(a*\upomega_{\lambda})(x)\leq\lambda$, $x\in{\mathbb{R}^{d}}$, as well. In particular, cf. (2.21),

[TABLE]

Next, by (2.20), for any $\delta>0$ there exists $\lambda=\lambda(\delta)\in(0,1)$ such that

[TABLE]

in particular, $(a*\upomega)(x)\leq(1+\delta)\eta\upomega(x)=(1+\delta)\eta\upomega_{\lambda}(x),\quad x\in\Omega_{\lambda}$. Therefore,

[TABLE]

for all $x\in\Omega_{\lambda}$; here we used that $\upomega\geq\upomega_{\lambda}$. Then (2.22)–(2.23) yield the statement.

We are ready to prove now Kesten’s bound on $\mathbb{R}$.

Theorem 25**.**

Let $b\in L^{1}(\mathbb{R}\to\mathbb{R}_{+})$ be a bounded weakly sub-exponential on $\mathbb{R}$ function with $\int_{\mathbb{R}}b(s)\,ds=1$, such that (2.9) holds. Then, for any $\delta\in(0,1)$, there exist $C_{\delta},s_{\delta}>0$, such that

[TABLE]

Proof 2.6**.**

Fix $\delta\in(0,1)$ and \varepsilon\in\bigl{(}0,\delta] with $(1+\varepsilon)^{3}\leq 1+\frac{\delta}{2}$. Let $s_{1},\lambda_{1}>0$ satisfy

[TABLE]

Define the following functions, for $s\in\mathbb{R}$,

[TABLE]

Here and below, $\|\cdot\|_{1}$ denotes the norm in $L^{1}(\mathbb{R})$. Then, by (2.25), (2.26), $\|b-\tilde{b}\|_{1}\leq\varepsilon$, and hence

[TABLE]

By Corollary 21, both functions $s\mapsto\tilde{b}(s)$ and $s\mapsto\tilde{b}(-s)$ are weakly sub-exponential on $\mathbb{R}$. Hence by Theorem 22, $\lim\limits_{s\to\pm\infty}\frac{\tilde{b}^{*m}(s)}{\tilde{b}^{*k}(s)}=\frac{m}{k}\|\tilde{b}\|_{1}^{m-k}$ for any $k,m\in\mathbb{N}$. In particular, there exist $m_{0}\in\mathbb{N}$ and $s_{2}\geq s_{1}$, such that, for $\upomega:=\tilde{b}^{*m_{0}}$ and $|s|\geq s_{2}$,

[TABLE]

For an $r>0$, $\operatorname*{ess\,inf}\limits_{s\in[-r,r]}\tilde{b}(s)>0$. It is straightforward to check then by induction, that

[TABLE]

On the other hand, by Remark 17 and (2.29), we have

[TABLE]

Hence there exists $\lambda_{2}\in(0,\lambda_{1}]$, such that for any $\lambda\in(0,\lambda_{2})$, the set $\Omega_{\lambda}$ defined by (2.19) will be a non-empty subset of $(-\infty,-s_{\lambda})\cup(s_{\lambda},\infty)$, where $s_{\lambda}\to\infty$, as $\lambda\searrow 0$. Therefore, by (2.28), the condition (2.20) holds with $a$ given by (2.26) and $\eta\leq 1+\varepsilon$. Then, by Proposition 24, there exists $\lambda_{3}\in(0,\min\{\lambda_{2},1\})$, such that (2.18) holds with

[TABLE]

By (2.29), (2.30) and since $\tilde{b}$ is bounded, we have $\|\tilde{b}\|_{\phi}<\infty$. Hence, by Proposition 23 and using that $\gamma\|\tilde{b}\|_{1}\leq 1+\frac{\delta}{2}$ because of (2.27) and the choice of $\varepsilon$, one easily gets

[TABLE]

By (2.31), (2.29), $\|\tilde{b}\|_{\phi}\min\{\lambda_{3},\upomega(s)\}\leq C_{\delta}\tilde{b}(s)=C_{\delta}b(s)$ for $s>s_{3}$, with some $s_{3}\geq s_{2}$ and $C_{\delta}=C_{\delta}(m_{0})>0$. As a result,

[TABLE]

By (2.26), $b\leq b_{1}+\tilde{b}$ and hence $b^{*n}\leq\sum_{k=0}^{n}\binom{n}{k}b_{1}^{*k}*\tilde{b}^{*(n-k)}$, $n\in\mathbb{N}$, pointwise. Since $b_{1}^{*k}(s)=0$ for all $s\geq-s_{1}$, $k\in\mathbb{N}$, then, by (2.32), one gets, for $s\geq s_{3}$,

[TABLE]

that yields (2.24).

Remark 26**.**

Following the scheme of the proof for [AFK2003, Proposition 8], it may be shown that Kesten’s bound (2.24) holds under a weaker assummption that $b\in L^{1}(\mathbb{R}\to\mathbb{R}_{+})$ is a bounded on $\mathbb{R}$ function with $\int_{\mathbb{R}}b(s)\,ds=1$ such that (2.16) holds, i.e.

[TABLE]

(recall that, in contrast to (1.5) $b$ is not necessary concentrated on $\mathbb{R}_{+}$). By Theorem 22, a sufficient condition for the latter asymptotic relation is that $b$ is weakly sub-exponential (i.e., cf. (2.4)–(2.5), its normalised restriction $b_{+}$ on $\mathbb{R}_{+}$ is subexponential) and the inequality (2.9) holds.

3 Regular sub-exponential densities on $\mathbb{R}$

We are going to obtain an analogue of Kesten’s bound on ${\mathbb{R}^{d}}$ with $d>1$, at least for radially symmetric functions. Our technique will require to deal with functions $b(|x|)^{\alpha}$, $x\in{\mathbb{R}^{d}}$, where $b$ is a sub-exponential function on $\mathbb{R}_{+}$ and $\alpha<1$ is close enough to $1$; in particular, we have to be sure that $b^{\alpha}$ is also sub-exponential on $\mathbb{R}_{+}$. Moreover, to weaken the condition of radial symmetry, we will allow double-side estimates by functions of the form $p(|x|)b(|x|)$ for appropriate $p$ on $\mathbb{R}_{+}$ (say, polynomial). Again, we will need have to check whether the functions $pb$ is also sub-exponential on $\mathbb{R}_{+}$. To check such a stability of the class of sub-exponential on $\mathbb{R}_{+}$ functions with respect to power and multiplicative perturbations, we have to reduce the class to appropriately regular sub-exponential functions. Then the mentioned stability takes place, see Theorem LABEL:thm:coolthm and Proposition LABEL:prop:tailprop2cool. The examples of regular sub-exponential functions are given in Subsection LABEL:subsec:examples. The analogues of Kesten’s bound on ${\mathbb{R}^{d}}$ are considered in Section LABEL:sec:KbddRd.

3.1 Main properties

Definition 27**.**

Let $\mathcal{S}_{\mathrm{reg}}$