Maximum on a random time interval of a random walk with infinite mean

TL;DR

This paper investigates the asymptotic behavior of the maximum of a random walk with infinite mean over a random time interval defined by the stopping time when the walk first drops below or equals zero.

Contribution

It provides new asymptotic results for the probability that the maximum exceeds a large threshold in the context of random walks with infinite mean.

Findings

Asymptotic characterization of P(M_τ > x) as x → ∞

Results applicable to heavy-tailed distributions with infinite mean

Insights into the maximum behavior of such random walks

Abstract

Let $\xi_1,\xi_2,\ldots$ be independent, identically distributed random variables with infinite mean $\mathbf E[|\xi_1|]=\infty.$ Consider a random walk $S_n=\xi_1+\cdots+\xi_n$, a stopping time $\tau=\min\{n\ge 1: S_n\le 0\}$ and let $M_\tau=\max_{0\le i\le \tau} S_i$. We study the asymptotics for $\mathbf P(M_\tau>x),$ as $x\to\infty$.

Equations

$$\tau :=\inf \{n\ge 1:S_n<0\}.$$

$$m(x)$$

$$K:={\int }_0^{\infty }\frac{x}{m(x)}F(dx)\text{mbox}isfinite,$$

$$\mathbf{P}(M_{\tau }>x)\sim \mathbf{E}\tau \overline{F}(x),$$

$$\frac{{\int }_0^x\overline{F}(x-y)\overline{F}(y)dy}{\overline{F}(x)}\to 2{\mu }_{+},\text{mbox}asx\to \infty$$

$$\mathbf{P}(\xi >x+y\mid \xi >x)=\frac{\overline{F}(x+y)}{\overline{F}(x)}\to 1,x\to \infty .$$

$$\frac{\mathbf{P}({\xi }_1+{\xi }_2>x)}{\mathbf{P}({\xi }_1>x)}=\frac{{\overline{F}}^{2\ast }(x)}{\overline{F}(x)}\to 2,\text{mbox}asx\to \infty ,$$

$$\underset{x>0}{\sup }\frac{\overline{F}(x/2)}{\overline{F}(x)}<\infty .$$

$$\frac{\overline{F}(\lambda x)}{\overline{F}(x)}\to {\lambda }^{-\alpha },x\to \infty .$$

$${\int }_0^x\pi (du)\overline{F}(x-u)\sim \overline{F}(x).$$

$${\int }_0^{x}G(du)\overline{F}(x-u)\sim \overline{F}(x).$$

$${\overline{G}}_1(x)=\frac{1}{K}{\int }_x^{\infty }\frac{t-x}{m(t-x)}F(dt),x\ge 0.$$

$$\mathbf{P}(M_{\tau }>x)\sim \mathbf{E}\tau \overline{F}(x).$$

$$\mathbf{P}(\xi +\psi >x,\psi \le x)\sim \mathbf{P}(\xi >x).$$

$${\int }_0^{x}G(du)\overline{F}(x-u)$$

$$1\ge \lim \underset{x\to \infty }{\sup }\frac{\overline{F}(x-t)}{\overline{F}(x)}\ge \lim \underset{x\to \infty }{\inf }\frac{\overline{F}(x-t)}{\overline{F}(x)}\ge 1$$

$$\underset{x\to \infty }{\mathrm{limsup}}\frac{G(x-t,x]}{\overline{F}(x)}=0$$

$${\int }_0^{h(x)}G(dy)\overline{F}(x-y)\sim \overline{F}(x){\int }_0^{h(x)}G(dy)\sim \overline{F}(x),$$

$$0\le {\int }_{x-h(x)}^{x}G(dy)\overline{F}(x-y)\le G(x-h(x),x]=o(\overline{F}(x)).$$

$${\int }_0^x{G}_1\ast G_2(du)\overline{F}(x-u)={\int }_0^x{G}_1(du){\int }_0^{x-u}{G}_2(dv)\overline{F}(x-u-v)$$

$$=\left({\int }_0^{x-h(x)}+{\int }_{x-h(x)}^x\right)G_1(du){\int }_0^{x-u}{G}_2(dv)\overline{F}(x-u-v)\equiv I_1(x)+I_2(x).$$

$$I_1(x)$$

$$\sim {\int }_0^{x-h(x)}{G}_1(du)\overline{F}(x-u)\sim \overline{F}(x)$$

$$I_2(x)\le G_1(x-h(x),x]=o(\overline{F}(x)).$$

$${\int }_0^x{G}^{\ast n}(du)\overline{F}(x-u)$$

$${\int }_0^{x}G(du)\overline{F}(x-u)\le (1+\epsilon )\overline{F}(x),x\ge x_0.$$

$$\frac{{\int }_0^x{G}^{\ast n}(du)\overline{F}(x-u)}{\overline{F}(x)}\le \frac{1}{\overline{F}(x)}\le \frac{1}{\overline{F}(x_0)}=A\le A(1+\epsilon {)}^n$$

$${\int }_0^x{G}^{\ast n}(du)\overline{F}(x-u)$$

$$G_{\nu }(x)\equiv \mathbf{P}(X_{\nu }\le x)=\sum _{n\ge 0}\mathbf{P}(\nu =n)G^{\ast n}(x).$$

$${\int }_{h(x)}^{x-h(x)}G(dy)\overline{F}(x-y)\le \sum _{k=[h(x)]-1}^{[x-h(x)]}G(k-1,k]\overline{F}(x-k),$$

$$\sum _{k=[h(x)]-1}^{[x-h(x)]}G(k-1,k]\overline{F}(x-k)\le \sum _{k=[h(x)]-1}^{[x-h(x)]}\overline{F}(k)\overline{F}(x-k-1)=o(\overline{F}(x)),$$

Full text

1. Introduction

Let $\xi,\xi_{1}$, $\xi_{2}$, … be independent random variables with a common distribution $F$. Consider a random walk $S_{0}=0$, $S_{n}=\xi_{1}+\ldots+\xi_{n}$ and a stopping time

[TABLE]

Let $M_{\tau}:=\max_{0\leq i\leq\tau}S_{i}$ and $M=\sup\ \{S_{n},\ n\geq 0\}$. We will consider the random walks with infinite or undefined mean ($\mathbf{E}[|\xi_{1}|]=\infty$) under the assumption that $S_{n}\to-\infty$ a. s. It is well known that the latter assumption is equivalent to $M<\infty$ a. s. and to $\mathbf{E}[\tau]<\infty$ (see Theorem 1 in [17, Chapter XII, Section 2]).

In the infinite-mean case, an important role is played by the negative truncated mean function

[TABLE]

where $\xi^{-}=\max\{-\xi,0\}$; the function $m(x)$ is continuous, increasing, $m(0)=0$ and $m(x)>0$ for any $x>0$. It is known that if $\mathbf{E}|\xi|=\infty$, then $S_{n}\to-\infty$ a.s. as $n\to\infty$ if and only if

[TABLE]

see Corollary 1 in [16].

The aim of this paper is to study the asymptotics for $\mathbf{P}(M_{\tau}>x)$ in the infinite-mean case. In the finite-mean case, under the assumption that $F\in\mathcal{S}^{*}$ it was shown by Asmussen [1], see also [22] for the regularly varying case, that

[TABLE]

where $\overline{F}(x)=1-F(x)=\mathbf{P}(\xi_{1}>x).$ The class $\mathcal{S}^{*}$ of strongly subexponential distributions was introduced by Klüppelberg [23] and is defined as follows,

Definition 1**.**

A distribution function $F$ with finite $\mu_{+}=\int_{0}^{\infty}\overline{F}(y)dy<\infty$ belongs to the class $\mathcal{S}^{*}$ of strong subexponential distributions if $\overline{F}(x)>0$ for all $x$ and

[TABLE]

This class is a proper subclass of class $\mathcal{S}$ of subexponential distributions. It is shown in [23] that the Pareto, lognormal and Weibull distributions belong to the class $\mathcal{S}^{*}$ as well.

The proof in [1] relied on the local asymptotics for $\mathbf{P}(M\in(x,x+T)$ found in [5] and, independently, in [3]. Foss and Zachary [19] pointed out the necessity of the condition $F\in\mathcal{S}^{*}$ and extended (2) to the case of an arbitrary stopping time $\sigma$ with the finite mean $\mathbf{E}\sigma<\infty$. Then, Foss, Palmowsky and Zachary [20] found the asymptotics $\mathbf{P}(M_{\sigma}>x)$, for a more general class of stopping times $\sigma$, including those that may take infinite values or have infinite mean. They also proved that these asymptotics hold uniformly in all stopping times. A short proof of (2) may be found in [9],[10] and [14]. The former proof relies on the local asymptotics for $\mathbf{P}(M_{\tau}\in(x,x+T])$ and the latter proof uses the martingale properties of $\mathbf{P}(M>x)$. The local asymptotics for $\mathbf{P}(M_{\tau}\in(x,x+T])$ were found in [13].

We will now introduce several subclasses of heavy-tailed distributions that will be used in the text.

Definition 2**.**

A distribution function $F$ is (right) long tailed ($F\in\mathcal{L}$) if, for any fixed $y>0$,

[TABLE]

An important subclass of heavy-tailed distributions is a class of subexponential distributions introduced independently by Chistyakov [7] and Chover et al [8].

Definition 3**.**

A distribution function $F$ on $\mathbf{R}^{+}$ is subexponential ($F\in\mathcal{S}$) if $\overline{F}(x)>0$ for all $x$ and

[TABLE]

where $\xi_{1},\xi_{2}$ are independent random variables with a common distribution function $F$.

Sufficient conditions for a distribution to belong to the class $\mathcal{S}\,$ may be found, or example, in [7], [23], [24] and [25]. The class $\mathcal{S}\,$ includes, in particular, the following distributions on $[0,\infty)$:

• (i)

the Pareto distribution with the tail $\overline{G}(x)=(\frac{\kappa}{\kappa+x})^{\alpha}$, where $\alpha,\kappa>0$;

• (ii)

the lognormal distribution with the density $e^{-(\ln x-\ln\alpha)^{2}/2\sigma^{2}}/x\sqrt{2\pi\sigma^{2}}$ with ${\alpha>0}$;

• (iii)

the Weibull distribution with the tail $\overline{G}(x)=e^{-x^{\alpha}}$ with ${\alpha\in(0,1)}$.

Another subclass of heavy-tailed distribution is a class of distributions with dominated varying tail.

Definition 4**.**

A distribution function $F$ is a dominated varying tail distribution function (${F\in\mathcal{D}}$) if

[TABLE]

A distribution function from $\mathcal{D}$ is not always subexponential. Indeed, all subexponential distributions are long-tailed, but there are some dominated varying distributions which are not long-tailed, see [15] and [21] for a counterexample. However, Klüppelberg [23] proved that if the mean $\int_{0}^{\infty}\overline{F}(y)dy$ is finite then $\mathcal{L}\cap\mathcal{D}\subset\mathcal{S}^{*}\subset\mathcal{S}$. All regularly varying distribution functions belong to $\mathcal{D}$.

Definition 5**.**

A distribution function $F$ is regularly varying with index $-\alpha$ if, for all $\lambda>0$,

[TABLE]

Examples of regularly varying distribution functions are the Pareto distribution function and $G$ with the tail $\overline{G}(x)\sim 1/x^{\alpha}\ln^{\beta}x$ An extensive survey of the regularly varying distributions may be found in [6]. It is shown in [7] that any subexponential distribution is long-tailed with necessity. The converse is not true, see [15] for a counterexample.

When the mean is finite, the derivation of (2) in [1],[19] and[10] heavily relied on the local asymptotics $\mathbf{P}(M\in(x,x+c])$ for a fixed $c>0$ as $x\to\infty$. In the infinite-mean case, these local asymptotics are not known. It seems that it can be found only in some particular cases. The reason for that are complications in the local renewal theorem in the infinite mean case, see [4] for the complete solution of the local renewal in the infinite mean case and its history. Therefore, we propose a slightly different approach: it appears that it is sufficient to prove directly that

[TABLE]

For that, we use a introduce a new class $\mathcal{S}_{F}$ of heavy-tailed distributions,

Definition 6**.**

Let $F$ be a distribution function. A distribution function $G$ on ${\bf R}^{+}$ belongs to ${\mathcal{S}}_{F}$ ($G\in{\mathcal{S}}_{F}$) if

[TABLE]

This class is a natural extension of the class of subexponential distributions. Indeed, it follows from the definition that $F$ is subexponential if and only if $F\in\mathcal{S}_{F}$. Then we study properties of this class. These properties (as well as its proofs) are rather close to that of subexponential distributions. Let $G_{1}$ be a distribution function on $\mathbb{R}^{+}$ with the distribution tail

[TABLE]

The following theorems are the main results of this paper.

Theorem 7**.**

Suppose ${\bf E}\xi^{-}=\infty$ and condition (1) holds. If the distribution function $G_{1}\in\mathcal{S}_{F}$ then asymptotics (2) hold, i. e.

[TABLE]

Theorem 8**.**

Let $\mathbf{E}\xi_{1}^{-}=\infty$ and either of the following conditions holds

1. (a)

$F\in{\mathcal{S}}^{*}$; 2. (b)

$F\in{\mathcal{L}}\cap{\mathcal{D}}$* and condition (1) holds.*

Then $G_{1}\in\mathcal{S}_{F}$.

2. Class ${\mathcal{S}}_{F}$ and its basic properties

Definition 6 may be rephrased as follows. Consider independent random variables $\psi\geq 0$ and $\xi$ with distributions $G$ and $F$ respectively. Then $G\in{\mathcal{S}}_{F}$ if and only if

[TABLE]

Basic properties of the class ${\mathcal{S}}_{F}$ are very close to those of the class of subexponential distributions(see Lemmas 9–13). For a fine account of the theory of subexponential and local subexponential distributions we refer to [2] and  [18].

Lemma 9**.**

Let $G$ be a distribution function on ${\mathbf{R}}^{+}$. Then $G\in{\mathcal{S}}_{F}$ if and only if there exists a function $h(x)\uparrow\infty,h(x)<x/2$ such that:

1. (1)

$\overline{F}(x-h(x))\sim\overline{F}(x)$; 2. (2)

$G(x-h(x),x]=o(\overline{F}(x))$; 3. (3)

$\int_{h(x)}^{x-h(x)}G(du)\overline{F}(x-u)=o(\overline{F}(x)).$

Proof of Lemma 9. It is clear that if such a function $h(x)$ exists, then $F\in\mathcal{L}$ and $G(x-t,x]=o(\overline{F}(x))$ for all fixed $t>0.$ Conversely, if $F\in\mathcal{L}$ and $G(x-t,x]=o(\overline{F}(x))$ for some fixed $t>0$, then one can construct a function $h(x)$ satisfying conditions ((1)) and ((2)).

First, assume $G\in S_{F}$. Fix any $t>0$. Then,

[TABLE]

By dividing both sides by $\overline{F}(x)$, letting $x$ to infinity and rearranging the terms, we obtain

[TABLE]

and

[TABLE]

Consequently, there exists a function $h(x)$ such that ((1)) and ((2)) hold. For this function, condition ((3)) holds.

Conversely, suppose that there is a function $h(x)$ satisfying ((1))–((3)). Condition ((1)) implies

[TABLE]

and condition ((2)) implies

[TABLE]

Using condition ((3)) we obtain the required result $G\in{\mathcal{S}}_{F}.$

∎

Lemma 10**.**

(convolution closure) Let distribution functions $G_{1},G_{2}$ belong to ${\mathcal{S}}_{F}$. Then $G_{1}*G_{2}\in{\mathcal{S}}_{F}.$

Proof. Take a function $h(x)$ satisfying conditions ((1))–((3)) of Lemma 9 for both distributions $G_{1}$ and $G_{2}$ simultaneously. Then,

[TABLE]

By conditions ((1)) and ((3)) of Lemma 9,

[TABLE]

and, by condition ((2)),

[TABLE]

∎

By induction, Lemma 10 yields

Corollary 11**.**

Let $G\in{\mathcal{S}}_{F}$. Then $G^{*n}\in{\mathcal{S}}_{F}$ for any $n\geq 1$.

Throughout, $G^{*0}$ denotes a distribution degenerated at [math].

Lemma 12**.**

Let $G\in{\mathcal{S}}_{F}$. Then, for any $\varepsilon>0$, there exists $A\equiv A(\varepsilon)>0$ such that, for any integer $n\geq 0$ and for any $x\geq 0$,

[TABLE]

Proof. Take any $\varepsilon>0$. Since $G\in{\mathcal{S}}_{F}$, there exists $x_{0}>0$ such that

[TABLE]

Put $A\equiv\frac{1}{\overline{F}(x_{0})}$. We use induction arguments. For $n=0$ the assertion clearly holds. Suppose that the assertion is true for $n-1$ and prove it for $n$. For $x<x_{0}$,

[TABLE]

Further, for $x\geq x_{0}$,

[TABLE]

The latter follows from (6). ∎

Let $\{\zeta_{n}\}$ be a sequence of i.i.d. non-negative random variables with a common distribution $G$, and let $\nu$ be a random stopping time independent of $\{\zeta_{n}\}$. Put $X_{n}=\zeta_{1}+\cdots+\zeta_{n}$. Then the distribution of the randomly stopped sum $X_{\nu}$ is

[TABLE]

Lemma 13**.**

*Let $G$ belong to ${\mathcal{S}}_{F}$. Assume that $\mathbf{E}(1+\delta)^{\nu}<\infty$ for some $\delta>0$. Then $G_{\nu}$ belongs to ${\mathcal{S}}_{F}$. *

Proof. The result follows from Corollary 11, Lemma 12 and from the dominated convergence theorem.

∎

It is known [23, Theorem 3.2] that if $F\in{\mathcal{S}}^{*}$ then $F$ is subexponential, i.e. $F\in\mathcal{S}_{F}$. In the following Lemma, we generalize this assertion to obtain sufficient conditions for $G\in{\mathcal{S}}_{F}$. Another extension may be found in [12, Lemma 9].

Lemma 14**.**

Let $F\in{\mathcal{S}}^{*}$ and $G$ on $\mathbf{R}^{+}$ be such that $G(x-1,x]=o(\overline{F}(x))$. Then $G\in{\mathcal{S}}_{F}$.

Proof. It follows from $F\in{\mathcal{S}}^{*}$ that $F\in{\mathcal{L}}$, see [23, Theorem 3.2]. Then there exists a function $h(x)$ satisfying conditions ((1)) and ((2)) of Lemma 9. Further,

[TABLE]

Since $G(x-1,x]=o(\overline{F}(x))$, for $x$ large enough $G(x-1,x]\leq\overline{F}(x)$. Then,

[TABLE]

the latter is due to $F\in\mathcal{S}^{*}$. Then, condition $(\ref{c3})$ of Lemma 9 is satisfied and $G~{}\in~{}\mathcal{S}_{F}.$ ∎

As a simple corollary of Lemma 14, we can obtain sufficient conditions for the convolution $F*G$ to belong to the class $\mathcal{S}^{*}\subset\mathcal{S}$.

Corollary 15**.**

Let $F$ and $G$ on $\mathbf{R}^{+}$ be such that $F\in{\mathcal{S}}^{*}$ and $G(x-1,x]=o(\overline{F}(x))$. Then $F*G$ belongs to $\mathcal{S}^{*}$.

Let $H$ be a non-decreasing function on $\mathbf{R}^{+}$ such that integral

[TABLE]

We assume that $H$ is subadditive, i.e. if $H(x+y)\leq H(x)+H(y)$ for all $x,y\geq 0$. Let $H(x,y]=H(y)-H(x)$. Consider a distribution function $G_{H}$ on $\mathbf{R}^{+}$ with the tail distribution

[TABLE]

Integrating (7) by parts, we obtain an equivalent representation for $G_{H}$,

[TABLE]

We now establish some properties of $G_{H}$, which will be used in the next Section.

Lemma 16**.**

Let $F\in\mathcal{L}$ and $H(x-1,x]\to 0$ as $x\to\infty$. Then,

[TABLE]

Proof. It follows from definition that, for all sufficiently large $x$,

[TABLE]

Since $F\in\mathcal{L}$, there exists a function $h(x)$ such that $\overline{F}(x)\sim\overline{F}(x+h(x))$. Then,

[TABLE]

and

[TABLE]

∎

Lemma 17**.**

Let $F\in\mathcal{L}$, and let $H_{1},H_{2}$ be subadditive functions such that $H_{1}(x-1,x]\to 0,H_{2}(x-1,x]\to 0$ and

[TABLE]

Then

[TABLE]

Proof. By Lemma 16, $G_{H_{1}}(x-1,x]=o(\overline{F}(x))$ and $G_{H_{2}}(x-1,x]=o(\overline{F}(x))$. Therefore, there exists a function $h(x)$ such that conditions ((1)) and ((2)) of Lemma 9 hold for both distribution functions $G_{H_{1}}$ and $G_{H_{2}}$. Integrating by parts ((3)) and using ((1)) and ((2)) , we obtain

[TABLE]

For any $y<x$, we have

[TABLE]

Due to the subadditive property,

[TABLE]

Hence condition ((3)) of Lemma 9 holds for distribution functions $G_{H_{1}}$ and $G_{H_{2}}$ if and only if

[TABLE]

Then the assertion of Lemma follows from (9). ∎

Lemma 18**.**

Assume that $F\in\mathcal{L}\cap\mathcal{D}$ and $H(x-1,x]\to 0$ as $x\to\infty$. Then $G_{H}\in\mathcal{S}_{F}.$

Proof. It follows from Lemma 16 that $G_{H}(x-1,x]=o(\overline{F}(x))$. Therefore, there exists a function satisfying conditions ((1)) and ((2)). From the proof of Lemma 17, it is clear that if (10) holds then $G_{H}\in\mathcal{S}_{F}$. Using the subadditive property of $H$, we obtain

[TABLE]

Further,

[TABLE]

It follows from $F\in\mathcal{D}$ that $o(1)\overline{F}(x/2)=o(\overline{F}(x))$. ∎

3. Proof of the main results

First recall a well-known construction of ladder moments and ladder heights [17, Chapter XII]. Let

[TABLE]

be the first (strict) ascending ladder epoch and put

[TABLE]

Let $\{\psi_{n}\}_{n\geq 1}$ be a sequence of i.i.d.r.v.’s distributed as

[TABLE]

Let $\nu$ be a random variable, independent of the above sequence, such that $\mathbf{P}(\nu=n)=p(1-p)^{n},n=0,1,2,\ldots$ Then

[TABLE]

We start with proving an auxiliary assertion.

Lemma 19**.**

Let $G_{+}\in{\mathcal{S}}_{F}$. Then asymptotics (2) hold, i. e.

[TABLE]

Proof. The proof is carried out via standard arguments: we obtain the lower and the upper bounds, which are asymptotically equivalent. Let us start with the lower bound, which is valid without any assumptions on $F$ and $G_{+}$. Fix a positive integer $N$, then for any $x>0$,

[TABLE]

Let $N\to\infty$ to obtain

[TABLE]

Now turn to the upper bound. For any $x\geq 0$,

[TABLE]

for the latter see [17, Chapter XII, (2.7)]. Finally, it follows from $G_{+}\in{\mathcal{S}}_{F}$, relation (11) and Lemma 13 that the distribution function of $M$ belongs to $\mathcal{S}_{F}$, that is

[TABLE]

∎

Now let us introduce a few more definitions. Let $\chi=-S_{\tau}$ be the absolute value of the first non-positive sum and let $G_{-}(x)\equiv\mathbf{P}(\chi\leq x)$ be its distribution function. Define a renewal function

[TABLE]

Then $\psi_{1}$ is distributed as follows [17, Chapter XII]:

[TABLE]

We will need the following asymptotic estimates for the renewal function $H$

Proposition 20**.**

(see [12, Corollary 2]) Suppose $\mathbf{E}[\xi_{1}^{-}]=\infty$ and condition (1) holds. Then,

[TABLE]

Proof of Theorem 7. We will prove that $G_{1}\in\mathcal{S}_{F}$ implies that $G_{+}\in\mathcal{S}_{F}$. For that, we verify the conditions of Lemma 17.

First, by the properties of renewal functions, $H_{-}$ is subadditive. It follows from $\mathbf{E}\xi_{1}^{-}=\infty$ that $\mathbf{E}\chi=\infty.$ This, the Key Renewal Theorem implies that $H(x-1,x]\equiv H(x)-H(x-1)\to 0,x\to\infty$.

Second, the function $x/m(x)$ is subadditive as well:

[TABLE]

The latter inequality holds since $m(x)$ is non-decreasing. Further, the function $x/m(x)$ is non-decreasing, since

[TABLE]

Hence,

[TABLE]

Then, Lemma 17 and (13) imply that $G_{+}\in\mathcal{S}_{F}$ if and only if $G_{1}\in\mathcal{S}_{F}.$ Consequently, $G_{+}\in\mathcal{S}_{F}$ and, by Lemma 19, asymptotics (2) hold. ∎

Proof of Theorem 8. Sufficiency of (a) follows from Lemma 14 and Lemma 16. Sufficiency of (b) follows from Lemma 18. ∎

Acknowledgements

The author is most grateful to Sergey Foss for drawing attention to this problem, valuable comments regarding the manuscript and a number of useful discussion.