Functional Central Limit Theorems for Occupancies and Missing Mass Process in Infinite Urn Models

Abstract

We study the infinite urn scheme when the balls are sequentially distributed over an infinite number of urns labeled 1,2,... so that the urn j at every draw gets a ball with probability $p_j$, where $\sum _j p_j=1$. We prove functional central limit theorems for discrete time and the Poissonized version for the urn occupancies process, for the odd occupancy and for the missing mass processes extending the known non-functional central limit theorems.

1 Introduction

In this paper, we study the following classical urn model first considered by Karlin [12]: $n\ge 1$ balls are distributed one by one over an infinite number of urns enumerated from 1 to infinity. The ball distributed at step $j=1,2\dots$, call it jth ball, gets into urn i with probability $p_i$, $\sum _{i=1}^\infty p_i=1$, independently of the other balls. Such multinomial occupancy schemes arise in many different applications, in Biology [11], Computer science [13, 14] and in many other areas, see, e.g., [10] and the references therein.

Let $X_j$ be the urn the jth ball gets into and let $J_i(n)$ be the number of balls the ith urn contains after n balls are distributed:

$$\begin{aligned} J_i(n)=\sum _{j=1}^{n} \mathbb {1}_{X_j=i}. \end{aligned}$$

Of a particular interest is the asymptotic behavior of the following quantities: the number of urns containing at least $k\ge 1$ balls and containing exactly k balls:

$$\begin{aligned} R^{*}_{n,k}=\sum _{i=1}^{\infty } \mathbb {1}_{J_i(n)\ge k},\quad R_{n,k}=\sum _{i=1}^{\infty } \mathbb {1}_{J_i(n)= k}=R^{*}_{n,k}-R^{*}_{n,k+1}, \end{aligned}$$

(1)

the number of urns with an odd number of balls and the scaled missing mass introduced in [12]:

$$\begin{aligned} U_n=\sum _{i=1}^\infty \mathbb {1}_{J_i(n)\equiv 1\ (\mathrm {mod} \ 2)}, \quad M_n=n \sum _{i=1}^\infty p_i \mathbb {1}_{J_i(n)=0}, \end{aligned}$$

(2)

We also use notation $R_n{\mathop {=}\limits ^{\mathrm{def}}}R^{*}_{n,1}=\sum _{k\ge 1} R_{n,k}$ for the number of non-empty urns. Renumbering the urns if necessary, we may assume that the sequence $(p_i)_{i\ge 1}$ is monotonely decaying. We further assume that it is regularly varying:

$$\begin{aligned} \alpha (x)=\max \{i:\ p_i\ge 1/x\}=x^{\theta } L(x) \ \text {with} \theta \in [0,1], \end{aligned}$$

(3)

where L(x) is a slowly varying function as $x\rightarrow \infty$.

Following Karlin’s [12] original approach, we will consider a Poissonized version of the model when the balls are put into urns at the times of jumps of a homogeneous Poisson point processes $\Pi (s), s\ge 0$ with intensity 1 on $\mathbb {R}_+$. According to the independent marking theorem for Poisson processes, $\{J_i(\Pi (s)){\mathop {=}\limits ^{\mathrm{def}}}\Pi _i(s), \ s\ge 0\}$ are independent homogeneous Poisson processes with intensities $p_i$. To ease the notation, we write simply

$$\begin{aligned} R(s){\mathop {=}\limits ^{\mathrm{def}}}R^{*}_{\Pi (s),1},\ U(s){\mathop {=}\limits ^{\mathrm{def}}}U_{\Pi (s)}, \end{aligned}$$

and we introduce the following Poissonized version of the scaled missing mass:

$$\begin{aligned} M(s){\mathop {=}\limits ^{\mathrm{def}}}s \sum _{i=1}^\infty p_i \mathbb {1}_{\Pi _i(s)=0}. \end{aligned}$$

It differs from $M_{\Pi (s)}$ by the scaling factor s vs. $\Pi (s)$, but, when properly scaled, it is asymptotically equivalent to it.

Ordinary (not functional) central limit theorems for the above quantities were established under various conditions in [2, 3, 9, 10, 12,13,14]. In particular, under rather general conditions on the sequence $(p_i)$ involving an unbounded growth of the variances, the following results are available: a strong law of large numbers and asymptotic normality of $R_n$, an asymptotic normality of the vector $(R_{n,1},\dotsc ,R_{n,\nu })$, local limit theorems, etc.

We acknowledge a novel method of a randomized decomposition for proving FCLTs developed in a recent paper [8], but we do not use it here. As a particular case of their Theorem 2.3, a FCLT holds for the processes $R_n$ and $U_n$ when $\theta \in (0,1)$.

Our goal here is to establish a FCLT for the triplet of processes: the occupancy, odd occupancy and the scaled missing mass when $\theta \in (0,1]$. In particular, we obtain previously unknown FCLT for $U_n$ for $\theta =1$ and for $M_n$ when $\theta \in (0,1]$. Up to a normalizing constant, the FCLT stated in Theorem 1 also holds for the original (non-scaled) missing mass $\sum _{i=1}^\infty p_i \mathbb {1}_{J_i(n)=0}$ on any interval $t\in [\varepsilon ,1]$, $\varepsilon >0$, separated from 0. The paper extends the results of [6] and [7], where a functional central limit theorem (FCLT) was shown under condition (3) for the vector process

$(R^{*}_{[nt],1},R^{*}_{[nt],2},\dots , R^{*}_{[nt],\nu })_{t\in [0,1]}$ in the case $\theta \in (0,1]$.

Extending the FCLT to the case $\theta =0$ would require additional to (3) conditions. As it was mentioned in [12] and in [2], $\theta =0$ does not imply that the variances grow to infinity and various asymptotic behavior is possible for different statistics. We also argue that even an infinite growth of variances does not guarantee per se the required relative compactness.

When $\theta =1$, we need a function

$$\begin{aligned} L^{*}(x)=\int _0^\infty L(xs) e^{-s}s^{-1} \hbox {d}s. \end{aligned}$$

It is known (see [12]) that $L^{*}(x)$ is slowly varying when $x \rightarrow \infty$.

Finally, for $t\in [0,1]$ introduce the following notation:

$$\begin{aligned} \beta (n)&=\left\{ \begin{array}{ll} \alpha (n), &{} \theta \in [0,1); \\ nL^{*}(n), &{} \theta =1, \end{array} \right. \quad&R_n(t) =\frac{R_{[nt]}-{{\,\mathrm{\mathbf{E}}\,}}R_{[nt]}}{(\beta (n))^{1/2}}, \end{aligned}$$

(4)

$$\begin{aligned} U_n(t)&=\frac{U_{[nt]}-{{\,\mathrm{\mathbf{E}}\,}}U_{[nt]}}{(\beta (n))^{1/2}},&M_n(t) =\frac{M_{[nt]}-{{\,\mathrm{\mathbf{E}}\,}}M_{[nt]}}{(\alpha (n))^{1/2}}. \end{aligned}$$

(5)

We are now ready to formulate the main result of the paper.

Theorem 1

When $\theta \in (0, 1]$, the vector process

$$\begin{aligned} (R_n(t),U_n(t),M_n(t)),\quad t\in [0,1], \end{aligned}$$

converges weakly in the uniform metric on $D([0,1]^3)$ to a three-dimensional Gaussian process $(\rho (t),\upsilon (t),\mu (t))$ with zero mean and the covariance function $c(\tau ,t)$ with the following components: when $\theta \in (0,1), \ \tau \le t$,

$$\begin{aligned} c_{\rho \rho }(\tau ,t)&=\Gamma (1-\theta )((\tau +t)^\theta -t^\theta ), \\ c_{\upsilon \upsilon }(\tau ,t)&= \Gamma (1-\theta )2^{\theta -2}((t+\tau )^\theta -(t-\tau )^\theta ),\\ c_{\mu \mu }(\tau ,t)&=\theta \Gamma (2-\theta ) \left( \frac{\tau }{t^{1-\theta }}-\frac{t\tau }{(t+\tau )^{2-\theta }} \right) , \\ c_{\rho \upsilon }(\tau ,t)&=\Gamma (1-\theta )((2t+\tau )^{\theta } - (2t-\tau )^{\theta })/2, \\ c_{\rho \upsilon }(t,\tau )&=\Gamma (1-\theta )((2t+\tau )^\theta - t^\theta )/2,\\ c_{\rho \mu }(\tau ,t)&=\theta \Gamma (1-\theta )\left( \frac{t}{(t+\tau )^{1-\theta }} - t^\theta \right) , \\ c_{\rho \mu }(t,\tau )&=\theta \Gamma (1-\theta )\left( \frac{\tau }{(t+\tau )^{1-\theta }} -\frac{\tau }{ t^{1-\theta }} \right) ,\\ c_{\mu \upsilon }(\tau ,t)&=\theta \Gamma (1-\theta )\left( \frac{\tau }{2(2t+\tau )^{1-\theta }} -\frac{\tau }{2(2t-\tau )^{1-\theta }} \right) ,\\ c_{\mu \upsilon }(t,\tau )&=\theta \Gamma (1-\theta )\left( \frac{t}{2(2\tau +t)^{1-\theta }} -\frac{t^\theta }{2} \right) . \end{aligned}$$

When $\theta =1$, $\tau \le t$, $c(\tau ,t)$ is given by

$$\begin{aligned} c_{\rho \rho }(\tau ,t)&= \tau , \ c_{\upsilon \upsilon }(\tau ,t)=2\tau , \ c_{\mu \mu }(\tau ,t)=\tau ^2, \\ c_{\rho \upsilon }(\tau ,t)&=\tau , \ c_{\rho \upsilon }(t,\tau )=(t+\tau )/2,\\ c_{\rho \mu }(\tau ,t)&=c_{\rho \mu }(t,\tau )=c_{\upsilon \mu }(\tau ,t)=c_{\upsilon \mu }(t,\tau )=0. \end{aligned}$$

Thus, when $\theta =1$, $\rho (t)$ and $\upsilon (t)$ are Wiener processes. For a general $\theta \in (0,1]$, the process $(\rho (t),\upsilon (t),\mu (t))$ is self-similar with the Hurst parameter $H=\theta /2$ which includes, in particular, a fractional Brownian motion, a bi-fractional Brownian motion with parameter $H = 1/2, K = \theta$ (see, e.g., [8]) with a new self-similar process $\mu (t)$.

2 Proof of Theorem 1

We start with formulating a couple of lemmas proved in [7]. We will generally use the letter C and its variants to denote a constant whose value is of no importance for us and note in parentheses the parameters it depends upon. This should not lead to a confusion when the same notation is used for, actually, different constants in different contexts, the same way O(1) notation is used.

Lemma 1

When $\theta >0$, there exist $n_0\ge 1$ and $C(\theta )<\infty$ such that

$$\begin{aligned} \frac{{{\,\mathrm{\mathbf{E}}\,}}R(n\delta )}{\beta (n)} \le C(\theta ) \delta ^{\theta /2} \end{aligned}$$

holds for any $\delta \in [0,1]$ and $n\ge n_0$.

Lemma 2

For any $\varepsilon , \delta \in (0,1)$ there exists an $N=N(\varepsilon ,\delta )$ such that for any $n\ge N$,

$$\begin{aligned} \mathbf {P}(\forall t\in [0,1] \ \ \exists \tau : |\tau -t|\le \delta , \ \Pi (n\tau )= [nt]) \ge 1-\varepsilon . \end{aligned}$$

In preparation of the proof, let us introduce some further notation and establish a few inequalities we will be using.

In view of (5), let

$$\begin{aligned} U_n^{*}(t)&=\frac{U (nt)-{{\,\mathrm{\mathbf{E}}\,}}U (nt)}{(\beta (n))^{1/2}},&U_n^{**}(t)=\frac{U ([nt])-{{\,\mathrm{\mathbf{E}}\,}}U ([nt])}{(\beta (n))^{1/2}} \end{aligned}$$

(6)

$$\begin{aligned} M_n^{*}(t)&=\frac{M (nt)-{{\,\mathrm{\mathbf{E}}\,}}M (nt)}{(\alpha (n))^{1/2}},&M_n^{**}(t)=\frac{M ([nt])-{{\,\mathrm{\mathbf{E}}\,}}M ([nt])}{(\alpha (n))^{1/2}}. \end{aligned}$$

(7)

For any two positive $\tau _1\le \tau _2$, define

$$\begin{aligned} U(\tau _2)-U(\tau _1)&=\sum _{i=1}^{\infty }\mathbb {1}\{\Pi _i(\tau _2)\ \text {is odd}\}-\mathbb {1}\{\Pi _i(\tau _1)\ \text {is odd}\}\\&=\sum _{i=1}^{\infty }\mathbb {1}\{\Pi _i(\tau _2) \ \text {is odd}, \Pi _i(\tau _1) \ \text {is even}\}\\&\quad - \mathbb {1}\{\Pi _i(\tau _2) \ \text {is even}, \Pi _i(\tau _1) \ \text {is odd}\}\\&{\mathop {=}\limits ^{\mathrm{def}}}\sum _{i=1}^{\infty } u_{i}(\tau _1,\tau _2) = \sum _{i=1}^{\infty } u_{i} =\sum _{i=1}^{\infty } u'_{i}-u''_i, \end{aligned}$$

and their expectations are denoted by

$$\begin{aligned} \overline{u}_i =\overline{u}_i'-\overline{u}_i''=\overline{u}_i(\tau _1,\tau _2){\mathop {=}\limits ^{\mathrm{def}}}{{\,\mathrm{\mathbf{E}}\,}}u'_i -{{\,\mathrm{\mathbf{E}}\,}}u''_i. \end{aligned}$$

Similarly for M,

$$\begin{aligned} M(\tau _2)-M(\tau _1)&=\sum _{i=1}^{\infty }(\tau _2-\tau _1)p_i\mathbb {1}\{\Pi _i(\tau _2)=0\}- \tau _1 p_i\mathbb {1}\{\Pi _i(\tau _1)=0,\Pi _i(\tau _2)>0\}\\&{\mathop {=}\limits ^{\mathrm{def}}}\sum _{i=1}^{\infty } m_{i}(\tau _1,\tau _2)=\sum _{i=1}^{\infty } m_{i}=\sum _{i=1}^{\infty } m'_{i}-m''_i,\\ \overline{m}_i&=\overline{m}_i'-\overline{m}_i''=\overline{m}_i(\tau _1,\tau _2){\mathop {=}\limits ^{\mathrm{def}}}{{\,\mathrm{\mathbf{E}}\,}}m'_i -{{\,\mathrm{\mathbf{E}}\,}}m''_i. \end{aligned}$$

Clearly, for all natural k,

$$\begin{aligned} {{\,\mathrm{\mathbf{E}}\,}}|u_i-\overline{u}_i|^k&= |1+\overline{u}_i|^k \overline{u}_i''+|\overline{u}_i|^k (1-\overline{u}_i'-\overline{u}_i'')+|1-\overline{u}_i|^k \overline{u}_i' \nonumber \\&\le 2^k (\overline{u}_i'+\overline{u}_i'')+|\overline{u}_i|^k\le (2^k+1) (\overline{u}_i'+\overline{u}_i'') \nonumber \\&=(2^k+1)\left[ \sum _{j=0}^\infty \mathbf {P}\{\Pi _i(\tau _1)=2j, \ \Pi _i(\tau _2)-\Pi _i(\tau _1)\ \text {is odd}\} \nonumber \right. \\&\quad \left. +\sum _{j=0}^\infty \mathbf {P}\{\Pi _i(\tau _1)=2j+1, \ \Pi _i(\tau _2)-\Pi _i(\tau _1)\ \text {is odd}\}\right] \nonumber \\&=(2^k+1)\mathbf {P}\{\Pi _i(\tau _2-\tau _1)\ \text {is odd}\} \nonumber \\&<(2^k+1) \mathbf {P}\{\Pi _i(\tau _2-\tau _1)>0\}. \end{aligned}$$

(8)

Similarly,

$$\begin{aligned} {{\,\mathrm{\mathbf{E}}\,}}|m'_i-\overline{m}'_i|^k&\le 2^{k-1}({{\,\mathrm{\mathbf{E}}\,}}|m'_i|^k+|\overline{m}'_i|^k)=2^{k-1}(\tau _2-\tau _1)^kp^k_i(e^{-\tau _2 p_i}+e^{-k\tau _2 p_i})\\&<2^k k! (1-e^{-(\tau _2-\tau _1)p_i})= 2^k k!\, \mathbf {P}\{\Pi _i(\tau _2-\tau _1)>0\},\\ {{\,\mathrm{\mathbf{E}}\,}}|m''_i-\overline{m}''_i|^k&\le 2^{k-1}({{\,\mathrm{\mathbf{E}}\,}}|m''_i|^k+|\overline{m}''_i|^k)<2^{k}\tau _1^kp^k_i e^{-\tau _1 p_i}(1-e^{-(\tau _2-\tau _1)p_i})\\&<2^k k! (1-e^{-(\tau _2-\tau _1)p_i})= 2^k k! \,\mathbf {P}\{\Pi _i(\tau _2-\tau _1)>0\}. \end{aligned}$$

As a result,

$$\begin{aligned} {{\,\mathrm{\mathbf{E}}\,}}|m_i-\overline{m}_i|^k<4^{k}k! \,\mathbf {P}\{\Pi _i(\tau _2-\tau _1)>0\}. \end{aligned}$$

(9)

We are using the same notation $u_i, \ m_i$ and $\overline{u}_i, \ \overline{m}_i$ without explicitly specifying the corresponding values of $\tau _1<\tau _2$; this should not create a confusion. The following lemma will be used in the proof of the relative compactness of the process $M^{*}_n(t)$.

Lemma 3

Let $\theta \in (0,1]$ and $\delta \in [0,1]$. Then, there exist $n_0\ge 1$ and $C(\theta )<\infty$ such that

$$\begin{aligned} \frac{{{\,\mathrm{\mathbf{var}}\,}}(M(nt_2)-M(nt_1))}{\alpha (n)} \le C(\theta )\delta ^{\theta /2} \end{aligned}$$

for all $t_2-t_1=\delta \ge 0$ and $n\ge n_0$.

Proof

Put $\tau _2=n t_2$ and $\tau _1=n t_1$. Since the variance of an indicator does not exceed its expectation, we have that

$$\begin{aligned}&{{\,\mathrm{\mathbf{var}}\,}}(M(\tau _2)-M(\tau _1))=\sum _{i=1}^{\infty }{{\,\mathrm{\mathbf{E}}\,}}(m_i-\overline{m}_i)^2 = \sum _{i=1}^{\infty } {{\,\mathrm{\mathbf{E}}\,}}(m'_i)^2-(\overline{m}'_i-\overline{m}_i'')^2+{{\,\mathrm{\mathbf{E}}\,}}(m''_i)^2 \\&\quad \le \sum _{i=1}^{\infty } (\tau _2{-}\tau _1)^2p_i^2 e^{-\tau _2 p_i} +\tau _1^2 p_i^2 e^{-\tau _1 p_i}(1-e^{{-}(\tau _2{-}\tau _1)p_i}\pm (\tau _2{-}\tau _1)p_i e^{-(\tau _2-\tau _1)p_i})\\&\quad \le 2\frac{(\tau _2-\tau _1)^2}{\tau _2^2}{{\,\mathrm{\mathbf{E}}\,}}R_{\Pi (\tau _2),2}+{{\,\mathrm{\mathbf{E}}\,}}R^{*}_{\Pi (\tau _2-\tau _1),2} +6\frac{\tau _1^2(\tau _2-\tau _1)}{\tau _2^3}{{\,\mathrm{\mathbf{E}}\,}}R_{\Pi (\tau _2),3}. \end{aligned}$$

By [12, Th. 2.1 and (23)],

$$\begin{aligned} \lim \limits _{x\rightarrow \infty }\frac{{{\,\mathrm{\mathbf{E}}\,}}R^{*}_{\Pi (x),2}}{\alpha (x)}=\Gamma (2-\theta )<2, \end{aligned}$$

and therefore, there exists an $x_1>1$ such that for all $x\ge x_1$,

$$\begin{aligned} {{\,\mathrm{\mathbf{E}}\,}}R_{\Pi (x),2}+{{\,\mathrm{\mathbf{E}}\,}}R_{\Pi (x),3}<{{\,\mathrm{\mathbf{E}}\,}}R^{*}_{\Pi (x),2}< 2 \alpha (x). \end{aligned}$$

According to Karamata (see, e.g., [5, Th. 2,1, Eq. A6.2.10]), there exists an $x_2>0$ such that for all x and $\delta \in (0,1]$ satisfying $x\delta \ge x_2$, one has

$$\begin{aligned} \frac{L(x\delta )}{L(x)}\le 2 \delta ^{-1/2}. \end{aligned}$$

Let $n\delta >\max \{x_1,x_2\}=x_0$, then

$$\begin{aligned} \frac{{{\,\mathrm{\mathbf{E}}\,}}R^{*}_{\Pi (n \delta ),2}}{\alpha (n)} \le 2\frac{(n \delta )^\theta L(n \delta )}{n^\theta L(n)} \le 4 \delta ^{\theta /2}, \ \ \ \frac{\max ({{\,\mathrm{\mathbf{E}}\,}}R_{\Pi (n t_2),2},{{\,\mathrm{\mathbf{E}}\,}}R_{\Pi (n t_2),3})}{\alpha (n)} \le 4 t_2^{\theta /2}. \end{aligned}$$

Choose $n_0$ such that for all $n\ge n_0$ we have $n^\theta L(n)\ge n^{\theta /2}$. Then, provided $n t_2 \le x_0$,

$$\begin{aligned}&\frac{{{\,\mathrm{\mathbf{E}}\,}}R^{*}_{\Pi (n\delta ),2}}{\alpha (n)} \le \frac{{{\,\mathrm{\mathbf{E}}\,}}{\Pi (n\delta )}}{\alpha (n)} \le \frac{n\delta }{n^{\theta /2}}=(n\delta )^{1-\theta /2}\delta ^{\theta /2}\le x_0 \delta ^{\theta /2}, \\&\frac{\max ({{\,\mathrm{\mathbf{E}}\,}}R_{\Pi (n t_2),2},{{\,\mathrm{\mathbf{E}}\,}}R_{\Pi (n t_2),3})}{\alpha (n)} \le x_0 t_2^{\theta /2}. \end{aligned}$$

Now, take $c=\max \{4, x_0 \}$. Since $t_2-t_1=\delta \ge 0$, for all $n\ge n_0$ we obtain

$$\begin{aligned} \frac{{{\,\mathrm{\mathbf{var}}\,}}(M(nt_2)-M(nt_1))}{\alpha (n)}\le 2 c \frac{\delta ^2}{t_2^{2-\theta /2}}+\delta ^{\theta /2}+6 c \frac{t_1^2 \delta }{t_2^{3-\theta /2}}\le 9c \cdot \delta ^{\theta /2}. \end{aligned}$$

$\square$

We are ready to prove Theorem 1. The proof is broken into four steps.

Step 1: Covariance The first rather technical step consists in establishing a formula for the covariances which is put in Appendix.

Step 2: Convergence of finite-dimensional distributions Along the lines of the proof of [9, Th. 12], one can show that for

$$\begin{aligned} m\ge 1, \ \ \ 0<t_1<t_2<\ldots <t_m\le 1 \end{aligned}$$

the triangular array of m-dimensional vectors (i.e., independent in k for every n)

$$\begin{aligned} \left\{ \frac{\mathbb {1}(\Pi _k(nt_j)\ \text {is odd})-\mathbf {P}(\Pi _k(nt_j)\ \text {is odd})}{\sqrt{\beta (n)}}, \ j \le m, \ k\le n \right\} _{n\ge 1} \end{aligned}$$

satisfies the Lindeberg condition (see, e.g., [5, Th. 6.2]). Similarly, the convergence of the finite-dimensional distributions is shown for the process $M^{*}_n(t)$.

Step 3: Relative compactness

We shall follow the following plan:

• (a) prove the continuity of the limiting process;

• (b) prove that $U_n^{*}$ and $U_n^{**}$ ($M_n^{*}$ and $M_n^{**}$) are sufficiently close;

• (c) prove the relative compactness of $U_n^{**}$ ($M_n^{**}$).

• a(U) Take $\tau _1=n t_1, \ \tau _2=n t_2$ for $0<t_1<t_2<0$. Then,

  $$\begin{aligned} {{\,\mathrm{\mathbf{E}}\,}}(U^{*}_n(t_2)-U^{*}_n(t_1))^2= & {} {{\,\mathrm{\mathbf{E}}\,}}\Big (\sum \limits _{i=1}^{\infty } (u_i-\overline{u}_i) \Big )^2/\beta (n) = \sum \limits _{i=1}^{\infty } {{\,\mathrm{\mathbf{E}}\,}}(u_i-\overline{u}_i)^2/\beta (n)\\&\quad \le 5 \sum \limits _{i=1}^{\infty } \mathbf {P}(\Pi _i(\tau _2-\tau _1){>}0) /\beta (n){=} 5 {{\,\mathrm{\mathbf{E}}\,}}R_{\Pi (\tau _2{-}\tau _1)}/\beta (n)\\&\quad \le 5 C(\theta ) (t_2-t_1)^{\theta /2}. \end{aligned}$$

  We have used above the independence of the summands, inequality (8) and Lemma 1. Since the covariance function has a limit, [1, Th. 1.4] will imply that the limiting Gaussian process a.s. has a continuous modification on [0, 1]. Since the trajectories of the limiting Gaussian process belong a.s. to the class C(0, 1), the weak convergence in the Skorohod topology implies the weak convergence in the uniform metric, see, e.g., [4]. Therefore, it is sufficient to prove the relative compactness of $\{U^{*}_n\}_{n\ge n_0}$ (with $n_0$ as in Lemma 1) in the Skorohod topology.

• b(U) Since with probability one we have

  $$\begin{aligned} |U(nt)-U([nt])|\le \Pi (nt)-\Pi ([nt])\le \Pi ([nt]+1)-\Pi ([nt]), \end{aligned}$$

  then

  $$\begin{aligned} {{\,\mathrm{\mathbf{E}}\,}}|U(nt)-U([nt])|\le 1. \end{aligned}$$

  Hence, for all $\eta >0$,

  $$\begin{aligned}&\mathbf {P}(\sup \limits _{0\le t\le 1} |U^{*}_n(t)-U^{**}_n(t)|>\eta )\\&\quad \le \mathbf {P}(\sup \limits _{0\le t \le 1} (|U(nt)-U([nt])|+ {{\,\mathrm{\mathbf{E}}\,}}|U(nt)-U([nt])|)>\eta \sqrt{\beta (n)})\\&\quad \le \mathbf {P}(\sup \limits _{0\le t \le 1} (\Pi ([nt]+1)-\Pi ([nt])+ 1)>\eta \sqrt{\beta (n)})\\&\quad = \mathbf {P}(\sup \limits _{0\le m \le n} (\Pi (m+1)-\Pi (m)+1)>\eta \sqrt{\beta (n)})\\&\quad \le \sum \limits _{m=0}^n \mathbf {P}(\Pi (m+1)-\Pi (m)+1>\eta \sqrt{\beta (n)})\\&\quad \le \sum \limits _{m=0}^n \frac{{{\,\mathrm{\mathbf{E}}\,}}e^{\Pi (m+1)-\Pi (m)+1}}{ e^{\eta \sqrt{\beta (n)}}} =(n+1)\frac{{{\,\mathrm{\mathbf{E}}\,}}e^{\Pi (1)}}{e^{\eta \sqrt{\beta (n)}-1}} =(n+1)e^{e-\eta \sqrt{\beta (n)}} \rightarrow 0 \end{aligned}$$

  when $n\rightarrow \infty$. Therefore, it is sufficient to show the relative compactness of $\{U^{**}_n\}_{n\ge n_0}$ (with $n_0$ as in Lemma 1) in the Skorohod topology.

• c(U) For any $t_1, \ t_2 \in [0,1]$ satisfying $\frac{1}{2n}\le t_2-t_1$ we have that

  $$\begin{aligned}{}[nt_2]-[nt_1]\le n(t_2-t_1)+1 \le n(t_2-t_1)+2n(t_2-t_1)= 3n(t_2-t_1) \end{aligned}$$
  $$\begin{aligned} \le 3n(t_2-t_1)\cdot (2n(t_2-t_1))^3=24n^4(t_2-t_1)^4. \end{aligned}$$

  (10)

  Put $k=[16/\theta ]+1$, $\tau _1=[n t_1], \ \tau _2=[n t_2]$.

Recall the Rosenthal inequality [15]: if $\varphi _i$ are independent random variables with ${{\,\mathrm{\mathbf{E}}\,}}\varphi _i=0$, then for all $k\ge 2$ there exists a constant c(k) such that

$$\begin{aligned} {{\,\mathrm{\mathbf{E}}\,}}\Big |\sum _i \varphi _i\Big |^k\le c(k)\max \bigg \{\sum _i{{\,\mathrm{\mathbf{E}}\,}}|\varphi _i|^k, \Big (\sum _i {{\,\mathrm{\mathbf{E}}\,}}\varphi _i^2\Big )^{k/2}\bigg \}. \end{aligned}$$

(11)

For all $n\ge n_0$ (with $n_0$ as in Lemma 1), we then have

$$\begin{aligned}&{{\,\mathrm{\mathbf{E}}\,}}|U^{**}_n(t_2)-U^{**}_n(t_1)|^k=\frac{{{\,\mathrm{\mathbf{E}}\,}}\Big |\sum \limits _{i=1}^{\infty } (u_i-\overline{u}_i)\Big |^k}{(\beta (n))^{k/2}} \\&\quad \le \frac{c(k)}{(\beta (n))^{k/2}} \bigg ( \sum \limits _{i=1}^{\infty } {{\,\mathrm{\mathbf{E}}\,}}| u_i-\overline{u}_i|^{k}+ \Big ( \sum \limits _{i=1}^{\infty } {{\,\mathrm{\mathbf{E}}\,}}( u_i-\overline{u}_i)^2\Big )^{k/2}\bigg )\\&\quad \le \frac{C(k)}{(\beta (n))^{k/2}} \bigg (\sum \limits _{i=1}^{\infty } \mathbf {P}(\Pi _i(\tau _2-\tau _1)>0) + \Big ( \sum \limits _{i=1}^{\infty } \mathbf {P}(\Pi _i(\tau _2-\tau _1)>0) \Big )^{k/2}\bigg )\\&\quad =\frac{C(k)}{(\beta (n))^{k/2}} \left( {{\,\mathrm{\mathbf{E}}\,}}R(\tau _2-\tau _1) + \left( {{\,\mathrm{\mathbf{E}}\,}}R(\tau _2-\tau _1) \right) ^{k/2}\right) \\&\quad \le \frac{C(k)}{(\beta (n))^{k/2}}\left( 24n^4(t_2-t_1)^4+ ({{\,\mathrm{\mathbf{E}}\,}}R(3n(t_2-t_1)))^{k/2}\right) \le \widetilde{C}(\theta )(t_2-t_1)^4, \end{aligned}$$

where c(k), C(k) and $\widetilde{C}(\theta )$ depend only on their arguments.

Above, we have used (11) in the first inequality, (8) in the second and finally (10) and Lemma 1 alongside with the bound

$$\begin{aligned} {{\,\mathrm{\mathbf{E}}\,}}R(\tau _2-\tau _1) \le {{\,\mathrm{\mathbf{E}}\,}}(\Pi ([nt_2])-\Pi ([nt_1]))=[nt_2]-[nt_1]. \end{aligned}$$

(12)

If $0\le t_2-t_1<\frac{1}{n}$, then $[nt_1]=[nt]$ or $[nt_2]=[nt]$ for all $t\in [t_1, t_2]$; therefore,

$$\begin{aligned} D{\mathop {=}\limits ^{\mathrm{def}}}{{\,\mathrm{\mathbf{E}}\,}}(|{U}^{**}_n(t)-{U}^{**}_n(t_1)|^{k/2} |{U}^{**}_n(t_2)-{U}^{**}_n(t)|^{k/2})=0\le (t_2-t_1)^2. \end{aligned}$$

If $t_2-t_1\ge 1/n$, then there are the following three cases:

• 1. if $t_2-t\ge \frac{1}{2n}$, $t-t_1\ge \frac{1}{2n}$, then the Cauchy–Schwarz inequality implies

  $$\begin{aligned} D\le \widetilde{C}(\theta ) (t_2-t)^2\cdot (t-t_1)^2 \le \widetilde{C}(\theta )(t_2-t_1)^2. \end{aligned}$$

• 2. If $t_2-t\ge \frac{1}{2n}$, $t-t_1< \frac{1}{2n}$, then since

  $$\begin{aligned} |U([nt])-U([nt_1])|\le _{\text {a.s.}} \Pi ([nt])-\Pi ([nt_1])\le _{st}\Pi (1), \end{aligned}$$

  the same inequality yields

  $$\begin{aligned} D\le \left( \widetilde{C}(\theta ) (t_2-t)^4 \cdot {{\,\mathrm{\mathbf{E}}\,}}\left( \frac{\Pi (1)+1}{\sqrt{\beta (n)}}\right) ^{k}\right) ^{1/2}\le \widehat{C}(\theta )(t_2-t_1)^2. \end{aligned}$$

• 3. If $t_2-t< \frac{1}{2n}$, $t-t_1\ge \frac{1}{2n}$, then since

  $$\begin{aligned} |U([nt_2])-U([nt])|\le _{\text {a.s.}} \Pi ([nt_2])-\Pi ([nt])\le _{st}\Pi (1), \end{aligned}$$

  we have that

  $$\begin{aligned} D\le \left( {{\,\mathrm{\mathbf{E}}\,}}\left( \frac{\Pi (1)+1}{\sqrt{\beta (n)}}\right) ^{k}\cdot \widetilde{C}(\theta ) (t-t_1)^4\right) ^{1/2}\le \widehat{C}(\theta )(t_2-t_1)^2. \end{aligned}$$

  Now, the relative compactness follows from, for example, [4, Th. 13.5].

• a(M) Because the covariance function has a limit, it is sufficient to appeal to Lemma 3 and [1, Th. 1.4] to establish existence of an almost sure continuous on [0, 1] modification of the limiting Gaussian process. Since the trajectories of this process are a.s. in C(0, 1), the weak convergence in the Skorohod topology implies the uniform convergence, see [4]. Thus, it is sufficient to prove a relative compactness of the family $\{M^{*}_n\}_{n\ge n_0}$ in the Skorohod topology (here, $n_0$ is the same as in Lemma 1).

• b(M) Set $\tau _2=nt$ and $\tau _1=[nt]$. Since $\tau _2-\tau _1\le 1$,

  $$\begin{aligned} {{\,\mathrm{\mathbf{E}}\,}}|M(\tau _2)-M(\tau _1)|\le & {} \sum _{i=1}^{\infty }(\tau _2-\tau _1)p_i e^{-p_i \tau _2} \\&\quad + \tau _1 p_i e^{-p_i \tau _1}(1-e^{-p_i (\tau _2-\tau _1)})\\&\le \sum _{i=1}^{\infty } p_i e^{-p_i \tau _2} + e^{-1}p_i (\tau _2-\tau _1)<\sum _{i=1}^{\infty } 2p_i=2. \end{aligned}$$

Let $m'''_i = m''_i(\tau _1,\tau _1+1)$ and $\overline{m}'''_i={{\,\mathrm{\mathbf{E}}\,}}m'''_i$. Then, we have almost surely

$$\begin{aligned} |M(\tau _2)-M(\tau _1)|\le & {} \sum _{i=1}^{\infty } (m'_i+m''_i) \le \sum _{i=1}^{\infty } (p_i +m'''_i)\\&\quad =1+\sum _{i=1}^{\infty } (m'''_i + \overline{m}'''_i - \overline{m}'''_i) <2+\left| \sum _{i=1}^{\infty } (m'''_i - \overline{m}'''_i) \right| . \end{aligned}$$

We know that for any integer $k\ge 2$

$$\begin{aligned} {{\,\mathrm{\mathbf{E}}\,}}| m'''_i - {{\,\mathrm{\mathbf{E}}\,}}m'''_i |^k<2^k k! \mathbf {P}(\Pi _i(\tau _1+1-\tau _1)>0)=2^k k!(1-e^{-p_i})< 2^k k! p_i. \end{aligned}$$

Using the independence of the terms and Rosenthal inequality, for any $k\ge 2$,

$$\begin{aligned}&{{\,\mathrm{\mathbf{E}}\,}}\left| \sum _{i=1}^{\infty } (m'''_i - \overline{m}'''_i )\right| ^k\\&\quad \le c(k) \left( \sum \limits _{i=1}^{\infty } {{\,\mathrm{\mathbf{E}}\,}}| m'''_i - \overline{m}'''_i |^k+ \left( \sum \limits _{i=1}^{\infty } {{\,\mathrm{\mathbf{E}}\,}}( m'''_i - \overline{m}'''_i )^2\right) ^{k/2}\right) \\&\quad <c(k)(2^kk! + 4^k)=C(k). \end{aligned}$$

Hence, for $k\ge [2/\theta ]+1$ and all $\eta >0$

$$\begin{aligned}&\mathbf {P}\left( \sup \limits _{0\le t\le 1} |M^{*}_n(t)-M^{**}_n(t)|>\eta \right) \\&\quad \le \mathbf {P}(\sup \limits _{0\le t \le 1} (|M(nt)-M([nt])|+ {{\,\mathrm{\mathbf{E}}\,}}|M(nt)-M([nt])|)>\eta \sqrt{\alpha (n)})\\&\quad \le \mathbf {P}\left( \max \limits _{0\le [nt] \le n} \left( \left| \sum _{i=1}^{\infty } m'''_i - {{\,\mathrm{\mathbf{E}}\,}}m'''_i \right| + 4\right)>\eta \sqrt{\alpha (n)}\right) \\&\quad \le \sum \limits _{[nt]=m\in \{0,1,\ldots ,n\}} \mathbf {P}\left( \left| \sum _{i=1}^{\infty } m'''_i - {{\,\mathrm{\mathbf{E}}\,}}m'''_i \right| + 4>\eta \sqrt{\alpha (n)}\right) \\&\quad \le \sum \limits _{m=0}^n \frac{C(k)}{ (\eta \sqrt{\alpha (n)}-4)^k} =\frac{C(k)(n+1)}{ (\eta \sqrt{\alpha (n)}-4)^k} \rightarrow 0 \ \text {when} \ n\rightarrow \infty . \end{aligned}$$

Therefore, it is sufficient to show the local compactness of $\{M^{**}_n\}_{n\ge n_0}$ in the Skorohod topology.

• c(M) Let $t_1, \ t_2 \in [0,1]$ and $\frac{1}{2n}\le t_2-t_1$, then (10) holds. Set $k=[16/\theta ]+1$, $\tau _1=[n t_1], \ \tau _2=[n t_2]$.

Again, by independence and the Rosenthal inequality,

$$\begin{aligned}&{{\,\mathrm{\mathbf{E}}\,}}|M^{**}_n(t_2)-M^{**}_n(t_1)|^{k}=\frac{{{\,\mathrm{\mathbf{E}}\,}}\left| \sum \nolimits _{i=1}^{\infty }(m_i-\overline{m}_i)\right| ^k}{(\alpha (n))^{k/2}} \\&\quad \le \frac{c(k)}{(\alpha (n))^{k/2}} \left( \sum \limits _{i=1}^{\infty } {{\,\mathrm{\mathbf{E}}\,}}| m_i-\overline{m}_i|^{k}+ \left( \sum \limits _{i=1}^{\infty } {{\,\mathrm{\mathbf{E}}\,}}( m_i-\overline{m}_i)^2\right) ^{k/2}\right) \\&\quad \le \frac{C(\beta )}{(\alpha (n))^{k/2}} \left( \sum \limits _{i=1}^{\infty } \mathbf {P}(\Pi _i(\tau _2-\tau _1)>0) + \left( {{\,\mathrm{\mathbf{var}}\,}}(M(\tau _2)-M(\tau _1)) \right) ^{k/2}\right) \\&\quad =\frac{C(k)}{(\alpha (n))^{k/2}} \left( {{\,\mathrm{\mathbf{E}}\,}}R(\tau _2-\tau _1) + \left( {{\,\mathrm{\mathbf{var}}\,}}(M(\tau _2)-M(\tau _1)) \right) ^{k/2}\right) \\&\quad \le \frac{C(k)}{(\alpha (n))^{k/2}}\left( 24n^4(t_2-t_1)^4+ (C(\theta ) \alpha (n)(\tau _2-\tau _1)/n)^{k/2}\right) \le \widetilde{C}(\theta )(t_2-t_1)^4, \end{aligned}$$

where c(k), C(k) and $\widetilde{C}(\theta )$ depend only on their arguments.

Above, we have used inequalities (9), (10) and Lemmas 3, 1 alongside with the bound

$$\begin{aligned} {{\,\mathrm{\mathbf{E}}\,}}R(\tau _2-\tau _1) \le {{\,\mathrm{\mathbf{E}}\,}}(\Pi ([nt_2]-[nt_1]))=[nt_2]-[nt_1]. \end{aligned}$$

When $0\le t_2-t_1<\frac{1}{n}$, then $[nt_1]=[nt]$ or $[nt_2]=[nt]$ for any $t\in [t_1, t_2]$. Thus,

$$\begin{aligned} B{\mathop {=}\limits ^{\mathrm{def}}}{{\,\mathrm{\mathbf{E}}\,}}(|{M}^{**}_n(t)-{M}^{**}_n(t_1)|^{k/2} |{M}^{**}_n(t_2)-{M}^{**}_n(t)|^{k/2})=0\le (t_2-t_1)^2. \end{aligned}$$

When $t_2-t_1\ge 1/n$, we have the following three cases:

• 1. if $t_2-t\ge \frac{1}{2n}$, $t-t_1\ge \frac{1}{2n}$, then the Cauchy–Schwarz inequality gives

  $$\begin{aligned} B\le \widetilde{C}(\theta ) (t_2-t)^2\cdot (t-t_1)^2 \le \widetilde{C}(\theta )(t_2-t_1)^2; \end{aligned}$$

• 2. if $t_2-t\ge \frac{1}{2n}$, $t-t_1< \frac{1}{2n}$, then since for any $l\ge 2$,

  $$\begin{aligned}&{{\,\mathrm{\mathbf{E}}\,}}|M([nt])-M([nt_1])-{{\,\mathrm{\mathbf{E}}\,}}(M([nt])-M([nt_1])|^l\\&\quad \le {{\,\mathrm{\mathbf{E}}\,}}\left( 4+\left| \sum _{i=1}^{\infty } m''_i([nt_1]+1,[nt_1]) - {{\,\mathrm{\mathbf{E}}\,}}m''_i([nt_1]+1,[nt_1]) \right| \right) ^l< C(l), \end{aligned}$$

  the Cauchy–Schwarz inequality yields the bound

  $$\begin{aligned} B\le \left( \widetilde{C}(\theta ) (t_2-t)^4 \cdot \frac{C(k)}{\alpha (n)^{k/2}}\right) ^{1/2}\le \widehat{C}(\theta )(t_2-t_1)^2; \end{aligned}$$

• 3. finally, $t_2-t< \frac{1}{2n}$, $t-t_1\ge \frac{1}{2n}$, is similar to the previous case.

Thus, the required compactness follows from [4, Th. 13.5].

Finally, for the next step we need to show that M(s), when time scaled, is close to its fully Poissonized version

$$\begin{aligned} \widetilde{M}(s){\mathop {=}\limits ^{\mathrm{def}}}M_{\Pi (s)}=\sum _{i=1}^\infty \Pi (s) p_i\mathbb {1}_{\Pi _i(s)=0}. \end{aligned}$$

Namely, we aim to show that

$$\begin{aligned} \sup \limits _{0\le t\le 1} |M^{*}_n(t)-\widetilde{M}_n(t)|\rightarrow 0\quad \text {in probability,} \end{aligned}$$

(13)

where

$$\begin{aligned} \widetilde{M}_n(t)=\frac{\widetilde{M}(nt) -{{\,\mathrm{\mathbf{E}}\,}}\widetilde{M}(nt)}{(\alpha (n))^{1/2}}. \end{aligned}$$

Introduce $\Pi '_i(s)=\Pi (s)-\Pi _i(s)$ and $\widetilde{\Pi }(s)=(\Pi (s)-s)/\sqrt{s}$. Since $\widetilde{M}(s)=\sum _{i=1}^\infty \Pi '_i(s) p_i \mathbb {1}_{\Pi _i(s)=0}$,

$$\begin{aligned}&|{{\,\mathrm{\mathbf{E}}\,}}\widetilde{M}(s)-{{\,\mathrm{\mathbf{E}}\,}}M(s)|=|{{\,\mathrm{\mathbf{E}}\,}}\sum _{i=1}^\infty (\Pi '_i(s)-s) p_i \mathbb {1}_{\Pi _i(s)=0}| \\&\quad = \Big |\sum _{i=1}^\infty (s(1-p_i)-s) p_i e^{-s p_i}\Big |=\frac{2 {{\,\mathrm{\mathbf{E}}\,}}R_{\Pi (s),2}}{s}\rightarrow 0 \end{aligned}$$

as $s\rightarrow \infty$ and it is bounded by 1. Thus, there exists a sufficiently small $\varepsilon =\varepsilon (\theta )>0$ such that for $\delta _n=n^{\varepsilon -1}$

$$\begin{aligned} \sup \limits _{0\le t\le \delta _n} |M^{*}_n(t)-\widetilde{M}_n(t)|< \frac{\Pi (n\delta _n)+n\delta _n+1}{(\alpha (n))^{1/2}} \rightarrow 0\ \text {a.s.} \end{aligned}$$

when $n\rightarrow \infty$.

By the strong law of large numbers for M(s) and the well-known asymptotic behavior of ${{\,\mathrm{\mathbf{E}}\,}}M(s)$ (see, e.g., [12][Eq. (23)]), we conclude that for any $\theta \in (0,1]$, $M(s)/(s \alpha (s))^{1/2}\rightarrow 0$ a.s. when $s\rightarrow \infty$. Moreover, according to the central limit theorem $\widetilde{\Pi }(s)$ is asymptotically standard normal for large s.

Finally, we have almost surely

$$\begin{aligned} |M^{*}_n(t)-\widetilde{M}_n(t)|\le \frac{|\widetilde{\Pi }(nt)| M(nt)}{(nt \alpha (n))^{1/2}}+\frac{1}{(\alpha (n))^{1/2}}. \end{aligned}$$

Using this inequality, and the fact that $\sup \nolimits _{0\le t\le 1}(\cdot )\le \sup \nolimits _{0\le t\le \delta _n}(\cdot )+\sup \nolimits _{\delta _n\le t\le 1}(\cdot )$ and that $\sup \limits _{0\le t\le 1}(\cdot )$ is a continuous functional, we readily obtain 13.

Step 4: Approximation of the initial process Since $\Pi (t)$ is monotone, the strong law of large numbers implies that for any $\varepsilon , \delta \in (0,1)$ there is an integer $N=N(\varepsilon ,\delta )$ such that for all $n\ge N$ one has

$$\begin{aligned} \mathbf {P}(\forall t\in [0,1] \ \ \exists \tau : |\tau -t|\le \delta , \ \Pi (n\tau )= [nt]) {\mathop {=}\limits ^{\mathrm{def}}}\mathbf {P}(A(n))\ge 1-\varepsilon , \end{aligned}$$

see Lemma 2. Here and below, F stands for R, U or M. The relative compactness of the distributions $\{F_n^{*}\}_{n\ge n_0}$ implies that for any $\varepsilon \in (0,1)$ and $\eta >0$ there exist $\delta \in (0,1)$ and an integer $N_1=N_1(\varepsilon ,\eta )$ such that for all $n\ge N_1$,

$$\begin{aligned} \mathbf {P}\left( \sup \limits _{|t-\tau | \le \delta } \left| F^{*}_n(\tau )-F^{*}_n(t)\right| \ge \eta \right) \le \varepsilon . \end{aligned}$$

Hence, since

$$\begin{aligned} \mathbf {P}(F_n(t)=F^{*}_n(\tau )|\Pi (n\tau )=[nt])=1, \end{aligned}$$

for all $n\ge \max (N,N_1)$,

$$\begin{aligned}&\mathbf {P}\left( \sup \limits _{0 \le t \le 1} \left| F_n(t)-F^{*}_n(t)\right| \ge \eta \right) \\&\quad \le \mathbf {P}\left( \sup \limits _{0 \le t \le 1} \left| F_n(t)-F^{*}_n(t)\right| \ge \eta , A(n)\right) +\varepsilon \\&\quad \le \mathbf {P}\left( \sup \limits _{|t-\tau | \le \delta } \left| F^{*}_n(\tau )-F^{*}_n(t)\right| \ge \eta \right) +\varepsilon \le 2 \varepsilon . \end{aligned}$$

which proves Theorem 1.

Appendix

An explicit expression for the covariance between $R(\tau )$ and R(t) can be found in [7]. Take $\tau \le t$. The

$$\begin{aligned} c^{*}_{UU} (\tau ,t)= & {} {{\,\mathrm{\mathbf{cov}}\,}}(U(\tau ), U(t)) \\= & {} \sum _{k=1}^{\infty } \mathbf {P}(\Pi _k(\tau ), \Pi _k(t)\ \text {is odd} )- \mathbf {P}(\Pi _k(\tau )\ \text {is odd}) \mathbf {P}(\Pi _k(t)\ \text {is odd})\\= & {} \frac{1}{4} \sum _{k=1}^{\infty } \bigg ( (1-e^{-2p_k\tau })(1+e^{-2p_k(t-\tau )})-(1-e^{-2p_k\tau })(1-e^{-2p_kt}) \bigg )\\= & {} \frac{1}{4} \sum _{k=1}^{\infty } e^{-2p_k(t-\tau )}-e^{-2p_k(t+\tau )} =\frac{1}{2}{{\,\mathrm{\mathbf{E}}\,}}(U(t+\tau )-U(t-\tau )). \end{aligned}$$

Hence (since $\frac{\beta (nt)}{\beta (n)} \rightarrow t^{\theta }$ as $n \rightarrow \infty$)

$$\begin{aligned} c_{\upsilon \upsilon }(\tau ,t)&=\lim _{n\rightarrow \infty } \frac{c_{UU}^{*}(n\tau , nt)}{\alpha (n)} =\Gamma (1-\theta )2^{\theta -2}((t+\tau )^\theta -(t-\tau )^\theta ), \theta \in (0,1), \\ c_{\upsilon \upsilon }(\tau ,t)&=\lim _{n\rightarrow \infty } \frac{c_{UU}^{*}(n\tau , nt)}{n L^{*}(n)} =2\tau , \ \theta =1, \end{aligned}$$

cf. [12][Eq. (21)].

Next,

$$\begin{aligned} c_{MM}^{*} (\tau ,t)= & {} {{\,\mathrm{\mathbf{cov}}\,}}(M(\tau ), M(t))\\= & {} \sum _{k=1}^{\infty } {{\,\mathrm{\mathbf{E}}\,}}(tp_i \mathbb {1}(\Pi _i(t)=0)- tp_i e^{-tp_i}) (\tau p_i \mathbb {1}(\Pi _i(\tau )=0)- \tau p_i e^{-\tau p_i})\\= & {} \sum _{k=1}^{\infty } t\tau p_i^2 e^{-tp_i}(1-e^{-\tau p_i}) =\frac{2\tau }{t}{{\,\mathrm{\mathbf{E}}\,}}R_{\Pi (t),2}-\frac{2t \tau }{(t+\tau )^2}{{\,\mathrm{\mathbf{E}}\,}}R_{\Pi (t+\tau ),2}. \end{aligned}$$

Since $\frac{\alpha (nt)}{\alpha (n)} \rightarrow t^{\theta }$ when $n \rightarrow \infty$,

$$\begin{aligned} c_{\mu \mu }(\tau ,t)= & {} \lim _{n\rightarrow \infty } \frac{c_{MM}^{*}(n\tau , nt)}{\alpha (n)}\\= & {} \theta \Gamma (2-\theta )\left( \frac{\tau }{t^{1-\theta }} - \frac{t\tau }{(t+\tau )^{2-\theta }} \right) , \end{aligned}$$

cf. [12][Eq. (23)].

Continuing,

$$\begin{aligned} c^{*}_{RU} (\tau ,t)= & {} {{\,\mathrm{\mathbf{cov}}\,}}(R(\tau ), U(t)) = \sum _{k=1}^{\infty } {{\,\mathrm{\mathbf{cov}}\,}}(1-\mathbb {1}(\Pi _k(\tau )=0), \mathbb {1}(\Pi _k(t)\ \text {is odd}))\\= & {} \quad -\sum _{k=1}^{\infty } {{\,\mathrm{\mathbf{cov}}\,}}(\mathbb {1}(\Pi _k(\tau )=0), \mathbb {1}(\Pi _k(t)\ \text {is odd}))\\= & {} \quad -\sum _{k=1}^{\infty } \mathbf {P}(\Pi _k(\tau )=0, \Pi _k(t)\ \text {is odd} )- \mathbf {P}(\Pi _k(\tau )=0) \mathbf {P}(\Pi _k(t)\ \text {is odd})\\= & {} -\frac{1}{2} \sum _{k=1}^{\infty } \bigg ( e^{-p_k\tau }(1-e^{-2p_k(t-\tau )})- e^{-p_k\tau }(1-e^{-2p_kt}) \bigg )\\= & {} \frac{1}{2}\sum _{k=1}^{\infty } \bigg ( e^{-p_k(2t-\tau )} - e^{-p_k(2t+\tau )}\pm 1\bigg ) =\frac{1}{2}{{\,\mathrm{\mathbf{E}}\,}}(R(2t+\tau )-R(2t-\tau )). \end{aligned}$$

Similarly,

$$\begin{aligned} c^{*}_{RU}(t,\tau )= & {} {{\,\mathrm{\mathbf{cov}}\,}}(R(t), U(\tau )) = -\sum _{k=1}^{\infty } {{\,\mathrm{\mathbf{cov}}\,}}(\mathbb {1}(\Pi _k(t)=0), \mathbb {1}(\Pi _k(\tau )\ \text {is odd}))\\= & {} \frac{1}{2} \sum _{k=1}^{\infty } e^{-p_kt}(1-e^{-2p_k\tau })\\= & {} \frac{1}{2} \sum _{k=1}^{\infty } \bigg ( e^{-p_k t} - e^{-p_k(2\tau +t)}\pm 1\bigg )\\= & {} \frac{1}{2}{{\,\mathrm{\mathbf{E}}\,}}(R(2t+\tau )-R(t)). \end{aligned}$$

Because $\frac{\beta (nt)}{\beta (n)} \rightarrow t^{\theta }$ when $n \rightarrow \infty$, for $\theta \in (0,1)$ we have that

$$\begin{aligned} c_{\rho \upsilon }(\tau ,t)&=\lim _{n\rightarrow \infty } \frac{c_{RU}^{*}(n\tau , nt)}{\alpha (n)} =\Gamma (1-\theta )((2t+\tau )^{\theta } - (2t-\tau )^{\theta })/2,\\ c_{\rho \upsilon }(t,\tau )&=\lim _{n\rightarrow \infty } \frac{c^{*}_{RU}(nt, n\tau )}{\alpha (n)}=\Gamma (1-\theta )((2t+\tau )^\theta - t^\theta )/2. \end{aligned}$$

For $\theta =1$, this reduces to

$$\begin{aligned} c_{\rho \upsilon }(\tau ,t)&=\lim _{n\rightarrow \infty } \frac{c_{RU}^{*}(n\tau , nt)}{n L^{*}(n)}=\tau ,\\ c_{\rho \upsilon }(t,\tau )&=\lim _{n\rightarrow \infty } \frac{c^{*}_{RU}(nt, n\tau )}{nL^{*}(n)}=(t+\tau )/2, \end{aligned}$$

cf. [12, Th. 1].

Next,

$$\begin{aligned} c^{*}_{MU} (\tau ,t)= & {} {{\,\mathrm{\mathbf{cov}}\,}}(M(\tau ), U(t))\\= & {} \sum _{k=1}^{\infty } \tau p_k {{\,\mathrm{\mathbf{cov}}\,}}(\mathbb {1}(\Pi _k(\tau )=0), \mathbb {1}(\Pi _k(t)\ \text {is odd}))\\= & {} \frac{1}{2} \sum _{k=1}^{\infty }\tau p_k \bigg ( e^{-p_k(2t+\tau )}-e^{-p_k(2t-\tau )}\bigg )\\= & {} \frac{\tau }{2(2t+\tau )}{{\,\mathrm{\mathbf{E}}\,}}M(2t+\tau )-\frac{\tau }{2(2t-\tau )}{{\,\mathrm{\mathbf{E}}\,}}M(2t-\tau ), \end{aligned}$$

and

$$\begin{aligned}&c^{*}_{MU} (t,\tau )={{\,\mathrm{\mathbf{cov}}\,}}(M(t), U(\tau )) \\&\quad = \frac{1}{2} \sum _{k=1}^{\infty }t p_k \bigg (e^{-p_k(2\tau +t)}-e^{-p_k t}\bigg )\\&\quad = \frac{t}{2(2\tau +t)}{{\,\mathrm{\mathbf{E}}\,}}M(2\tau +t)-\frac{1}{2}{{\,\mathrm{\mathbf{E}}\,}}M(t). \end{aligned}$$

Finally,

$$\begin{aligned} c^{*}_{RM} (\tau ,t)= & {} {{\,\mathrm{\mathbf{cov}}\,}}(R(\tau ), M(t))\\= & {} \sum _{k=1}^{\infty } {{\,\mathrm{\mathbf{cov}}\,}}(1-\mathbb {1}(\Pi _k(\tau )=0), t p_k\mathbb {1}(\Pi _k(t)=0))\\= & {} -\sum _{k=1}^{\infty } t p_k {{\,\mathrm{\mathbf{cov}}\,}}(\mathbb {1}\{\Pi _k(\tau )=0\}, \mathbb {1}\{\Pi _k(t)=0)\}\\= & {} -\sum _{k=1}^{\infty }t p_k \bigg ( e^{-p_k t}- e^{-p_k(\tau +t)} \bigg ) =\frac{t}{\tau +t}{{\,\mathrm{\mathbf{E}}\,}}M(\tau +t)-{{\,\mathrm{\mathbf{E}}\,}}M(t), \end{aligned}$$

and

$$\begin{aligned} c^{*}_{RM} (t,\tau ) ={{\,\mathrm{\mathbf{cov}}\,}}(R(t), M(\tau )) =\frac{\tau }{\tau +t}{{\,\mathrm{\mathbf{E}}\,}}M(\tau +t)-\frac{\tau }{t}{{\,\mathrm{\mathbf{E}}\,}}M(t). \end{aligned}$$

Because $\frac{\alpha (nt)}{\alpha (n)} \rightarrow t^{\theta }$ when $n \rightarrow \infty$, for $\theta \in (0,1)$ we obtain

$$\begin{aligned} c_{\rho \mu }(\tau ,t)&=\lim _{n\rightarrow \infty } \frac{c^{*}_{RM} (n\tau , nt)}{\alpha (n)}=\theta \Gamma (1-\theta ) \left( \frac{t}{(t+\tau )^{1-\theta }} - t^\theta \right) , \\ c_{\rho \mu }(t,\tau )&=\lim _{n\rightarrow \infty } \frac{c^{*}_{RM} (nt, n\tau )}{\alpha (n)} =\theta \Gamma (1-\theta )\left( \frac{\tau }{(t+\tau )^{1-\theta }} -\frac{\tau }{ t^{1-\theta }} \right) , \\ c_{\mu \upsilon }(\tau ,t)&=\lim _{n\rightarrow \infty } \frac{c^{*}_{MU} (n\tau , nt)}{\alpha (n)} =\theta \Gamma (1-\theta )\left( \frac{\tau }{2(2t+\tau )^{1-\theta }} -\frac{\tau }{2(2t-\tau )^{1-\theta }} \right) , \\ c_{\mu \upsilon }(t,\tau )&=\lim _{n\rightarrow \infty } \frac{c^{*}_{MU} (nt, n\tau )}{\alpha (n)} =\theta \Gamma (1-\theta )\left( \frac{t}{2(2\tau +t)^{1-\theta }} -\frac{t^\theta }{2} \right) , \end{aligned}$$

cf. [12][Eq. (23)].

Clearly, $L(n)\rightarrow 0$ as $n\rightarrow \infty$. According to [12][Lem. 4], in the case $\theta =1$ the function $L^{*}(n)\rightarrow 0$ when $n\rightarrow \infty$ is slowly varying and

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{L(n)}{L^{*}(n)}{\mathop {=}\limits ^{\mathrm{def}}}\lim _{n\rightarrow \infty }\delta _n=0. \end{aligned}$$

(14)

Therefore, in the case $\theta =1$,

$$\begin{aligned} c_{\rho \mu }(\tau ,t)&=\lim _{n\rightarrow \infty } \frac{c^{*}_{RM} (n\tau , nt)}{\alpha (n)} \sqrt{\delta _n}=0,\\ \quad c_{\rho \mu }(t,\tau )&=\lim _{n\rightarrow \infty } \frac{c^{*}_{RM} (nt, n\tau )}{\alpha (n)}\sqrt{\delta _n}=0, \\ c_{\mu \upsilon }(\tau ,t)&=\lim _{n\rightarrow \infty } \frac{c^{*}_{MU} (n\tau , nt)}{\alpha (n)} \sqrt{\delta _n}=0, \\ \quad c_{\mu \upsilon }(t,\tau )&=\lim _{n\rightarrow \infty } \frac{c^{*}_{MU} (nt, n\tau )}{\alpha (n)} \sqrt{\delta _n}=0. \end{aligned}$$