Introduce the following notation. Let *𝓔*_{n –1} denote the -algebra of observable events after the (*n* – 1)th step. Let *V*_{n} denote the number of vertices after the *n*th step. Let *j* ≥ 0 and 1 ≤ *M* ≤ *N* be fixed integers. Let *I*[*n*, *M*, *j*] be the indicator of the event that the *j* th *M*-clique exists after *n* steps, that is

$$I[n,M,j]=\{\begin{array}{l}1,\text{if}\text{\hspace{0.17em}}W[n,M,j]>0,\hfill \\ 0,\text{if}\text{\hspace{0.17em}}W[n,M,j]=0.\hfill \end{array}$$

Let *J*[*n*, *M*, *j*] be the indicator of the event that the *j* th *M*-clique is born at the *n*th step. Then *J*[*n*, *M*, *j*] = *I*[*n*, *M*, *j*] – *I*[*n* – 1, *M*, j]. For all fixed positive integers *j*, *k*, *l*, *M*, 0 ≤ *j* ≤ *l*, 1 ≤ *k*, 1 ≤ *M* ≤ *N*, we consider the following sequences:

$$b[n,M,k]={\displaystyle \prod _{i=1}^{n}{\left(1+\frac{{\alpha}^{\prime}k}{i}\right)}^{-1},}$$(15)

$$d[n,M,k,j]=-{\displaystyle \sum _{i=1}^{n-1}b[i+1,M,k]\frac{\left(\begin{array}{c}N\\ M\end{array}\right)}{N\left(\begin{array}{c}{V}_{i}\\ M\end{array}\right)}p{\beta}^{\prime}\left(\begin{array}{c}W[i,M,j]+k-1\\ k-1\end{array}\right),}$$(16)

$${e}_{n,M}={\displaystyle \prod _{i=1}^{n}{\left(1-\frac{{\alpha}^{\prime}}{i}\right)}^{-1}}$$(17)

We can see that *b*[*n*, *M*, *k*] and *e*_{n}_{,M} are deterministic, while *d*[*n*, *M*, *k*, j] is an *𝓔*_{n – 1}-measurable random variable for any *n*, *M*, *k* and *j*. Using the definition of *b*[*n*, *M*, *k*] and the Stirling-formula for the Gamma function, we can show that

$$b[n,M,k]\sim {b}_{M,k}{n}^{-k{\alpha}^{\prime}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{as}\phantom{\rule{thinmathspace}{0ex}}n\to \mathrm{\infty},$$(18)

where *b*_{M}_{,k} = Γ(1 + *α*′*k*) > 0, *k* and *M* are fixed. Moreover, we can easily see that

$${e}_{n,M}\sim \mathrm{\Gamma}(1-{\alpha}^{\prime}){n}^{{\alpha}^{\prime}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{as}\phantom{\rule{thinmathspace}{0ex}}n\to \mathrm{\infty}.$$(19)

In the following lemma we introduce a martingale which will play an important role in the paper.

Let *j*, *k*, *l*, *M*, 0 ≤ *j* ≤ *l*, 1 ≤ *M* ≤ *N be fixed nonnegative integers and let*

$$Z[n,M,k,j]=\left(b[n,M,k]\left(\begin{array}{c}W[n,M,j]+k-1\\ k\end{array}\right)+d[n,M,k,j]\right)I[l,M,j].$$(20)

Then (*Z*[*n*, *M*, *k*, *j*]; *𝓔*_{n}) *is a martingale for n* ≥ *l*.

*Proof*. At each step, the weight of a fixed *M*-clique is increased by 1 if and only if it takes part in an interaction. The total weight of *N*-cliques after n steps is *n* + 1. The total weight of (*N* – 1)-cliques after *n* steps is *N*(*n* + 1). When a new vertex is born and we choose *N* – 1 vertices uniformly, the probability that the vertices of a given *M*-clique are selected is

$$\frac{\left(\begin{array}{c}{V}_{n}-M\\ N-1-M\end{array}\right)}{\left(\begin{array}{c}{V}_{n}\\ N-1\end{array}\right)}=\frac{\left(\begin{array}{c}N-1\\ M\end{array}\right)}{\left(\begin{array}{c}{V}_{n}\\ M\end{array}\right)}.$$

When we choose *N* vertices uniformly at random, the probability that the vertices of a given *M*-clique are chosen is

$$\frac{\left(\begin{array}{c}{V}_{n}-M\\ N-M\end{array}\right)}{\left(\begin{array}{c}{V}_{n}\\ N\end{array}\right)}=\frac{\left(\begin{array}{c}N\\ M\end{array}\right)}{\left(\begin{array}{c}{V}_{n}\\ M\end{array}\right)}.$$

Therefore, as in [15], it is easy to show that the probability that the j th *M*-clique takes part in interaction at step (*n* + 1) is

$$\begin{array}{c}pr\frac{(N-M)W[n,M,j]}{N(n+1)}+p(1-r)\frac{\left(\begin{array}{c}N-1\\ M\end{array}\right)}{\left(\begin{array}{c}{V}_{n}\\ M\end{array}\right)}+(1-p)q\frac{W[n,M,j]}{n+1}+(1-p)(1-q)\frac{\left(\begin{array}{c}N\\ M\end{array}\right)}{\left(\begin{array}{c}{V}_{n}\\ M\end{array}\right)}=\\ =\frac{W[n,M,j]}{n+1}{\alpha}^{\prime}+\frac{\left(\begin{array}{c}N\\ M\end{array}\right)}{N\left(\begin{array}{c}{V}_{n}\\ M\end{array}\right)}p{\beta}^{\prime},\end{array}$$(21)

provided that the *j* th *M*-clique exists at the *l*th step. Using this fact, we can see for *n* ≥ l

$$\begin{array}{c}E\left\{\left(\begin{array}{c}W[n+1,M,j]+k-1\\ k\end{array}\right)I[l,M,j]|{\mathcal{F}}_{n}\right\}=\\ =I[l,M,j]\left(1-\left(\frac{W[n,M,j]}{n+1}{\alpha}^{\prime}+\frac{\left(\begin{array}{c}N\\ M\end{array}\right)}{N\left(\begin{array}{c}{V}_{n}\\ M\end{array}\right)}p{\beta}^{\prime}\right)\right)\left(\left(\begin{array}{c}W[n,M,j]+k-1\\ k\end{array}\right)\right)+\\ +I[l,M,j]\left(\frac{W[n,M,j]}{n+1}{\alpha}^{\prime}+\frac{\left(\begin{array}{c}N\\ M\end{array}\right)}{N\left(\begin{array}{c}{V}_{n}\\ M\end{array}\right)}p{\beta}^{\prime}\right)\left(\begin{array}{c}W[n,M,j]+k\\ k\end{array}\right)=\\ =I[l,M,j]\frac{\left(\begin{array}{c}N\\ M\end{array}\right)}{\left(\begin{array}{c}{V}_{n}\\ N\end{array}\right)}p{\beta}^{\prime}\left(\begin{array}{c}W[n,M,j]+k-1\\ k-1\end{array}\right)+I[l,M,j]\left(1+{\alpha}^{\prime}\frac{k}{n+1}\right)\left(\begin{array}{c}W[n,M,j]+k-1\\ k\end{array}\right).\end{array}$$

Multiplying both sides by *b*(*n* + 1, *M*, *k*, we see that

$$\begin{array}{c}E\left\{b[n+2,M,k]\left(\begin{array}{c}W[n+1,M,j]+k-1\\ k\end{array}\right)I[l,M,j]|{\mathcal{F}}_{n}\right\}=\\ =I[l,M,j]\left(\left(\begin{array}{c}W[n,M,j]+k-1\\ k\end{array}\right)b[n,M,k]+d[n,M,k,j]-d[n+1,M,k,j]\right).\end{array}$$(22)

Using that *d*[*n* + 1, *M*, *k*, *j*] is *𝓔*_{n}-measurable, we obtain the desired result. □

*The following sequence is a nonnegative supermartingale*

$$\left(\frac{{e}_{n,M}I[k,M,j]}{W[n,M,j]-1}|{\mathcal{F}}_{n}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n=j,j+1,\mathrm{...}\text{\hspace{0.17em}}.$$(23)

*Proof*. In a similar way as in the proof of Lemma 3.1, we have for *n* ≥ *k*

$$\begin{array}{c}E\left\{\frac{I[k,M,j]}{W[n+1,M,j]-1}|{\mathcal{F}}_{n}\right\}=\\ =\left(\frac{W[n,M,j]}{n+1}{\alpha}^{\prime}+\frac{\left(\begin{array}{c}N\\ M\end{array}\right)}{N\left(\begin{array}{c}{V}_{n}\\ M\end{array}\right)}p{\beta}^{\prime}\right)\frac{I[k,M,j]}{W[n,M,j]}+\left(1-\left(\frac{W[n,M,j]}{n+1}{\alpha}^{\prime}+\frac{\left(\begin{array}{c}N\\ M\end{array}\right)}{N\left(\begin{array}{c}{V}_{n}\\ M\end{array}\right)}p{\beta}^{\prime}\right)\right)\frac{I[k,M,j]}{W[n,M,j]-1}=\\ =\left(\frac{W[n,M,j]}{n+1}{\alpha}^{\prime}+\frac{\left(\begin{array}{c}N\\ M\end{array}\right)}{N\left(\begin{array}{c}{V}_{n}\\ M\end{array}\right)}p{\beta}^{\prime}\right)\left(\frac{I[k,M,j]}{W[n,M,j]}-\frac{I[k,M,j]}{W[n,M,j]-1}\right)+\frac{I[k,M,j]}{W[n,M,j]-1}\le \\ \le -\frac{{\alpha}^{\prime}I[k,M,j]}{(n+1)(W[n,M,j]-1)}+\frac{I[k,M,j]}{W[n,M,j]-1}=\frac{I[k,M,j]}{W[n,M,J]-1}\left(1-\frac{{\alpha}^{\prime}}{n+1}\right).\end{array}$$(24)

Multiplying both sides of (24) by *e*_{n}_{+1,M}, we obtain the result. □

The proof contains two parts. First, we will show that the result is valid with non-negative *γ*_{M, j}. Then we will show that *γ*_{M, j} is positive with probability 1.

Let *B*_{n}_{+1} = {*W*[*n* + 1, *M*, *j*] = *W*[*n*, *M*, *j*] + 1}. Consider the event that the *j*th *M*-clique exists after *n* steps. On this event, by (21),

$$\text{P}\left({B}_{n+1}|{\mathcal{F}}_{n}\right)\ge \frac{{\alpha}^{\prime}}{n+1}.$$(25)

The sequence (*B*_{n}, *n* ∈ ℕ) is adapted to the sequence of *σ*-algebras (*𝓔*_{n}; *n* ∈ ℕ). Using Corollary VII-2-6 of [19] and (25), we have

$$W[n,M,j]\to \infty \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}\text{.s}\text{.}\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}\to \infty .$$(26)

Consider the martingale (*Z*[*n*, *M*, *k*, *j*], *𝓔*_{n}) introduced in Lemma 3.1 and let *k* = 1. Then

$$Z[n,M,j]=(b[n,M,1]W[n,M,j]+d[n,M,j])I[l,M,j].$$(27)

By the Marcinkiewicz strong law of large numbers, we have

$${V}_{n}=pn+o\left({n}^{1/2+\epsilon}\right)$$(28)

almost surely, for any *ɛ* > 0. By this fact and (18), we obtain that

$$d[n,M,1,j]=-\sum _{i=1}^{n-1}b[i+1,M,1]\frac{\left(\begin{array}{c}N\\ M\end{array}\right)}{N\left(\begin{array}{c}{V}_{i}\\ M\end{array}\right)}p{\beta}^{\prime}\sim -\frac{1}{{p}^{M-1}}\left(\begin{array}{c}N\\ M\end{array}\right)\frac{M!}{N}{\beta}^{\prime}\mathrm{\Gamma}(1+{\alpha}^{\prime})\sum _{i=1}^{n-1}{i}^{-({\alpha}^{\prime}+M)}(1+\text{o(1))}\text{.}$$

Using that *α*′ > 0 and *M* > 0, we see that *d*[*n*, *M*, 1, *j*] converges as *n* → ∞. Therefore the martingale *Z*[*n*, *M*, *1*, *j*] is bounded from below. Now, we shall see that the martingale *Z*[*n*, *M*, 1, *j*] has bounded differences. The sequence *b*[*n*, *M*, 1] is decreasing, hence

$$Z[n+1,M,1,j]-Z[n,M,1,j]\le b[n,M,1](W[n+1,M,j]-W[n,M,j])\le 1.$$

It is also easy to compute that

$$\begin{array}{c}Z[n,M,1,j]-Z[n+1,M,1,j]\le (b[n,M,1]-b[n+1,M,1])W[n,M,j]+(d[n,M,1,j]-d[n+1,M,1,j])=\\ =(b[n,M,1]-b[n+1,M,1])W[n,M,j]+b[n+1,M,1]\frac{\left(\begin{array}{c}N\\ M\end{array}\right)}{N\left(\begin{array}{c}{V}_{n}\\ M\end{array}\right)}p{\beta}^{\prime}\le \\ \le b[n+1,M,1]\left({\alpha}^{\prime}+\frac{1}{N}p{\beta}^{\prime}\right)\le {\alpha}^{\prime}+p\frac{{\beta}^{\prime}}{N}.\end{array}$$

So the martingale *Z*[*n*, *M*, 1, *j*] is bounded from below and it has bounded differences. Therefore, by Proposition VII-3-9 of [19], it is convergent almost surely as *n* → ∞. By the definition of *Z*[*n*, *M*, 1; *j*], we see that *b*[*n*, *M*, 1]*W*[*n*, *M*, *j*] also converges almost surely on the event {*W*[*l*, *M*, *j*] > 0}. This fact, (26) and (18) imply that (8) is true with non-negative *γ*_{M}_{,j}.

Now we will show that *γ*_{M}_{,j} is positive with probability 1. Consider the supermartingale $\left(\frac{{e}_{n,M}I[k,M,j]}{W[n,M,j]-1},{\mathcal{F}}_{n}\right)$, *n* ≥ *j*, in Lemma 3.2. This supermartingale is nonnegative therefore, according to the submartingale convergence theorem, it converges almost surely. lim_{t→∞} *I*[*l*, *M*, *j*] = 1 almost surely, hence $\frac{{e}_{n,M}}{W[n,M,j]-1}$ also converges almost surely as *n* → ∞. This and (19) imply that *γ*_{M}_{,j} is positive almost surely. □

*Let j* ≥ 0 *be fixed. For n* ≥ *k*, *we have*

$$\begin{array}{ll}I[k,1,j]\left(D[n,j]+{\alpha}_{2}\frac{W[n,1,j]}{n+1}\right)\hfill & \le \text{E}\{I[k,1,j]D[n+1,j]|{\mathcal{F}}_{n}\}\hfill \\ \hfill & =I[k,1,j]\left(D[n,j]+{\alpha}_{2}\frac{W[n,1,j]}{n+1}+{R}_{n}\right).\hfill \end{array}$$

*where* $0\le {R}_{n}\le (N-1)\frac{p\beta}{{V}_{n}}.$

*Proof*. Consider the conditional expectation E{*D*[*n* + 1, *j*] – *D*[*n*, *j*]|*𝓔*_{n}} provided that the *j*th vertex exists after *k* steps.

When the model evolves according to the preferential attachment rule the degree of a fixed vertex can be increased by 0 or 1 at each step. Therefore, the expected growth of the degree of the *j*th vertex in the (*n* +1)th step when the choice is PA is $\frac{W(n,1,j]}{n+1}$. At steps when the choice is UNI the degree of a fixed vertex can be increased at most by *N* – 1. Moreover, using(21), the probability that the growth of the degree of the *j*th vertex is positive when the choice is UNI is not greater than $\frac{p\beta}{{V}_{n}}$. Therefore, the expected growth of the degree of the *j*th vertex in the (*n* + 1)th step when the choice is UNI is less than or equal to (*N* – 1) $\frac{p\beta}{{V}_{n}}$. □

*Proof of Theorem 2.5*. Consider the following bounded random variable: ${\xi}_{n}=\frac{I[k,1,j]}{N-1}(D[n,j]-D[n-1,j])$. By the Remark 3.3, we have 0 ≤ *ξ*_{n} ≤ 1. Applying an appropriate version of Corollary VII-2-6 of [19] (see Proposition 2.4 of [20]), then using Lemma 3.4 and (8), we have

$$\begin{array}{c}D[n,j]=(N-1){\displaystyle \sum _{i=1}^{n}{\xi}_{i}\sim (N-1){\displaystyle \sum _{i=1}^{n}\text{E(}{\xi}_{i}|{\mathcal{F}}_{i}{}_{-1}})={\displaystyle \sum _{i=1}^{n}\left({\alpha}_{2}\frac{W[i-1,1,j}{i}+{R}_{i-1}\right)\sim}}\\ \sim \frac{1}{\Gamma (1+\alpha )}\frac{{\alpha}_{2}}{\alpha}{\gamma}_{1,j}{n}^{\alpha},\end{array}$$(29)

provided that the *j*th vertex exists after *k* steps. As lim_{k→∞} *W*[*k*, 1, *j*] = ∞ a.s., we obtain the statement. □

The following lemma will be used to study the maximal weight. It is an extension of Lemma 5.2 in [14].

*For all fixed nonnegative integers k* ≥ 0, 1 ≤ *m* ≤ *n*, *let*

$$S[m,n,k]={\displaystyle \sum _{j=m}^{n}\text{E}\left(b[n,1,k]\left(\begin{array}{c}W[n,1,j]+k-1\\ k\end{array}\right)I[n,1,j]\right).}$$(30)

*Then there exists a positive constant C*_{k} such that

$$S[m,n,k]\le {C}_{k}{\displaystyle \sum _{j=m}^{n}{j}^{-\alpha k}.}$$(31)

*Proof*. We use induction on *k*. Let *k* = 0. Then

$$S[m,n,0]={\displaystyle \sum _{j=m}^{n}\text{E}(b[n,1,0]I[n,1,j])={\displaystyle \sum _{j=m}^{n}\text{P(}W[n,1,j]>0)\le n-m+1.}}$$

Suppose that the statement is true for *k* – 1. By Lemma 3.1, *Z*[*n*, 1, *k*, *j*] is a martingale. The difference of two martingales is also a martingale. So, in the definition of *Z*[*n*, 1, *k*, *j*] changing *I*[*l*, 1, *j*] for *J*[*l*, 1, *j*], we obtain again a martingale. Using the definitions of *J*[*n*, 1, *j*] and *Z*[*n*, 1, *k*, *j*], we have

$$\begin{array}{c}S[m,n,k]={\displaystyle \sum _{j=m}^{n}\text{E}\left({\displaystyle \sum _{l=j}^{n}(Z[l,1,k,j]-d[n,1,k,j])J[l,1,j]}\right)=}\\ =\text{E}\left({\displaystyle \sum _{j=m}^{n}{\displaystyle \sum _{l=j}^{n}b[l,1,k]J[l,1,j]}}\right)+\text{E}\left({\displaystyle \sum _{j=m}^{n}{\displaystyle \sum _{l=j}^{n}(d[l,1,k,j]-d[n,1,k,j])J[l,1,j]}}\right).\end{array}$$(32)

In the last step we used that *W*(*l*, 1, *j*) = 1 if *J*[*l*, 1, *j*] = 1. Now, we give upper bounds for the two terms in (32) separately. We have already seen that, for a fixed *k*, the sequence *b*[*n*, 1, *k*] is decreasing. Therefore, applying also (18),

$$\text{E}\left({\displaystyle \sum _{j=m}^{n}{\displaystyle \sum _{l=i}^{n}b[l,1,k]J[l,1,j]}}\right)\le {\displaystyle \sum _{j=m}^{n}b[j,1,k]\text{E}\left({\displaystyle \sum _{l=j}^{n}J[l,1,j]}\right)\le {C}_{k}^{(1)}{\displaystyle \sum _{j=m}^{n}{j}^{-\alpha k}.}}$$(33)

For the second term in (32), changing the order of summations and using that $I[i,1,j]={\displaystyle {\sum}_{l=j}^{i}J[l,1,j]}$, we have

$$\begin{array}{c}\text{E}\left({\displaystyle \sum _{j=m}^{n}{\displaystyle \sum _{l=j}^{n}(d[l,1,k,j]-d[n,1,k,j]}})J[l,1,j]\right)=\\ \text{E}\left({\displaystyle \sum _{i=m}^{n}\frac{b[i+1,1,k]}{b[i+1,k-1]}\frac{p}{{V}_{i}}\beta}{\displaystyle \sum _{j=m}^{n}b[i,1,k-1]\left(\begin{array}{c}W[i,1,k]+k-2\\ k-1\end{array}\right)\frac{W[i,1,j]+k-1}{W[i,1,j]}}I[i,1,j]\right)\le \\ \le k{\displaystyle \sum _{i=m}^{n-1}\frac{b[i+1,1,k]}{b[i,1,k-1]}\text{E}\left(\frac{p}{{V}_{i}}\beta {\displaystyle \sum _{j=m}^{i}b[i,1,k-1]\left(\begin{array}{c}W[i,1,k]+k-2\\ k-1\end{array}\right)}I[i,1,j]\right).}\end{array}$$(34)

In the last step we applied that $\frac{W[i,1,j]+k-1}{W[i,1,j]}\le k$ if *I*[*i*, 1, *j*] > 0.

Now, we give upper bounds on the events $\left\{{V}_{i2}<\frac{pi}{2}\right\}$ and $\left\{{V}_{i}<\frac{pi}{2}\right\}$ separately. Using the induction hypothesis, we have

$$\text{E}\left(\frac{p}{{V}_{i}}\beta {\mathbb{I}}_{\left\{{V}_{i}\ge \frac{pi}{2}\right\}}\sum _{j=m}^{i}b[i,1,k-1]\left(\begin{array}{c}W[i,1,j]+k-2\\ k-2\end{array}\right)I[i,1,j]\right)\le \frac{2\beta}{i}{C}_{k-1}\sum _{j=m}^{i}{j}^{-\alpha (k-1)}.$$(35)

(Here 𝕀_{A} is the indicator of the set *A*.) On the other hand, by(18),

$$\begin{array}{c}\text{E}\left(\frac{p}{{V}_{i}}\beta {\mathbb{I}}_{\left\{{V}_{i}<\frac{pi}{2}\right\}}{\displaystyle \sum _{j=m}^{i}b[i,1,k-1]\left(\begin{array}{c}W[i,1,j]+k-2\\ k-2\end{array}\right)}I[i,1,j]\right)\le \\ \le \frac{P}{N}\beta \text{P}\left\{{V}_{i}<\frac{pi}{2}\right\}{\displaystyle \sum _{j=m}^{i}b[i,1,k-1]\left(\begin{array}{c}i+k-2\\ k-1\end{array}\right)=\text{o}\left(\frac{1}{i}{\displaystyle \sum _{j=m}^{i}{j}^{-\alpha (k-1)}}\right)}\end{array}$$(36)

as *i* → ∞. In the above computation we used Hoeffding’s exponential inequality (Theorem 2 in [21]) to obtain the following upper bound: $\text{P}\left\{{V}_{i}<\frac{pi}{2}\right\}\le {e}^{-\epsilon i}$, where $\epsilon =\frac{{p}^{2}}{2}$. Therefore, by(34),(36) and(35), we have

$$\begin{array}{c}\text{E}\left({\displaystyle \sum _{j=m}^{n}{\displaystyle \sum _{l=j}^{n}(d[l,1,k,j]-d[n,1,k,j])J[l,1,j]}}\right)\le \\ \le k{C}_{k}^{(2{)}^{\prime}}{\displaystyle \sum _{i=m}^{n-1}{i}^{-\alpha}{\displaystyle \sum _{j=m}^{i}\frac{1}{i}{j}^{-\alpha (k-1)}\le {C}_{k}^{(3)}{\displaystyle \sum _{j=m}^{n}{j}^{-\alpha k}}.}}\end{array}$$(37)

Above we applied that, by(18), $\frac{b[i+1,1,k]}{b[i,1,k-1]}=\text{O}\left({i}^{-\alpha}\right)$ as *i* → ∞. Finally,(32),(33) and(37) give the result. □

*Proof of Theorem 2.6*. Let *M*[*m*, *n*] = max{*W*[*n*, 1, *j*] : –(*N* – 1)*j* < *m*}, where 1 ≤ *m* ≤ *n* fixed. From Corollary 2.4, we have

$$\Gamma (1+\alpha )\underset{n\to \infty}{\mathrm{lim}}{n}^{-\alpha}M[m,n]=\mathrm{max}\{{\gamma}_{1,j}:(N-1)\le j<m\}$$(38)

almost surely. By (22), the following process is a submartingale:

$$b[n,1,k]\left(\begin{array}{c}W[n,1,j]+k-1\\ k\end{array}\right)=b[n,1,k]\left(\begin{array}{c}W[n,1,j]+k-1\\ k\end{array}\right)I[n,1,j],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\ge j.$$

Let *Q*[*m*, *n*] = max_{m ≤ j ≤ n}*W*[*n*, 1, *j*]. Then 0 ≤ *𝓦*_{n} – *M*[*m*, *n*] ≤ *Q*[*m*, *n*]. The maximum of increasing numbers of submartingales is also a submartingale, so

$$b[n,1,k]\left(\begin{array}{c}Q[m,n]+k-1\\ k\end{array}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\ge m,$$

is a submartingale. For non-negative numbers the maximum is majorized by the sum. Therefore, and by Lemma 3.5, we obtain

$$\text{E}\left(b[n,1,k]\left(\begin{array}{c}Q[m,n]+k-1\\ k\end{array}\right)\right)\le S[m,n,k]\le {C}_{k}{\displaystyle \sum _{j=m}^{n}{j}^{-\alpha k}.}$$(39)

Since

$$0\le {(b[n,1,1]Q[m,n])}^{k}\le \frac{b{[n,1,1]}^{k}}{b[n,1,k]}k!b[n,1,k]\left(\begin{array}{c}Q[m,n]+k-1\\ k\end{array}\right),$$(40)

we see that the submartingale *b*[*n*, 1, 1]*Q*[*m*, *n*] is bounded in *L*^{k} for all *kα* > 1. Hence, this submartingale converges almost surely and in *L*^{k} for every $k>\frac{1}{\alpha}$. Moreover, by (18), (39) and (40), we have

$$\text{E}{\left(\underset{n\to \infty}{\mathrm{lim}\mathrm{sup}}({n}^{-\alpha}Q[m,n]\right)}^{k}\le k!{C}_{k}\frac{1}{\Gamma (1+\alpha k)}{\displaystyle \sum _{j=m}^{\infty}{j}^{-\alpha k}.}$$(41)

$$\text{E}\left(\underset{m\to \infty}{\mathrm{lim}}\underset{n\to \infty}{\mathrm{lim}\mathrm{sup}}{({n}^{-\alpha}Q[m,n])}^{k}\right)=0,$$

for $k>\frac{1}{\alpha}$. As *Q*[*m*, *n*] is decreasing as *m* increases, so

$$\underset{m\to \mathrm{\infty}}{lim}\underset{n\to \mathrm{\infty}}{limsup}\left({n}^{-\alpha}\left({\mathcal{W}}_{n}-M[m,n]\right)\right)=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{a}\text{.s}\text{.}$$(42)

Therefore, as 0 ≤ *𝓦n* – *M*[*m*, *n*] ≤ *Q*[*m*, *n*],

$$\underset{m\to \mathrm{\infty}}{lim}\underset{n\to \mathrm{\infty}}{limsup}\phantom{\rule{thinmathspace}{0ex}}{n}^{-\alpha}Q[m,n]=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{a}\text{.s}\text{.}$$

This relation and (38) imply (11). By relation *μ* is a.s. finite. □

The evolution mechanism of the graph implies that *D*[*n*, *j*] ≤ (*N* –1)*W*[*n*, 1, *j*]. Therefore we have

$$\begin{array}{c}\mathrm{max}\{D[n,j]:-(N-1)\le j\le m\}\le {\mathcal{D}}_{n}\le \\ \le \mathrm{max}\{D[n,j]:-(N-1)\le j<m\}+\mathrm{max}\{(N-1)W[n,1,j]:m\le j\le n\}.\end{array}$$

Multiplying both sides by *n*^{-α} and then considering the limit as *n* → ∞,
Theorem 2.5 implies

$$\begin{array}{c}\mathrm{max}\left\{\frac{1}{\Gamma (1+\alpha )}\frac{{\alpha}_{2}}{\alpha}{\gamma}_{1,j}:-(N-1)\le j<m\right\}\le \\ \le \underset{n\to \infty}{\mathrm{lim}\mathrm{inf}}{\mathcal{D}}_{n}{n}^{-\alpha}\le \underset{n\to \infty}{\mathrm{lim}\mathrm{sup}}{\mathcal{D}}_{n}{n}^{-\alpha}\le \\ \le \mathrm{max}\left\{\frac{1}{\Gamma (1+\alpha )}\frac{{\alpha}_{2}}{\alpha}{\gamma}_{1,j}:-(N-1)\le j<m\right\}+(N-1)\underset{n\to \infty}{\mathrm{lim}\mathrm{sup}}{n}^{-\alpha}Q[m,n]\end{array}$$

as *n* → ∞. As *m* → ∞, by (42), we obtain the desired result. □

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