Exponential Inequalities for the Tail Probabilities of the Number of Cycles in Generalized Random Graphs

Abstract

Let $R_n$ be the centered and normalized number of cycles of fixed length contained in a generalized random graph with $n$ vertices. We obtain a Höffding-type exponential inequality for the tail probability of $R_n$.

1. INTRODUCTION

In recent years, many authors are interested in mathematical models of networks of various nature. As a rule, such models have the form of random graphs of different types. A well-known classical model is the Erdös-Rényi graph with $n$ vertices, where each edge appears with a fixed probability $p\in (0,1)$ independently of all other edges. Along with the Erdös-Rényi graphs, more general constructions are of interest. Such a construction, for example, is a generalized graph model in which each vertex has either a random or deterministic weight and the probability of an edge appearing between two vertices is a function of the weights. Thus, vertices with large weights are more likely to have a large number of neighbors than vertices with small weights.

Objects that arouse the increased attention of researchers are the distributions of the number of subgraphs of some fixed structure in a random graph. Many authors established limit theorems for these distributions. Thus, the asymptotic normality of the number of such subgraphs in the Erdös-Rényi graph was studied in [14]. The large deviations principle for the number of triangles in the Erdös-Rényi graph was obtained in [5] in the case $p=p(n)\rightarrow 0$, and in [6] for a fixed $p$ (see also [11], [13]). As for more general models, the weak convergence of the distribution of the number of triangles in a generalized random graph to the Poisson law is proved in [12]. A similar result for the number of cycles in a generalized random graph was obtained in [1] (a cycle is a closed continuous path (a polygon), in which there are no repeating vertices, except for the first and the last ones). The convergence rate is also studied there.

In the present paper, a different model of a generalized random graph is considered. Namely, we study the case when the probability of the appearance of each edge depends only on the weights of the vertices, which are connected by this edge (all possible edges arise independently of each other). The aim of this work is to obtain Höffding-type exponential inequalities for the distribution tails of the centered and normalized number of cycles in a generalized graph.

The classical Höffding inequality for sums of independent bounded random variables [9] has the following form:

$$ \mathbb {P}(n^{-1}(S_n-\mathbb {E}S_n)\geq t)\leq e^{-2nt^2/(b-a)^2} $$

for all positive $t$. Here $S_n$ is the sum of $n$ independent random variables with values in $[a,b]$. Upper bounds of this kind have been widely studied for decades. In particular, Höffding-type exponential inequalities are well known for such general objects as $U$- and $V$-statistics including the case of weakly dependent observations. Unfortunately, studying the number of subgraph with a fixed structure, for example, the number of cycles with $k$ vertices ( $k$-cycles), we deal with multifold sums of strongly dependent random variables.

We notice that similar inequalities were obtained in [10] for the Erdös-Rényi model, but only for large deviations of order $n$ (after normalization) and without an explicit constant in the exponent:

$$\mathbb {P}(T_n\geq \mathbb {E}T_n+\varepsilon n^kp^k)\leq \exp (-\alpha (\varepsilon ,k) n^2p^2),$$

where $T_n$ is the number of subgraphs in the Erdös-Rényi graph, and $\alpha (\varepsilon )$ has an implicit form. A close result for the number of triangles was obtained in [7].

In [3] and [4], we obtained the desired estimates for the number of $k$-cycles and arbitrary subgraphs, respectively, for the Erdös-Rényi graphs. In this paper, the result for $k$-cycles is extended to the case of generalized graphs.

2. THE MAIN RESULT

Let $V(G)=\{1, 2, ..., n\}$ be the vertex set of graph $G$, and for every $1\leq i\leq n$, let $W_i$ be a random weight of the $i$-th vertex. Our model is defined by the following conditions:

(A1) $W_1,\ldots ,W_n$ are independent and identically distributed on the open interval $(0,1)$.

(A2) Given the collection $W=(W_1, W_2, ..., W_n)$), the edge between the vertices $i$ and $j$ appears with probability $p_{ij}=p_{ij}(W_i, W_j)$ independently of other edges.

(A3) There exist constants $c_1, c_2$, with $0 < c_1 \leq c_2$, such that, for all $(i,j)$,

$$ c_1 W_i W_j \leq p_{ij}(W_i, W_j) \leq c_2 W_i W_j. $$

For $k\geq 3$, denote by $I(k)$ the set of possible $k$-cycles in $G$. The cardinality of $I(k)$ is equal to $(n)_k/(2k),$ where $(n)_k=n(n-1)...(n-k+1)$ is the number of ways to choose $k$ different vertices in order, and $2k$ is the number of options to choose the starting point and the cycle orientation (all sets of $k$ ordered vertices with different orientations and different initial vertices are the same cycle). For each potential cycle $\alpha \in I(k)$ denote by $Y_{\alpha }$ the indicator of the event “$G$ contains $\alpha$”. More precisely, $Y_\alpha =\prod _{i<j: (i,j) \in \alpha } z_{ij},$ where $z_{ij}$ are independent Bernoulli random variables equal to one if $e_{ij} \in E(G)$ ($E(G)$ is the set of edges of $G$).

In this paper, exponential inequalities are obtained for the tail probabilities of the centered and normalized multifold sum

$$ R_n=b_n^{-1/2}\sum _{\alpha \in I(k)}(Y_{\alpha }-\mathbb {E}Y_{\alpha }),$$

where

$$ b_n=\mathbb {E} \;{\bf Var} (\sum _{\alpha \in I(k)}Y_{\alpha }|W)$$

and ${\bf Var}(\ldots |W)$ is the conditional variance given the collection of weights $\{W_i\}$.

Lemma 1 $.$ Denote $m_j= \mathbb {E} W_1^j$ . Under conditions $(A1)$ – $(A3)$ , for any $C \in (0,\frac {1}{2})$ there exists a number $n_k=n_k(C)$ such that, for all positive integers $n \ge n_k$ , the lower bound

$$ b_n\geq C \rho n^{2k-2} $$

(1)

is valid, where

$$ \rho = m_2^{2(k-2)} \min _{2\leq l\leq k} (c_1^{2k-1} m_3^l- c_2^{2k} m_4^l). $$

Here the constant $C \in (0,\frac {1}{2})$ can be arbitrarily chosen, and $n_k=n_k(C)$ is the largest root of a certain polynomial defined in the proof.

We need the following additional condition:

(A4) $\quad \rho > 0.$

Theorem $.$ Let conditions $(A1)$ – $(A4)$ be satisfied. Then for all $n \ge n_k$ and all $x > 0$ the following upper bound holds $:$

$$ \mathbb {P}(|R_n|>x)\leq \exp \left \{-\frac {C \rho k^{2/3} x^2}{2^{5+2(k-1)/3} e}\right \},$$

where $\rho , C$ , and $n_k$ are defined in Lemma $1$ .

Note that in the special case $c_1=c_2=1$ condition (A4) always holds in view of the fact that the sequence $\{m_j\}$ is strongly decreasing.

Corollary $.$ Under conditions $(A1)$ , $(A2)$ , and $p_{ij}=W_i W_j$ the assertion of the theorem is valid.

3. PROOFS

To derive exponential estimates for the distribution tails of $R_n$, we use a well-known method based on Markov’s power inequality for multiple sums of dependent random variables. Our main goal is to obtain upper bounds (which are uniform in $n$) for even moments $\mathbb {E}R_n^{2m},\;\; m=1,2,\ldots$. First we need to estimate $b_n$ from below.

Proof of Lemma 1 . Given the weights, the conditional expectation for the cycle indicators introduced above is represented as

$$ \mathbb {E} (Y_{\alpha }|W) = \mathbb {P}\big (Y_{\alpha }=1|W \big )=\prod _{i<j:(i,j) \in \alpha }p_{ij}. $$

Next, for any two cycles $\alpha ,\alpha ^{\prime }\in I(k)$ having common edges, one has

$$ \mathrm {cov}\big (Y_{\alpha },Y_{\alpha ^{\prime }}|W\big )=\mathbb {E} (Y_{\alpha }Y_{\alpha ^{\prime }}|W) - \mathbb {E} (Y_{\alpha }|W)\mathbb {E} (Y_{\alpha ^{\prime }}|W) $$

$$ = \prod _{(i,j) \in \alpha \bigtriangleup \alpha ^{\prime }} p_{ij} \prod _{(i,j) \in \alpha \cap \alpha ^{\prime }}p_{ij} - \prod _{(i,j) \in \alpha \bigtriangleup \alpha ^{\prime }} p_{ij} \prod _{(i,j) \in \alpha \cap \alpha ^{\prime }}p_{ij}^2.$$

Here and below, in order not to obstruct the notation, we do not indicate $i<j$ in the products meaning that the edge $(i,j)$ is included in the graph once, regardless of the order of the vertices forming this edge. We also write $(i,j)\in \alpha \cap \alpha ^{\prime }$ meaning that the product is taken over all edges belonging simultaneously to both cycles (with a similar convention in the case of a symmetric difference). If two cycles have no common edges, the corresponding indicators are independent random variables.

Using (A3), we can estimate this conditional covariance from below:

$$ \mathrm {cov}\big (Y_{\alpha },Y_{\alpha ^{\prime }}|W\big ) \ge c_1^{2k-1} \prod _{(i,j) \in \alpha \bigtriangleup \alpha ^{\prime }} W_i W_j \prod _{(i,j) \in \alpha \cap \alpha ^{\prime }} W_i W_j - c_2^{2k} \prod _{(i,j) \in \alpha \bigtriangleup \alpha ^{\prime }} W_i W_j \prod _{(i,j) \in \alpha \cap \alpha ^{\prime }}W_i^2 W_j^2. $$

Here we take the maximum possible power of $c_1$ in the minuend since $c_1\leq 1$. The power of $c_2$ in the subtrahend coincides with the number of edges in the corresponding product and always is equal to $2k$. The degree of each $W_i$ in the product

$$ \prod _{(i,j) \in \alpha \bigtriangleup \alpha ^{\prime }}W_iW_j\prod _{(i,j) \in \alpha \cap \alpha ^{\prime }}W_iW_j $$

(2)

is equal to the degree of the vertex $i$ in $\alpha \cup \alpha ^{\prime }$. The power of $W_i$ in the product

$$ \prod _{(i,j) \in \alpha \bigtriangleup \alpha ^{\prime }}W_iW_j \prod _{(i,j) \in \alpha \cap \alpha ^{\prime }}W_i^2 W_j^2 $$

(3)

is equal to the sum of the degrees of the vertex $i$ in $\alpha$ and $\alpha ^{\prime }$.

Next, we want to go from products over all edges to products over the vertices of the cycles. In this representation, we can divide all the vertices of two cycles into four disjoint groups. In the example below, two cycles of length $6$ are shown with all four types presented and labeled. The first group (I) includes vertices connecting exactly two edges which belong to exactly one of the cycles. For vertices of this type, the degree of the corresponding weight $W_i$ equals two for both (2) and (3). The second group (II) includes vertices connecting exactly two edges which are included in both cycles. For vertices of this type, the degree of $W_i$ is two for (2) and four for (3). The third group (III) includes vertices that connect exactly three edges, one of which is included in both cycles. For such vertices, the degree $W_i$ is three for (2) and four for (3). Finally, the last group (IV) consists of the vertices that belong to both cycles, but not to the common edges of the cycles (touch points). For vertices from this group, the weight degree $W_i$ is four for both (2) and (3):

Fig. 1.

.

Since $W_i$ are i.i.d. random variables, the expectations of the products in (2) and (3) are equal to the products of the moments of $W_i$. Denote by $l_s$, $s=1,2,3,4$, the number of vertices belonging to the $s$-th group in a given fixed pair of cycles. We have

$$ \begin {gathered} \mathbb {E}\mathrm {cov}\big (Y_{\alpha },Y_{\alpha ^{\prime }}|W\big )\geq m_2^{l_1} m_4^{l_4} (c_1^{2k-1}m_3^{l_3} m_2^{l_2} - c_2^{2k} m_4^{l_2+l_3}) \\ \geq m_2^{l_1 + 2l_4} (c_1^{2k-1} m_3^l - c_2^{2k} m_4^l) \geq m_2^{2(k-2)} \min _{2\leq l\leq k} (c_1^{2k-1} m_3^lc_ 2^{2k} m_4^l)=\rho . \end {gathered}$$

(4)

Here $l=l_2+l_3$ is the number of vertices included in the common edges of two cycles. The second inequality in the chain above uses $m_2 \geq m_3$ (nonincreasing of the moments follows from the fact that $W_i$ belong to $(0,1)$ almost surely) and inequality $m_4 \geq m_2^2$.

We now estimate from below (for sufficiently large $n$) the value

$$ b_n=\mathbb {E} \;{\bf Var} (\sum _{\alpha \in I(k)}Y_{\alpha }|W) = \mathbb {E} \sum _{\alpha ,\alpha ^{\prime }\in I(k)}\mathrm {cov}(Y_{\alpha },Y_{\alpha ^{\prime }}|W)=\sum _{\alpha ,\alpha ^{\prime }\in I(k)} \mathbb {E} \;\mathrm {cov}(Y_{\alpha },Y_{\alpha ^{\prime }}|W). $$

(5)

To estimate the last sum in (5), it is necessary to analyze the number of $(\alpha ,\alpha ^{\prime })$ pairs with nonzero expectations of the conditional covariances. This means that cycles $\alpha$ and $\alpha ^{\prime }$ have at least one common edge. One can consider the number of such pairs under the assumption that two sets of vertices have exactly $l$ common elements ($l$ runs from $2$ to $k$) and all common vertices belong to common edges. For each such $l$, the corresponding number of terms in the sum (5) is the certain polynomial of $n$, and the main term (of order $n^{2k-2}$ as we will see) corresponds to the case $l=2$: $\alpha$ and $\alpha ^{\prime }$ have exactly two common vertices forming a common edge. The number of such summands equals

$$ C_n^2(n-2)_{k-2}(n-k)_{k-2}=\frac {n(n-1)...(n-2k+3)}{2}, $$

where $C_n^2$ is the number of ways to choose common two vertices (say, in increasing order), $(n-2)_{k-2}$ is the number of ways to complete the cycle $\alpha$, and $(n-k)_{k-2}$ is the number of ways to complete the cycle $\alpha ^{\prime }$. It is evident that the number of summands with cycles having more than one common edge has a lesser order in $n$. For example, the number of summands with cycles having exactly two common edges is equal to $C_1(k)C_n^3 C_{n-3}^{k-3} C_{n-k}^{k-3} + C_2(k)C_n^2 C_{n-2}^2 C_{n-4}^{k-4}C_{n-k}^{k-4}$ and is a polynomial of degree $2k-4$ of the variable $n$.

For two cycles that have a common edge, $\mathbb {E} \;\mathrm {cov}(Y_{\alpha },Y_{\alpha ^{\prime }}|W) \geq \rho$ in virtue of (4). Hence,

$$ b_n\geq \frac {\rho }{2}n^{2k-2}+\rho r_n, $$

where $r_n$ is a polynomial of $n$ of degree less than $2k-2$. We can rewrite derived lower estimate for $b_n$ in the following form:

$$ b_n \geq C \rho n^{2k-2}+ ({1}/{2}-C) \rho n^{2k-2} + \rho r_n.$$

Here we can arbitrarily choose $C \in (0, \frac {1}{2})$. We notice that $({1}/{2}-C) n^{2k-2} + r_n \geq 0$ for all $n \ge n_k$, where $n_k$ is the ceiling function of the largest zero of the polynomial. Note that the closer $C$ is to $\frac {1}{2}$, the greater $n_k$ is. For example, for $k=3$ one can choose $C=\frac {1}{6}$ and obtain $n_3=7$. Thus, for $n \geq n_k$, we have a desired estimate from below:

$$ b_n\geq C \rho n^{2k-2}.$$

(6)

Thus, Lemma 1 is proved.

Proof of Theorem. Now we are going to derive an upper bound for $\mathbb {E}R_n^{2m}$. This part of the proof mostly repeats [3]. Denote

$$\tilde {Y}_{\alpha }=Y_{\alpha }-\mathbb {E}Y_{\alpha }.$$

Then

$$ \mathbb {E}R_n^{2m}=b_n^{-m}\mathbb {E}\sum _{\alpha _s\in I(k)}\mathbb {E} \Big (\tilde {Y}_{\alpha _1}...\tilde {Y}_{\alpha _{2m}}|W\Big ).$$

(7)

Consider a summand in the sum above. It does not vanish if only, for every cycle $\alpha _s$ from the mixed moment, another cycle can be found with at least one common edge (otherwise, due to the conditional independence, a factor of the form $\mathbb {E}(\tilde {Y}_{\alpha _s}|W)$ “kills” the summand). Note that $|\tilde {Y}_{\alpha }| \leq 1$ a.s. for all $\alpha$. Let us estimate the number of nonzero summands in (7).

First, consider the case

$$ m \leq \frac {n(n-1)}{8}. $$

(8)

Our aim is to derive an upper bound for the number of nonzero summands, i.e., such that each cycle in the set $\{\alpha _1, \ldots , \alpha _{2m}\}$ contains at least one common edge with one or several cycles from this set. Let us fix an arbitrary summand. Note that some of the cycles in the set can coincide. To derive such a bound we are going to divide all edges into two certain types. Also we are going to divide into three types all cycles in the set. This classification is not unique, but we are interested only in the fact that it is always possible.

We begin with forming a list of edges, which are contained in at least two different cycles in the set $\{\alpha _1, \ldots , \alpha _{2m}\}$. This is a first type of edges, which we will call common edges. We will use the term “arbitrary” for the second type of edges and we are going to move some of the edges from “common” to “arbitrary” using the following algorithm. First, if two cycles contain two or more common edges, then we move to the “second-type” list all this common edges but one, which is chosen arbitrarily. Hence, each pair of cycles from the set contains one common edge at most. Second, we consider subsequently all the cycles in our set (in any order). The cycle $\alpha _s$ under consideration at the moment can have several “neighbors”, i.e., cycles having a common edge with $\alpha _s$. If a “neighbor” also has a common edge with some other cycle from the set, we move its common edge with $\alpha _s$ to the “arbitrary” type, if only it is not the last common edge in $\alpha _s$. We now have final lists of common and “arbitrary” edges.

Further, we will call centers (the cycles of the first type) those cycles from the set, which contain two or more common edges. We will call petals (the cycles of the second type) those cycles from the set, which contain a common edge with one of the centers. Note that centers do not have common edges with each other. All the cycles which are not centers or petals will be called third-type cycles. All the third-type cycles can be divided into pairwise disjoint clusters, where all the cycles in each cluster contain exactly one common edge.

Such a classification of edges and cycles depends on the two things: Which exactly of several common edges of two cycles we left in the list of common edges, and what was the order of considering cycles in the set when we moved again some of the common edges to the list of “arbitrary”.

The following example for $4$ triangles illustrates two different classifications. Due to two different orders of considering subgraphs we obtain either one “center” with three petals or two clusters of third-type cycles:

Fig. 2.

.

The following upper bound holds for any described classification. Let $l$ be the number of common edges, $s$ be the number of centers, $Q$ be the number of common edges in centers. Every center has at least two petals, i.e., contains at least two edges of $Q$ above. Thus, the following inequalities hold:

$$ 0 \leq 2s \leq Q \leq l \leq 2m, \;\;3s \leq 2m.$$

Note that the total number of centers and petals is greater or equal then $s+Q$. The number of third-type cycles is greater or equal than $2(l-Q)$ due to the fact that each of $l-Q$ common edges not belonging to centers is contained in at least two third-type cycles. Hence, $2m \geq s+Q +2(l-Q)$, with $Q \geq s$, that yields

$$ m \geq s+l-Q.$$

(9)

Given $l,s,Q$, we are going to estimate from above the number of nonzero summands in (7) and to sum up then over all possible $l,s,Q$. We will use the following technical

Lemma 2 $.$ Let $t_1,...,t_l$ be nonnegative integers such that $t_1+...+t_l=u \geq l$ . Then

$$ \frac {u!}{t_1!...t_l!}\leq l! l^{u-l}.$$

Proof. The polynomial coefficient on the left side of the inequality reaches its maximum when all multiplicities $t_i$ coincide or, if $u$ can not be exactly divided by $l$, the differences between them do not exceed $1$. Denote by $\lfloor x\rfloor$ the integer part of $x$, and let

$$ j=u-\lfloor {u}/{l}\rfloor l.$$

Then the maximum value of the function $\frac {u!}{(t_1!...t_l!)}$ in $t_1,...,t_l$ under the condition $t_1+...+t_l=u \geq l$, equals to

$$ \frac {u!}{\bigg (\big (\big \lfloor (u)/l\big \rfloor +1\big )!\bigg )^j\bigg (\big \lfloor (u)/l\big \rfloor !\bigg )^{l-j}}. $$

(10)

Note that

$$ u!=l!\prod _{i=1}^j(i+l)(i+2l)\cdot ...\cdot \big (i+\big \lfloor u/l\big \rfloor l\big )\prod _{i=j+1}^l(i+l)(i+2l)\cdot ...\cdot \big (i+\big (\big \lfloor u/l\big \rfloor -1\big )l\big ) $$

$$ \leq l!\prod _{i=1}^j\bigg (l^{\lfloor u/l\rfloor }\big (\big \lfloor u/l\big \rfloor +1\big )!\bigg ) \prod _{i=j+1}^l\bigg (l^{\lfloor u/l\rfloor -1}\big \lfloor u/l\big \rfloor !\bigg ) $$

$$ =l!l^{u-l}\bigg (\big (\big \lfloor u/l\big \rfloor +1\big )!\bigg )^j\bigg (\big \lfloor u/l\big \rfloor ! \bigg )^{l-j}. $$

Hence, the maximum in (10) does not exceed $l! l^{u-l}$. $\quad \square$

The number of ways to choose vertices and edges for $s$ centers does not exceed $((n)_k/(2k))^s$ and is bounded by the value

$$ \bigg (\frac {n^k}{2k}\bigg )^s. $$

(11)

The number of ways to choose which of $2m$ cycles in the set $\{\alpha _1, \ldots , \alpha _{2m}\}$ will be centers equals

$$ C_{2m}^s.$$

(12)

Now we are going to choose which of the edges in centers are of the first type (common edges). Wee need an estimate for the number of ways to divide $Q$ into $s$ summands, $Q=q_1+\ldots + q_s$, where $q_i \geq 2$ is the number of petals connected to the $i-$th center. It equals $C_{Q-2s+(s-1)}^{s-1}$ and is bounded by the value

$$ 2^{Q-s-1}.$$

(13)

Further, the number of ways to choose $q_i$ common edges between $k$ edges of each center is bounded by

$$ \sum _{q_1+\ldots +q_s=Q} \prod _{i=1}^s C_k^{q_i}. $$

(14)

The number of ways to choose common edges in the third-type cycles equals $C_{n(n-1)/2-Q}^{l-Q}$ and is bounded by

$$ \frac {n^{2l-2Q}}{2^{l-Q}(l-Q)!}. $$

(15)

The next step is to estimate the number of ways to divide cycles of the second and third types into $l$ pairwise disjoint clusters (one cluster for each common edge). Let $t_i, i=1,\ldots l$ be the cardinality of each cluster, i.e., the “multiplicity” of $i-$th common edge. Let $t_1,\ldots ,t_Q$ correspond to petals (so these multiplicities are greater or equal than one)and $t_{Q+1},\ldots ,t_l$ correspond to third-type cycles (so they are greater or equal than two as at least two cycles of the third type contain each corresponding common edge). The number of ways to choose possible values of $t_1,\ldots ,t_l$ and to divide $2m-s$ cycles of the second and third types into corresponding clusters is equal to $C_{2m-s-l+Q-1}^{l-1} \;\frac {(2m-s)!}{t_1!...t_l!}$ and is bounded from above by the value

$$ 2^{2m-s-l+Q-1} l! l^{2m-s-l} $$

(16)

in virtue of Lemma 2 and the estimate $C_a^b \leq 2^a$ that was already used in (13).

Finally, we need to take into consideration the number of ways to complete (i.e., choose other vertices and edges) $2m-s$ cycles (divided into clusters above) given one fixed common edge for each cycle. It is bounded by

$$ (n^{k-2})^{2m-s}.$$

(17)

Now we are ready to provide an upper estimate for the number of nonzero summands in (7) by multiplying all combinatorial expressions (11-17). This estimate equals

$$ \sum _{s,l,Q} \sum _{q_1+\ldots +q_s=Q} \prod _{i=1}^s C_k^{q_i} \;2^{2m-2s +3Q-2l-2} n^{2m(k-2)+2s+2(l-Q)} C_{2m}^s l^{2m-s-l} \frac {l!}{(2k)^s(l-Q)!}. $$

Taking into account the estimate $\frac {l!}{(l-Q)!} \leq l^Q$, with $Q\leq l$, and the normalization in (6), we get the upper bound

$$ \mathbb {E}R_n^{2m}\leq \sum _{s,l,Q} C_{2m}^s \bigg (\sum _{q_1+\ldots +q_s=Q} \prod _{i=1}^s C_k^{q_i} \bigg ) 2^{2m-2} \frac {1}{C^m \rho ^m (2k)^s} \bigg (\frac {4l}{n^2}\bigg )^{m-s-l+Q} \;l^m.$$

Using (8) (which yields $4l \leq 8m \leq n^2$) and (9), we have

$$ \bigg (\frac {4l}{n^2}\bigg )^{m-s-l+Q} \leq 1. $$

Further, using the inequalities

$$ \sum _Q \sum _{q_1+\ldots +q_s=Q} \prod _{i=1}^s C_k^{q_i} \leq 2^{ks}, \; \sum _s C_{2m}^s \leq 2^{2m}, $$

$$ \bigg (\frac {2^k}{2k}\bigg )^s \leq \bigg (\frac {2^{k-1}}{k}\bigg )^{2m/3},$$

we obtain

$$ \mathbb {E}R_n^{2m}\leq \frac {2^{4m-2+2m(k-1)/3} }{C^m \rho ^m k^{2m/3} } \sum _l l^m. $$

Now we can use the estimate

$$ \sum _{l=1}^{2m} l^m \leq \int _0^{2m+1}t^mdt=\frac {1}{m+1}(2m+1)^{m+1}=\frac {2m+1}{m+1}(2m+1)^m $$

$$ \leq 2(2m)^m\Big (1+\frac {1}{2m}\Big )^m\leq 2\sqrt {e}(2m)^m<2^2(2m)^m$$

Finally,

$$ \mathbb {E}R_n^{2m}\leq \frac {2^{4m+2m(k-1)/3}}{k^{2m/3}C^m \rho ^m} (2m)^m=C_0^{2m} (2m)^m, $$

(18)

where

$$ C_0=\frac {2^{2+(k-1)/3}}{k^{1/3}(C\rho )^{1/2}} > 0. $$

Using the estimate (18), one can derive an exponential inequality for the two-sided distribution tails of $R_n$. The corresponding arguments can be found for example in [2] (Lemma $1$, Remark $1$). Give an assertion from [2] we need.

Lemma 3 $.$ Let $\zeta$ be an arbitrary random variable with finite moments of all orders $r\ge 0$ , which satisfy the following relations:

$$ {\bf E}|\zeta |^r\le AC_o^rr^{r/2}\thinspace \thinspace \thinspace \thinspace \mbox {for all natural}\thinspace \thinspace \thinspace r, $$

where the constants $A\ge 1$ and $C_o>0$ do not depend on $r$ . Then the following estimate is valid for all $x\ge 0$ :

$$ {\bf P}(|\zeta |\ge x)\le Ae^{-\frac {1}{2e}(x/C_o)^2}. $$

We now use this result with $\zeta =R_n$, $A=1$, and $r=2m$ and finally obtain the following inequality:

$$ \mathbb {P}(|R_n|>x)\leq \exp \left \{-\frac {C \rho k^{2/3} x^2}{2^{5+2(k-1)/3} e}\right \}.$$

Remind that this estimate holds for $n \geq n_k$, with $n_k$ increasing as $C$ is chosen closer to ${1}/{2}$. An explicit value of $n_k$ can be found for given $k$ and $C$. The theorem is proved.