1. Introduction and the Main Results
Let be independent random variables (r.v.s) taking values in a separable Banach space with respective distributions . In the i.i.d. case, we will denote by P the common distribution.
denotes the compound Poisson distribution with Lévy measure
:
where
is the
k-fold convolution of a finite measure
with itself;
is the Dirac measure with the atom at zero.
, where
by definition. The compound Poisson distribution with Lévy measure
is called the accompanying infinitely divisible law for
(see [
1]); here and everywhere in the future, the symbol
denotes the distribution of a random variable (r.v.)
. We denote by
a r.v. having this distribution.
For every natural
, let
be independent copies of the random variable
We assume that all the sequences
are independent. Additionally, let
,
be independent Poisson random processes with unit intensity which do not depend on the sequences of r.vs above. From (
1), it follows that
The characteristic functional of a
-valued r.v.
is defined as follows:
where
is a bounded linear functional on
, i.e., it is an element of the conjugate space
. So, one obtains
Next, the characteristic functional of the accompanying infinitely devisable law is calculated by the formula
In other words,
where the sums
,
(with
) are independent, and
by virtue of (
1). In the i.i.d. case, from (
3), we obtain that Formula (
4) can be rewritten as follows:
The main goal of this paper is to obtain upper and lower moment inequalities for some measurable functions of or of the collection via the analogous moments of the accompanying compound Poisson laws as well as to obtain upper bounds for the probability tail of these functionals. Results of this kind are related to Kolmogorov’s problem of approximating sums of independent r.vs by various infinitely divisible laws, in particular, by the accompanying ones, as well as with an improvement of the classical probability inequalities for these sums.
In what follows, we consider functions of one or several (say,
n)
-valued arguments. In the latter case, we consider functions of the
n-variate argument
from the new Banach space
with the norm
. So, for arbitrary Borel functions
,
and
, with
and
, introduce the following notation under appropriate moment conditions:
where
, with
(
is the set of all nonnegative integers). It is clear that the function
is a particular example of the function
. In turn, the latter function is a particular case of
if
.
We say that a function is convex (concave) if the difference is a nondecreasing (nonincreasing, respectively) function in n.
The following two theorems, in particular, contain some results from [
2] together with some new results.
Theorem 1. The following assertions are valid:
1.
For all and naturals m, let the functions be convex. Then,provided that the expectation on the right-hand side of this inequality exists.In the i.i.d. case, inequality (9) holds if only the function is convex. 2.
Let the function be convex with respect to each coordinate . Then, for every vector ,if the expectation on the right-hand side of (10) exists, where , are independent r.v-s with respective distributions , . Remark 1. If the functions and in Theorem 1 are concave, then, inequalities (9) and (10) are changed to the opposite. It follows from the well-known connection between convex and concave functions: concave = −convex. For a r.v.
with values in
,
denotes the minimal closed subset of
such that
with probability 1. We need the notion of convexity in the direction determined by a subset of
. We say that a measurable function
f is convex in direction
if for all
and all
, this function satisfies the inequality
Notice that, in the one-dimensional case, convexity in direction
(nonnegative summands) or
(integer-valued nonnegative summands) is the classical convexity. But in the multivariate case, the convexity of
f does not imply the relation (
11). As a counterexample, we consider the three-dimensional case and the convex function
. Put
,
and z=(0,0,3). It is clear that inequality (
11) for these parameters is false. However, this function satisfies the relation (
11) in the direction determined by any one-dimensional subspace of
.
Proposition 1. In the i.i.d. case, let a measurable function f satisfy (11). Then, under the moment conditions above, inequality (9) holds. Example 1. If a.s., and f is an arbitrary convex function on , then, inequalities (9) and (11) are valid. We now consider some particular cases of the scheme described in Proposition 1. Let be the empirical distribution function based on a sample from the -uniform distribution. Then, the normalized empirical process can be represented as the nth partial sum of the indicator-type i.i.d. random processes taking values in a Banach space, say, . It is well known that the accompanying compound Poisson r.v. for this sum is a Poisson random process with intensity n, which coincides in distribution with the Poisson random process , . Notice that the finite-dimensional distributions of the random process are multinomial. In particular, for each , the distribution is binomial with parameters .
As consequences of Proposition 1, we obtain the following two assertions.
Corollary 1. Let be a convex function. Then, for any ,whenever the right-hand side in (12) is well-defined. Corollary 2. Let G and f be nondecreasing convex functions on R. Then,if the right-hand side of this inequality is well defined, where is an arbitrary finite measure on . One can slightly weaken the convexity property in Corollary 1 by studying power moments of the r.vs under consideration.
Proposition 2. For every and any naturals n and m, the following inequalities hold:and Remark 2. It is worth noting that, from Corollary 2, one can easily obtain similar inequalities for all even moments and . Additionally, for all , inequalities (13) and (14) coincide and also follow from (12). So, the only nontrivial cases in (13) and (14) are and , respectively. We note that, for , the right-hand sides in (13) and (14) may be negative (say, for ). A direct consequence of Proposition 2 is as follows.
Corollary 3. Let be an entire function on .
1.
Assume that there is a point such that, for all , the values of k-th derivatives at the point are nonnegative. Then, for every and all ,provided that the expectation on the right-hand side of (15) is well defined.2.
Assume that there is a point such thatis an entire function on as well. Then, for every and all ,provided that the expectation on the right-hand side of (16) is well defined. Example 2. Let , , where . Put . Then, the conditions in item 1 of Corollary 3 are fulfilled, and inequality (15) is valid for all . But the function is convex only for ; otherwise, it is concave. Theorem 2. Suppose that at least one of the following two conditions is fulfilled:
1.
The function f is continuously differentiable in the Fréchet sense(
i.e., is continuous in x for each fixed h)
, and for each and all 2. for all k, f is twice continuously differentiable in the Fréchet sense, and is convex in x for each fixed
Then, the function is convex, i.e., inequality (9) is valid. Corollary 4. If a.s. and is a convex function, then, inequality (
11)
is valid. If are random vectors in or in the Hilbert space , with nonnegative coordinates, then, the function where is the corresponding Euclidean norm and , satisfies inequalities (
11)
and (
17)
. For the zero-mean Hilbert-space-valued r.vs , the function where or satisfies condition 2
of Theorem 2.
Therefore, in these cases, inequality (
9)
holds under the restriction Remark 3. There exist functions which do not satisfy the conditions of Theorem 2 but the corresponding function is convex. For example, in the i.i.d. one-dimensional case, let us consider the function and the centered summands It is clear that the conditions of Theorem 2 are not fulfilled. In this case, we havei.e., it is a quadratic function with respect to the variable n. Thus, if then, the function is convex; otherwise, it is concave. In other words, in this case, we obtain upper and lower Poissonization inequalities in dependence on the sign of the moment . The exactness of inequality (
9) is characterized by the following two assertions.
Corollary 5. For independent one-dimensional centered r.v.s , consider the function . Since, for any fixed , the second derivative of the function is convex and concave simultaneously, then, by item 2 of Theorem 2, Given a finite measure
on
satisfying the condition
we denote by
the function
in (
6) defined in the i.i.d. case for the summand distribution
Theorem 3 ([
2]).
In the i.i.d. case, let the function be convex. Then,whenever the expectation on the right-hand side of (5)
is well defined, where , and the supremum is taken over all n and P such that for all Borel subsets . Remark 4. Taking inequality (9) into account, we can easily reformulate Theorem 2 for the non-i.i.d. case. The idea of employing compound Poisson distributions for constructing upper bounds for the moments of the sums was first proposed by Prokhorov ([3,4]). In particular, relations (9) and (19) were obtained in [4] for the functions (m is an arbitrary natural) and , in the case of one-dimensional symmetric . Moreover, in the case of zero-mean one-dimensional summands, these relations for the functions can be easily deduced from [3] (see also [5]). A more general result in this direction was obtained by Utev [
6]. Under condition 2 of Theorem 2, he proved extremal equality (
19) for nonnegative functions
having an exponential majorant, using a technique by Kemperman [
7]. In our opinion, the proof of item 2 of Theorem 2 (see
Section 3) is much simpler than that in [
6] and needs no additional restrictions on
and the sample space.
Relations like (
9) and (
19) can also be applied for obtaining sharp moments and the tail probability inequalities for sums of independent r.vs (for details, see [
5,
6,
7,
8,
9,
10,
11,
12]).
We now consider the centered empirical point process
,
, that one can interpret as a sum of
n i.i.d. centered r.vs
taking values, say, in the Hilbert space
, where
is an arbitrary finite measure on
. The accompanying compound Poisson random process can be represented in the form
,
, which may be called as a “Poissonian bridge” with intensity
n on the unit interval. By Corollary 5, we then obtain
where
or
. If
is the Dirac measure with atom at a point
t, then, a univariate analog of inequality (
20) is as follows:
with an arbitrary
or
and any
. But compared to (
20), we have here less restrictive conditions on
due to item 2 of Theorem 2. It is clear that we can replace the power functions in (
21) with any function
f having a convex second derivative under appropriate moment conditions:
It is interesting to compare inequalities (
21) and (
22) with (
12) and (
13) taking Remark 2 into account and setting
in (
13). Put
.
Proposition 3. For every and any even convex function f on R, the following two-sided inequality is valid:if only the Poissonian moments exist. Moreover, if , then, Proposition 4. For any , and every natural number m, Thus, inequalities (
21)–(
25) improve the estimates (
12)–(
15).
We supplemented Corollary 2 and Theorem 2 with an example of an infinitely dimensional function space
. Let
, with
. Consider an integral-type functional of the form
where
is a smooth function on
R. In this case, the first two Fréchet derivatives of
f are defined as follows:
For example, if the continuous random processes , , are nonnegative and the function g is convex (or the first derivative is nondecreasing in the positive direction), then, item 1 of Theorem 2 will be fulfilled. On the other hand, if are centered random processes on and the second derivative is convex on R, then, item 2 of Theorem 2 will also be satisfied.
We can easily reformulate condition (
11) and Theorem 2 for the functions
in (
7) if for any
and fixed
, we put
and, under the conditions of Theorem 2, replace the function
with
for
from an appropriate subset.
Corollary 6. For every fixed , let the functions satisfy (11) for all and , . Then, under the moment conditions above, inequality (10) holds. Corollary 7. Let the functions , defined in (26) by the function in (27), satisfy the conditions of Theorem 2. Then,where the independent r.v-s are defined in Theorem 1.
The above results deal with some type of convexity. However, one can obtain moment inequalities close to those mentioned above without any convexity conditions. The following result is valid for the r.vs such that for all i.
Theorem 4. In the i.i.d. case, for every nonnegative measurable function f, the following inequality holds:where Corollary 8. For any measurable nonnegative function in (7),with , where . In particular, in the non-i.i.d. case, the factor in (29) may be replaced with . For an arbitrary vector ,where . In Theorem 4 and Corollary 8, we do not require the existence of the expectations considering their values on the extended real line. It is clear that, in the non-i.i.d. case, inequalities (
29) and (
30) provide a sufficiently good upper bound under the so-called Poissonian setting when the summand distributions have large atoms at zero, i.e., the probabilities
are such that the constant
is not too large.
Notice that some particular cases of inequality (
29) are contained in [
1,
13].
Remark 5. In the case , there exists a slightly better upper bound than that in (29). In this case, the factor on the right-hand side of (29) can be replaced with . However, in the special case when , where are independent Bernoulli r.vs with respective parameters , there exists a better upper bound than that in (29). In this case, we can replace the factor with where (see [14,15]). Corollary 9. Let g be a nonnegative function satisfying the condition for some Then, for every n and p satisfying the condition the following inequality holds:
Moreover, Remark 6. It is worth noting that, under the minimal moment condition above, we cannot replace the one-sided double limit in (33) with the classical double limit, and the condition in (32) cannot be omitted. Moreover, there exists a nonnegative function (see Section 3) such that and 2. Applications to Empirical Point Processes
In this section, we formulate some consequences of the above theorems as well as some new similar results for empirical point processes indexed by subsets of a measurable space. These processes generalize the scheme of univariate empirical point processes
from the previous section. These results are a basis for the so-called Poissonization method for generalizing empirical point processes. Sometimes, it is more convenient to replace an empirical point process under study with the corresponding accompanying Poisson point process having a simpler structure for analysis (for example, independent “increments”). Some versions of this sufficiently popular and very effective method can be found in many papers. In particular, some probability inequalities connecting the distributions of empirical processes (in various settings) and those of the corresponding Poisson processes are contained in [
13,
16,
17,
18], etc.
Let
be i.i.d. r.vs taking values in an arbitrary measurable space
and having a common distribution
P. The empirical point process is introduced as
and the accompanying Poisson point process
where
is a countable family of measurable sets, and a standard Poisson random process
is independent of the collection
.
We will consider these processes as r.vs taking values in the separable Banach space
of all functions
on
such that
for some
, endowed with the norm
In the case , we deal with the supnorm . It is clear that the Banach space is isomorphic to the Banach space , where is the set of natural numbers, and is a discrete probability measure on with . So, the point process is the accompanying compound Poisson process for the point process in the Banach space .
As a direct consequence of Proposition 1, the following assertion is valid.
Corollary 10. Let be a measurable functional on , which is convex in the positive direction with respect to the standard pointwise partial order in function spaces. Then, under appropriate moment conditions, the following inequality holds: As examples, one can consider functionals of the form
where
G and
are nondecreasing convex functions on
provided that
for all
. For such functionals, it is easy to verify the conditions of Corollary 10 (see the proof of Corollary 2).
By analogy with the univariate case, the centered empirical point process and the corresponding accompanying compound Poisson point process are introduced. For such processes, the second assertion of Theorem 2 can be reformulated as follows:
Corollary 11. Let be a measurable functional on having a convex second Fréchet derivative. Then,whenever the expectation on the right-hand side of this inequality exists. As examples of such functionals, one can cite .
We now introduce the so-called restricted empirical point processes. Let and . Consider the restrictions of the point processes and on the set , which is denoted by and , respectively. We call these processes -restricted point processes. In this case, , where , , are i.i.d. stochastic processes indexed by the elements from the family , with . We may consider as i.i.d. r.vs taking values in the Banach space . As a direct consequence of Theorem 4, we then obtain
Corollary 12. The following inequalities are valid:where and the expectations take their values on the extended real line. We now introduce a class of additive statistics of the empirical point processes. Let
be a finite or countable measurable partition of the sample space. We assume that
for all
i and
. Denote
. We study a class of additive functionals of the form
where
is an array of functions on
, with
.
Example 3. We now give a few examples of additive statistics.
(1) Given a finite partition , put , . Then, we deal with a -statistic of the form (2) The log-likelihood function can be represented as the following linear functional: (3) Fix a countable partition . Let . (For countable partitions, we assume that ). Then, the functionalis the number of cells , each of them contains a number of the sample points from the range defined by a subset A of naturals. It is the so-called infinite multinomial scheme of placing particles (balls) in cells (boxes) (for example, see [19,20,21,22,23,24]). For additive functionals (
38), one can also obtain Poissonization inequalities using the above mentioned inequalities for restricted empirical point processes. The next theorem is related to estimating the distribution tails of additive functionals (
38) via the probability tails of the same functionals of the accompanying Poisson point process
. The main property of the functionals
is that they have a structure of sums of independent r.vs.
Theorem 5. Let for all i. Then, for any ,where . If, additionally,for some natural number m, then, Remark 7. It is worth noting that, in (41), the constant may be interpreted as a level of truncation for the r.v. . In this case, we should add the probability to the right-hand side of inequality (41). Integrating both sides of inequality (
40) in
x on the positive half-line, we obtain
Corollary 13. Under the conditions of Theorem 5, let F be a nondecreasing function defined on , continuous at zero and . If , then, As an example, consider the functional
defined in (
39). Then, as a consequence of (
41) and Chernoff’s upper bound (see [
25]) for the probability tail of a sum of independent nonidentically distributed Bernoulli r.vs (the transition from finite sums to series in this case is obvious), we obtain the following result.
Corollary 14 ([
24]).
Put . Then, the following inequality holds for any :where . Remark 8. We note that one can replace the Poissonian mean in (43) with the mean , which differs from by no more than 1 due to Barbour–Hall’s estimate of the Poisson approximation to a binomial distribution (see [24,26]). Further, if the condition is met as , then, from (43), we obtain not only the law of large numbers (already formulated in Corollary 2) but at a certain growth rate of the sequence , the strong law of large numbers (SLLN). In particular, if , then, for any subset of natural numbers B (see [24]). If in the case , we consider the infinite intervals for any , then, the SLLN is valid only under the condition for all i. This follows from estimate (43), monotonicity of the functions and simple arguments in proving SLLN in [27] (see also [21]). Moreover, inequality (43) allows us to estimate the rate of convergence in SLLN. If , this rate of convergence has the order .