On the rate of Poisson approximation to partial sum processes

TL;DR

This paper studies how closely a Bernoulli partial sum process can be approximated by a Poisson process in non-i.i.d. cases, focusing on the rate of convergence in probability.

Contribution

It provides new insights into the rate at which non-i.i.d. Bernoulli sums approximate Poisson processes.

Findings

Derived bounds for the approximation rate in non-i.i.d. settings

Extended classical Poisson approximation results to dependent cases

Quantified the minimal probability distance for convergence

Abstract

We investigate approximation of a Bernoulli partial sum process to the accompanying Poisson process in the non-i.i.d. case. The rate of closeness is studied in terms of the minimal distance in probability.

Equations

$$Pois(\mu ):=e^{-\mu (\mathbf{B})}\sum _{k=0}^{\infty }\frac{{\mu }^{\ast k}}{k!},$$

$$S_n\equiv S_n(t):=\sum _{i\le nt}{X}_i,t\in [0,1],$$

$${\Pi }_n\equiv {\Pi }_n(t):=\sum _{i\le nt}{\hat{\pi }}_i(P_i),t\in [0,1],$$

$$d(z,S_{n},\Pi_{n}):=\inf_{\{X_{k},\hat{\pi}_{k}(\cdot);\ k\leq n\}}{\bf P}\biggl{(}\sup_{t\in[0,1]}\|S_{n}(t)-\Pi_{n}(t)\|>z\biggr{)}$$

$$=\inf_{\{X_{k},\hat{\pi}_{k}(\cdot);\ k\leq n\}}{\bf P}\biggl{(}\max_{k\leq n}\biggl{\|}\sum_{i=1}^{k}(X_{i}-\hat{\pi}_{i}(P_{i}))\biggr{\|}>z\biggr{)}$$

$$V(Y_1,Y_2):=\underset{A\in \mathcal{A}}{\sup }|\mathcal{L}(Y_1)(A)-\mathcal{L}(Y_2)(A)|.$$

$$V(S_n,{\Pi }_n)=d(0,S_n,{\Pi }_n).$$

$$V(X,Y)=\underset{X,Y}{\inf }\mathbf{P}(X\ne Y),$$

$$\frac{1}{32}\varepsilon\sum_{i\leq n}p_{i}^{2}\leq V\Bigl{(}\sum_{i\leq n}\nu_{i},\sum_{i\leq n}\pi_{i}\Bigr{)}\Bigr{)}\leq\varepsilon\sum_{i\leq n}p_{i}^{2},$$

$${\overline{\nu }}_n(k):={\sum }_{i\le k}{\nu }_i,{\bar{\pi }}_n(k):={\sum }_{i\le k}{\pi }_i,k\le n.$$

$$\mathcal{L}(X_i^0)=\mathcal{L}(X_i\mid X_i\ne 0).$$

$${\cal L}(X_{k})={\cal L}\biggl{(}\,\sum\limits_{i=\overline{\nu}_{n}(k-1)+1}^{\overline{\nu}_{n}(k)}X_{k,i}^{0}\biggr{)},\quad Pois(P_{k})={\cal L}\biggl{(}\,\sum\limits_{i=\bar{\pi}_{n}(k-1)+1}^{\bar{\pi}_{n}(k)}X_{k,i}^{0}\biggr{)},$$

$$\mathcal{L}(S_n)=\mathcal{L}\left\{\sum _{i=1}^{{\bar{\nu }}_n(k)}{X}_i^0;k=1,\dots ,n\right\},\mathcal{L}({\Pi }_n)=\mathcal{L}\left\{\sum _{i=1}^{{\bar{\pi }}_n(k)}{X}_i^0;k=1,\dots ,n\right\}.$$

$$V(S_n,{\Pi }_n)\le V({\bar{\nu }}_n(\cdot ),{\bar{\pi }}_n(\cdot )).$$

$$V(S_n,{\Pi }_n)=V({\bar{\nu }}_n(\cdot ),{\bar{\pi }}_n(\cdot )).$$

$$d(z,S_n,{\Pi }_n)\le d(z,{\bar{\nu }}_n(\cdot ),{\bar{\pi }}_n(\cdot )).$$

$$\Vert \sum _{i=m+1}^{m+k}{X}_i^0\Vert =k.$$

$$d(z,S_n,{\Pi }_n)=d(z,{\bar{\nu }}_n(\cdot ),{\bar{\pi }}_n(\cdot )).$$

$$d(z,{\bar{\nu }}_n(\cdot ),{\bar{\pi }}_n(\cdot ))\le (n{p}^2{)}^{\lfloor z\rfloor +1}\exp \{-C_{1}z\ln \ln (z+2)+C_2\}\text{if}np\ge 1,$$

$$d(z,{\bar{\nu }}_n(\cdot ),{\bar{\pi }}_n(\cdot ))\le n{p}^{\lfloor z\rfloor +2}\exp \{-C_{3}z+C_4\}\text{if}np\le 1,$$

$$d(z,S_n,{\Pi }_n)\le \Delta (z,n,p),$$

$$S_n\equiv S_n(t):=\lfloor nt\rfloor F_{\lfloor nt\rfloor }(\mathbf{z}),{\Pi }_n={\Pi }_n(t):=\sum _{i=1}^{\overline{\pi }(\lfloor nt\rfloor )}I(y_i\le \mathbf{z}),\mathbf{z}\le \mathbf{a},t\in [0,1];$$

$$d(z,{\bar{\nu }}_n(\cdot ),{\bar{\pi }}_n(\cdot ))<7\cdot 10^6{\left(p^{\ast }\right)}^{z+1}\exp \left\{-\frac{1}{2}z\log \log (z+8)\right\}\text{mbox}if\sum _{i=1}^n{p}_i^2\le 1,$$

$$d(z,{\bar{\nu }}_n(\cdot ),{\bar{\pi }}_n(\cdot ))$$

$$d(0,{\bar{\nu }}_n(\cdot ),{\bar{\pi }}_n(\cdot ))\le \sum _{i=1}^n{p}_i^2.$$

$$d(z,{\bar{\nu }}_n(\cdot ),{\bar{\pi }}_n(\cdot ))\ge \frac{1}{\sqrt{2\pi }}\exp \{-(z+3)\log (z+2)+z\}\sum _{k=1}^n{B}_k{p}_k^{z+2},$$

$$d(0,\bar{\nu}_{n}(\cdot),\bar{\pi}_{n}(\cdot))\geq 1-\exp\biggl{\{}-\frac{1}{2}\sum\limits_{i\leq n}p_{i}^{2}(1-p_{i})\biggr{\}}.$$

$${\nu }_i^{\ast }:=F_{{\nu }_i}^{-1}({\omega }_i),{\pi }_i^{\ast }:=F_{{\pi }_i}^{-1}({\omega }_i),$$

$$\underset{k\le n}{\max }\left|\sum _{i=1}^k{\nu }_i^{\ast }-\sum _{i=1}^k{\pi }_i^{\ast }\right|\le \sum _{i=1}^n{\zeta }_i,$$

$$\mathbf{P}({\zeta }_i=0)=1-p_i(1-e^{-p_i}),\mathbf{P}({\zeta }_i=1)=e^{-p_i}-1+p_i+\frac{p_i^2}{2}{e}^{-p_i},$$

Full text

Key words and phrases: partial sum process, Poisson approximation, Bernoulli random variables, minimal distance, total variation distance.

§1. Introduction and results

Let $X,X_{1},X_{2},\dots$  be independent random variables taking values in a measurable Banach space \big{(}\bf B,\cal A,\|\cdot\|\big{)}, where $\cal A$ is a $\sigma$-field of subsets of ${\bf B}$. Denote by $P,P_{1},P_{2},\dots,$ the respective distributions. We will assume that the linear operations and the norm $\|\cdot\|$ are ${\cal A}$-measurable. As a rule, if ${\bf B}$ is separable, then ${\cal A}$ is the Borel $\sigma$-field. The above-mentioned measurability conditions are fulfilled in this case. For nonseparable Banach spaces, the class $\cal A$ is usually smaller than the Borel $\sigma$-field (for example, $\cal A$ may be the cylindrical or the ball $\sigma$-field).

Denote by $Pois(\mu)$ the compound Poisson distribution with Lévy measure $\mu$:

[TABLE]

where $\mu^{*k}$  is the $k$-fold convolution of a finite measure $\mu$ with itself; $\mu^{*0}$ is the unit mass at zero. As is easy to see, $Pois(\mu)$ coincides with the distribution of the sum $\sum\nolimits_{i\leq\pi(\mu({\bf B}))}Y_{i}$, where $\{Y_{i}\}$ are independent identically distributed random variables with the distribution $\mu(\cdot)/\mu({\bf B})$, and $\pi(\mu({\bf B}))$  is a Poisson random variable with mean $\mu({\bf B})$, which is independent of $\{Y_{i}\}$. (Here $\sum\nolimits_{i\leq 0}=0$ by definition.)

The main subject of the paper is the two partial sum processes

[TABLE]

and

[TABLE]

where $\{\hat{\pi}_{k}(P_{k})\}$  are independent random variables with the respective distributions $\{Pois(P_{k})\}$. We consider these ${\bf B}$-valued stochastic processes as elements of the Banach space of ${\bf B}$-valued right-continuous functions on $[0,1]$ endowed with the sup-norm. We study the proximity in terms of the distance

[TABLE]

[TABLE]

where the infimum is taken over all collections of $\{X_{k},\,i\leq n\}$ and $\{\hat{\pi}_{k}(\cdot),\,i\leq n\}$ defined on a common probability space. It is worth noting that this distance belongs to the class of so-called minimal distances between distributions of ${\bf B}$-valued stochastic processes with paths in the above-mentioned Banach space (e.g., see Zolotarev, 1976).

Alongside with (3), we also need the total variation distance between the distributions ${\cal L}(Y_{i})$ of arbitrary ${\bf B}$-valued random variables $Y_{i}$, $i=1,2$:

[TABLE]

There is the following connection between these two distances:

[TABLE]

This equality is a consequence of the following duality theorem by Dobrushin (1970):

[TABLE]

where the infimum is taken over all pairs ($X,Y$) based on a common probability space. Relation (4) means that if the total variation distance between some random variables is sufficiently small, then there exist versions of these random variables coinciding with probability close to $1$.

Systematic investigation of the accuracy of Poisson approximation to the distributions of the random variable $S_{n}(\ref{eqno1})$ as well as of the stochastic process $S_{n}(\cdot)$ began in the middle of 20-th century. We would like to note the pioneer paper by Prohorov (1953) and the three papers by Le Cam (1960, 1965, 1970) where unimprovable estimates (up an absolute constant factor) for the total variation distance $V(S_{n}(\ref{eqno1}),\Pi_{n}(\ref{eqno1}))$ were obtained. Since that time, a significant number of published studies were devoted to this problem. Regarding estimation of the total variation distance we refer the reader to the references in the book by Barbour et al. (1992) and the recent review by Novak (2019b). Notice that there is a number of papers devoted to the Poisson approximation in terms of other probability distances like $d(z,\cdot,\cdot)$ (e.g., see Ruzankin, 2001), the information divergence (e.g., see Harremoës and Ruzankin, 2004), $\chi^{2}$-distance (e.g., see Borisov and Vorozheikin, 2008), and of other distances (e.g., see Novak, 2019a; Ruzankin, 2004, 2010).

We would like to recall the most significant relevant results. In the case ${\bf B}={\bf R}$, let $X_{i}\equiv\nu_{i}$  be Bernoulli random variables with the respective success probabilities $p_{i}$. Denote by $\pi_{i}$,  $i=1,\ldots,n$, independent Poissonian random variables with respective means $p_{i},\,i=1,\ldots,n$. There is the following two-sided estimate obtained by Barbour and Hall (1984):

[TABLE]

where $\varepsilon:=\min\left\{1,\,\left(\sum_{i\leq n}p_{i}\right)^{-1}\right\}$. It is worth noting that, in general, for non-Bernoulli sequences $\{X_{i}\}$ of nonidentically distributed random variables, upper and lower bounds for the total variation distance will be slightly worse than (5) and has the unimprovable order $O\left(\sum_{i\leq n}p_{i}^{2}\right)$ (see Le Cam, 1960; Borisov, 1996).

To formulate an analog of inequality (5) for non-Bernoulli sequences $\{X_{i}\}$ we need the notation

[TABLE]

Denote by $\{X_{i}^{0}\}$ independent random variables with the distributions

[TABLE]

Note that these random variables are well defined because, by the above assumptions, the event $\{X_{i}\neq 0\}=\{\|X_{i}\|\neq 0\}$ is measurable. It is proved in Borisov (1996) that, for every $k$, the following equalities hold:

[TABLE]

where \bigl{\{}X_{k,i}^{0};i=1,2,\dots\bigr{\}} are independent copies of $X_{k}^{0}$, $k\leq n$, which do not depend on the random processes $\overline{\nu}_{n}(\cdot)$ and $\overline{\pi}_{n}(\cdot)$, with $p_{k}={\bf P}(\|X_{k}\|\neq 0)$, $k\leq n$. Notice that the first relation in (7) can be rewritten as ${\cal L}(X_{k})={\cal L}(\nu_{k}X^{0}_{k})$ but the second one cannot. If $X_{i}^{0}$ are identically distributed (whereas $X_{i}$ may be nonidentically distributed with arbitrary $p_{i}$), then (12) and independence of the increments of the processes $\bar{\nu}_{n}(\cdot)$ and $\bar{\pi}_{n}(\cdot)$ readily imply more informative Khintchine’s representations for the distributions of $S_{n}$ and $\Pi_{n}$ in the Banach space $\bf B^{n}$:

[TABLE]

The relations (4), (7) and (8) imply the following results (see Borisov, 1996):

Lemma 1. For arbitrary distributions $\{P_{i}\}$,

[TABLE]

Lemma 2. If {\bf P}\bigl{(}\sum\nolimits_{i\leq m}X_{k,i}^{0}=0\bigr{)}=0 for all natural numbers $m$ and $k$ then

[TABLE]

Lemma 3. If $\|X_{i}\|\leq 1$ with probability $1$ for all $i\geq 1$, and the random variables $\{X_{i}^{0}\}$ are identically distributed then, for all $z\geq 0$,

[TABLE]

Finally, the last assertion is connected with a special structure of $\{X_{i}\}$. We will say that $X_{i}$ are elements of indicator type with conforming supports if, for all natural $m$ and $k$, the following identity is valid with probability 1:

[TABLE]

Lemma 4. If $\{X_{i}\}$ are random elements of indicator type with conforming supports and the random variables \bigl{\{}X_{i}^{0}\bigr{\}} are identically distributed then, for all $z\geq 0$,

[TABLE]

It is worth noting that, to study the generic problem of Poisson approximation, one can find one or another implicit form of reducing the problem to that for the Bernoulli case (e.g., see Borisov and Ruzankin, 2002). The Lemmas above allow us to do it the shortest way.

In the case of i.i.d. random variables $\{X_{i}\}$ one has

Theorem 1 (Borisov, 1996). Let $\{\nu_{i}\}$  be independent identically distributed Bernoulli random variables with success probability $p$. Then, for all $z\geq 0$,

[TABLE]

[TABLE]

*where $C_{1}$– $C_{4}$  are absolute positive constants. *

Adduce an example of random elements of indicator type with conforming supports. Consider the problem of uniform Poisson approximation of so-called local multivariate empirical processes. Let $F_{n}(\bf z)$, ${\bf z}\in{\bf R}^{k},$ be the empirical distribution function based on the sample $y_{1},\dots,y_{n}$ of i.i.d. vectors with an arbitrary distribution in ${\bf R}^{k}$. Put $X_{i}\equiv X_{i}({\bf z}):={I}(y_{i}\leq\bf z)$, where ${I}(\cdot)$  is the indicator function and the sign “$\leq$” means the coordinatewise partial order in $\bf R^{k}$. We consider $X_{i}$ as elements of the Banach space of all bounded right-continuous functions (in the sense of the introduced partial order) defined on the set $\{{\bf z}\in{\bf R}^{k}:{\bf z}\leq{\bf a}\}$ and endowed with the sup-norm. Denote by ${\cal A}$ the cylindrical $\sigma$-field. Notice that the sup-norm is ${\cal A}$-measurable although the space ${\bf B}$ is not separable. Without loss of generality, we may assume that ${\bf P}(X_{i}\neq 0)\equiv{\bf P}(y_{i}\leq{\bf a})\neq 0$. Then the indicator random variables $X_{i}^{0}$ are well defined and meet (11). Therefore, from (10) and (13) we obtain the estimate

[TABLE]

where

[TABLE]

$\Delta(z,n,p)$ is the right-hand side of (13), with $p:={\bf P}(y_{1}\leq\bf a)$.

The approximation of local empirical processes, in particular, in term of the distance $d(z,{S}_{n},{\Pi}_{n})$, was studied by a number of authors (e.g., see Adell and de la Cal, 1994; Borisov, 1993, 1996, 2000; Horváth, 1990; Major, 1990), where relations similar to (10) were applied as well, which allowed to reduce the problem to the one-dimensional Bernoulli case.

The main result of the paper is the following generalization of Theorem 1, which allows us to use Lemma 3 in the non-i.i.d. case as well.

Theorem 2. For all integer $z\geq 0$,

[TABLE]

where $p^{*}=\max\left\{\max_{i\leq n}p_{i},\sum_{i\leq n}p_{i}^{2}\right\}$,

[TABLE]

Moreover,

[TABLE]

The following theorem is devoted to lower bounds for the distance $d(z,\bar{\nu}_{n}(\cdot),\bar{\pi}_{n}(\cdot))$. This assertion generalizes Theorem 2 in Borisov (1996) where the i.i.d. case was considered.

Theorem 3. For all integer $z\geq 0$,

[TABLE]

where $B_{k}:=\exp\{-\sum_{i=1}^{k}p_{i}\}\prod_{j=1}^{k-1}p_{j}$, $k=1,\ldots,n$, and $\prod_{j=1}^{0}=1$ by definition.

Moreover,

[TABLE]

Theorems 2 and 3 will be proved in Section 2.

R e m a r k. The lower bound (19) shows that, in general, the index $z+2$ on the right-hand side of (17) cannot be improved. Moreover, observe that the lower bound (20) has the asymptotic $\frac{1}{2}\sum\limits_{k\leq n}p_{k}^{2}$ if $\sum\limits_{k\leq n}p_{k}^{2}\to 0$. In other words, the total variation distance $V(\bar{\nu}_{n}(\cdot),\bar{\pi}_{n}(\cdot))$ cannot decrease faster than $\sum\limits_{k\leq n}p_{k}^{2}$ in contrast to the estimates in (5).

§2. Proofs

*Proof of Theorem *2. We begin with constructing the families of random variables $\{\nu_{i};\ i\leq n\}$ and $\{\pi_{i};\ i\leq n\}$ on a common probability space. We will use an approach, in fact, proposed independently by Serfling (1975) and Borovkov (1976, English translation: 2013). But to describe this approach we will prefer somewhat another terminology. Given a random variable $\eta$ with distribution function $F_{\eta}(t)$, introduce the quantile transform $F_{\eta}^{-1}(\omega):=\inf\{t\in\overline{\bf R}:F_{\eta}(t)\geq\omega\}$, where $\overline{\bf R}$  is the extended real line and $\omega\in[0,1]$. It is well known that if $\omega$  has the uniform distribution on $[0,1]$, then ${\cal L}(F_{\eta}^{-1}(\omega))={\cal L}(\eta)$. Now we set

[TABLE]

where $\{\omega_{i};\ i\leq n\}$ are independent random variables with the uniform distribution on $[0,1]$. One has

[TABLE]

where $\zeta_{i}:=|\nu_{i}^{*}-\pi_{i}^{*}|=(\pi_{i}^{*}-1){\bf I}\{\pi_{i}^{*}\geq 2\}+{\bf I}\{\omega_{i}\in[1-p_{i},e^{-p_{i}}]\}$. It is easy to verify that the distribution of $\zeta_{i}$ is determined as follows:

[TABLE]

[TABLE]

Hence,

[TABLE]

[TABLE]

Besides, we will need the following inequality (see relation (28) in Borisov, 1996):

[TABLE]

for $t>0$.

Prove now the relation (15) Denote

[TABLE]

Then, by (22),

[TABLE]

[TABLE]

[TABLE]

that proves inequality (15).

The estimate (18) is immediate from the definition of random variables $\zeta_{k}$:

[TABLE]

[TABLE]

Next, notice that (16) for $z=0$ follows from (18). Thus, in order to prove (16), it remains to consider the case $z\geq 1$.

First, we will prove the estimate (16) by induction on $n$, under the condition

[TABLE]

Denote $Q_{n}(k):={\bf P}\left(\sum_{i\leq n}\zeta_{i}\geq k\right),\ k=1,2,\ldots$. By the total probability formula one has

[TABLE]

It suffices to establish that, under the condition $\sum_{i\leq n}p_{i}^{2}\leq\frac{1}{3}$, the estimate

[TABLE]

is valid for all integer $k\geq 2$. For $n=1$ the relation (24) is immediate by (21), since

[TABLE]

Next, let (24) be valid for all $n\leq N-1$. We then obtain from (23)

[TABLE]

with

[TABLE]

where we assumed, without loss of generality, that $b\leq 1/2$ which can be achieved, e.g., by ordering $p_{1}\geq p_{2}\geq p_{3}\geq\cdots$.

Thus, for $\sum_{i\leq n}p_{i}^{2}\leq\frac{1}{3}$ the estimate (16) is valid. If $\frac{1}{3}\leq\sum_{i\leq n}p_{i}^{2}\leq 1$ then, in order to prove the second estimate of the theorem, we use the first one:

[TABLE]

[TABLE]

The estimate (16) is proved.

Now, prove the estimate (17). It suffices to establish that, under the condition $\sum_{i\leq n}p_{i}\leq\frac{1}{2}$, the following inequality is valid:

[TABLE]

Without loss of generality we can assume that $p_{1}\leq p_{2}\leq\ldots\leq p_{n}$. For $n=1$ the correctness of (25) is obvious. Next, let (25) be valid for all $n\leq N-1$. From (23) we then obtain

[TABLE]

where

[TABLE]

Thus, the estimate (17) and Theorem 2 are proved.

Proof of Theorem 3. Introduce the finite family of pairwise disjoint events

[TABLE]

It is clear that for any construction of independent Bernoulli random variables $\{\nu_{i}\}$ on a common probability space with independent Poissonian random variables $\{\pi_{i}\}$ the following implication of the events is valid:

[TABLE]

Hence,

[TABLE]

It remains to use Stirling’s formula.

The last statement of the theorem is immediate from the simple lower bound

[TABLE]

[TABLE]

since

[TABLE]

Theorem 3 is proved.

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doi:10.1070/sm1976v030n03abeh002280