\vsize = 24cm
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\topmatter
\title
Small deviation probabilities of some random sums
\endtitle
\author 
{L. ~ V. ~ Rozovsky } 
\endauthor
\endtopmatter

\redefine\P{\,\bold P\,}
\define\E{\,\bold E\,}
\define\I{\,\bold I\,}
\define\V{\,\bold {Var}\,}

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\def\al{\alpha}
\def\si{\sigma}
\def\om{\omega}
\def\Om{\Omega}
\def\la{\lambda}
\def\La{\Lambda}
\def\ep{\varepsilon}
\def\ga{\gamma}
\def\Ga{\Gamma}
\def\dl{\delta}
\def\Dl{\Delta}
\def\be{\beta}
\def\ka{\kappa}

\def\publogo{\hfil}
\NoRunningHeads

\document


\head \bf 1. Introduction \endhead

Denote
$ S = \sum _ {j\ge 1} \la _ j\4 X _ j, $ 
where $ \la _ 1\ge \la _ 2\ge \cdots $ are non-negative numbers 
and $\{X _ i \} $ are independent copies of positive random variable $X$
with distribution function $V(x)$.
We assume that
$$1\le n=\#\{j|\la_j>0\}\le\infty\tag1$$
and that series $ S $ converges a.s. if  $n = \infty $.

Our main goal is to examine a behavior of the probabilities 
$ \P (r-s < S\le r) $ for positive $r$ and $s$.  
Similar problems, in the Gaussian case in the main,
were studied by many authors. Most general and exact 
(in a context of the present work)  results were obtained by 
M.A. Lifshits," On the lower tail probabilities of some random series",
Ann. Probab.(1997), v. 25, pp. 424 - 442. 

Let us formulate the basic result of his work.

In what follows, $ c _ 1, \4 c _ 2, \dots $ denote positive constants
depending only on $V$  and parameters  from conditions below,
connected with  $V$.

We assume that there exist constants $ b\in (0,1), \ c _ 1, \4 c _ 2> 1 $ 
and $ r _ 0> 0 $ such that for each $ r\le r _ 0 $
$$
c _ 1\4 V (br) \le V (r) \le c _ 2\4 V (br) .\tag $ L $
$$

Note that  condition $(L)$ implies
$$
c_ 3 r ^ \al \le V (r) \le c _ 4 r ^ \be, \ r\le r _ 0, \tag2
$$
with some positive constants $ \al, \be $.

As is easily seen,  $(L)$ follows if $V(r)$ is regular
varying at zero. In particular, $(L)$ includes the important
Gaussian case $ X = | \xi | ^ p, \ p> 0 $, where 
$ \xi $ is a standard normal variable.

For $ \ga\ge 0 $ define
$$\gathered
\La (\ga) = \E e ^ {-\ga S}, \quad m (\ga) = -\left (\log {\La (\ga)} \right)
', \quad \si ^ 2 (\ga) = \left (\log {\La (\ga)} \right) ", \\
Q (\ga) = -\ga\4 m (\ga) -\log {\La (\ga)}.
\endgathered\tag3 $$

\proclaim {Theorem} Let condition $(L)$ hold and let
random variable $ X $ have \newline 
a) a finite variance, \newline
b) absolute continuous distribution.\newline 
Also assume that  \newline 
c)  $ \{\la _ j \} $ is a nonincreasing positive sequence such that 
$\sum _ {j\ge 1} \la _ j < \infty. $\newline Then
$$ \P (S\le r) = (2\pi) ^ {-1/2} \4 (\ga\si (\ga)) ^ {-1} \4 e ^ {-Q (\ga)} \4
\big (1 + o\4 (1) \big), \ r\searrow 0, $$ 
where the parameter $ \ga = \ga (r) $
is the solution of the equation $ m (\ga) = r $.
\newline
Moreover, if $ \E X ^ 3 < \infty $, then for $ \al $ from $(2)$ and 
for each $\ka\in (0,2) $
$$\multline
\P (S\le r) = (2\pi) ^ {-1/2} \4 (\ga\si (\ga)) ^ {-1} \4 e ^ {-Q (\ga)} \\
\left (1 + O\4\big ((\ga\si (\ga)) ^ {-1} + (\ga\si (\ga)) ^ {- \ka /\al}
\big) \right), \ r\searrow 0.
\endmultline\tag4
$$
\endproclaim

Note that assumptions a) - c) are {\it essentially} used in proofs.

We prove that a {\it nonasymptotic} approximation, 
close to (4), remains valid without assumption b) and 
under significant relaxation of conditions a) and b).




\head \bf 2. Results
\endhead

\proclaim {Theorem 1} Let condition $(L)$ hold. Then
$$
e ^ {-Q (\ga)} \ge \P (S\le r) \ge
e ^ {-Q (\ga) -c _ 5\4\sqrt {1 + Q (\ga)}}, \quad 0 < r < \4 \E S, \tag 5
$$
where $ \ga $ is the unique solution of the equation $ m (\ga) = r $.
\endproclaim


From (5) it follows that  if $ Q (\ga) $ tends to infinity then
$\log {\P (S\le r)} $ {\it approximately} equals $ -Q (\ga) $.

To formulate more exact result we need some information on a behaviour of $V$ 
at infinity.  Let us introduce the following condition
$$\limsup\limits _ {r\to\infty} \frac {r ^ 2 \,\P (X\ge r)} {\E X ^ 2 \,\bold I
[X\le r]} < \infty.\tag $ F $ $$ 
Remark that condition $ (F) $, so-called condition of Feller's 
stochastic compactness, holds if $ X $ belongs to 
domain of attraction of any stable law, including normal.
One can also show that condition $ (F) $ is equivalent to relation
$$
r^{-\om} \E\bigl (1\land r \, X ^ 2\bigr) \nearrow, r> 0, \ \exists\ \om\in
(0,1). \tag6 $$

\proclaim {Theorem 2} Let conditions $(L)$ and $ (F) $ hold. 
Then for all positive $ r, \, s $ and $ \ga $
$$\multline
\P (r-s < S\le r) = \La (\ga) \4e ^ {\ga\4r} \4\frac {1-e ^ {-\ga \, s}}
{\tau\sqrt {2\pi}} \\ \left (e ^ {- (r-m (\ga)) ^ 2/2\si ^ 2 (\ga)} + \theta
\,\left (\tau ^ {-1} + \left (\frac {\ln {(1 + \tau)}} {\tau ^ 2} \right) ^
{1/\al} \bigl (1 + (\ga s) ^ {-1} \bigr) \right) \right), \endmultline
\tag7$$
where $ \tau = \ga \,\si (\ga)$ $($ see $ (3)) $, $ \al $ is defined in $(2)$  
and $ | \theta | \le c_6 $.

In particular, if $0 < r <\4 \E S $ and $ \ga $ is the solution of 
the equation $ m (\ga) = r $, then
$$\multline\P (r-s < S\le r) = e ^ {-Q (\ga)} \4\frac {1-e ^ {-\ga \, s}}
{\tau\sqrt {2\pi}} \\ \left (1 + \theta \, \left (\tau ^ {-1} + \left (\frac
{\ln {(1 + \tau)}} {\tau ^ 2} \right) ^ {1/\al} \bigl (1 + (\ga s) ^ {-1}
\bigr) \right) \right). \endmultline
\tag8$$
\endproclaim

Remark. In the case $n<\infty$ (see (1)) condition $(F)$, generally
speaking, can be removed: under conditions of theorem 2, 
relations (7), $\ga >\dl/\la _ n $,  and (8), $ 0 < r < m (\dl/\la _ n) $,
with $ | \theta | \le c (V, \dl) $ hold true for each positive $ \dl $  
without $(F)$.

We pay attention that the equalities  (7) and (8) are informative if value of 
parameter $s$ is not too small.
Consequently, the local version of theorem 2 is also obtained.

\newpage


Assume that random variable $ X $ has an absolute continuous distribution 
with a density $f$ such that for some positive $ C, x _ 0, \dl $ and $ p> 1 $
$$\gather
|df(x)|\le C\,V(x)\,x^{-2}\,dx,\ 0<x\le x_0,\tag9\\
\int\limits_{x_0}^\infty \left(x^{-\dl} f(x)\right)^p dx <\infty.\tag10\\
\endgather$$                       

\proclaim {Theorem 3} Let distribution function $ V $ and its density $ f $ 
satisfy conditions $(L)$, $ (F) $ and  $ (9),(10) $, respectively.
Assume also that $ n\ge n _ 0 $, where integer $n_0>(1+\al)/(1\land\be)$ 
and $n_0\ge {2 \lor p / (p-1)} $ .
Then for all positive $ r, \, s $ and all $ \ga\ge \dl/\la _ {n _ 0} $
$$
\P (r-s < S\le r) = \La (\ga) \4 e ^ {\ga\4r} \4
\frac {1-e ^ {-\ga \, s}} {\ga \,\si (\ga) \sqrt {2\pi}}
\left (e ^ {- (r-m (\ga)) ^ 2/2\si ^ 2 (\ga)} + \theta\4 (\ga \,\si (\ga)) ^
{-1} \right)
\tag11 $$
with $|\theta| \le c _ 8 \,\left (1 + \la^{-2}_{n_0} \, 
\sum\limits _ {j\ge 1} \E (1 \land \la _ j\4 X)^2\right).$ 

In particular, if $ 0 < r\le m (\dl/\la _ {n _ 0}) $ and $ m (\ga) = r $, then
$$
\P (r-s < S\le r) = e ^ {-Q (\ga)} \4\frac {1-e ^ {-\ga \, s}} {\tau\sqrt
{2\pi}}
\left (1 + \theta\4 (\ga \,\si (\ga)) ^ {-1} \right).
$$
\endproclaim

Note that equality (11) in contrast to (7) is nontrivial 
for arbitrary small values of a parameter $s$.
This allows, by dividing both parts of (11) on $s$ and running $ s $ to zero,
to find an appropriate asymptotics of a density function $q(r)$ of 
random variable $ S $. 

So, if the conditions of theorem 3 hold,
$$
q(r) = \bigl (\si (\ga) \4\sqrt {2\pi} \bigr) ^ {-1} \4
\La (\ga) \4 e ^ {\ga\4r} \4
\left (e ^ {- (r-m (\ga)) ^ 2/2\si ^ 2 (\ga)} + \theta\4 (\ga \,\si (\ga)) ^
{-1} \right),
$$
and, if $ 0 < r\le m (\dl/\la _ {n _ 0}) $, and $ m (\ga) = r $,
$$
q(r) = \bigl (\si (\ga) \4\sqrt {2\pi} \bigr) ^ {-1} \4 e ^ {-Q (\ga)} \4 
\left (1 + \theta\4 \left (\ga \,\si (\ga) \right) ^ {-1} \right).
$$

Remark. If in conditions of theorem 3 to replace (10) by more 
restrictive assumption
$$
\int\limits_{x_0}^\infty x^{-\dl} f(x) dx <\infty
$$                       
(and to reject the inequality $ n _ 0\ge {2 \lor p / (p-1)} $),
then relation (11) holds true with $ | \theta | \le c _ 8 $.


\newpage

\head \bf 3. Corollaries.
\endhead

We formulate several corollaries of theorems 1 and 2.
The case $ s = \infty $ is considered for a simplicity. 

Let  $\{\la _ j \} $ be a nonincreasing sequence of {\it positive} numbers 
such that 
$$
\sum\limits _ {j\ge 1} \E (1\land \la _ j\4 X) < \infty.\tag12
$$    
Note that condition (12) implies the convergence of  series 
$ \sum\limits _ j \la _ j $ and, if $ \E X < \infty $, conversely. 

It is simple to verify that
$$\gather
(12),\  \la _ j = j ^ {-\om}, \ \om > 1, \Longleftrightarrow 
\E X ^ {1/\om} < \infty, \\ 
(12), \ \la _ j = q ^ j, \ 0 < q < 1,  \Longleftrightarrow \
E\log {(1 + X)} < \infty, 
\endgather$$
and the last condition follows from $ (F) $.


\proclaim {Corollary 1} Let condition $(L)$ hold.
Then
$$-\log {\P (S\le r)} \sim Q (\ga), \ r\to 0, $$
where $ m (\ga) = r $.\newline
If conditions $(L)$ and $ (F) $ are satisfied then uniformly in $ r $ 
$$\multline
\P (S\le r) = \La (\ga) \4 e ^ {\ga\4r} \4 (\tau\sqrt {2\pi}) ^ {-1} \\
\left (e ^ {- ((r-m (\ga)) /\si (\ga)) ^ 2/2} + O\4 (1/\tau + 
(1/\tau) ^ {2/\al} \4\ln^{1/\al} {\tau)} \right), \ \ga\to \infty.
\endmultline\tag13 $$
Thus, if $ m (\ga) = r $,
$$
\P (S\le r) = e ^ {-Q (\ga)} \4 (\tau\sqrt {2\pi}) ^ {-1}
\left (1 + O\4 (1/\tau + (1/\tau)^{2/\al}\4\ln^{1/\al} {\tau} \right), \ r\to 0.
\tag14$$
\endproclaim
    
As one more example we consider small deviation probability of sum  
$ S=S(p) =(1-p) \sum _ {j\ge 1} p^{j-1}\4 X _ j$ as $p \nearrow 1$.

Let constant $\mu<\E X$. 

\proclaim {Corollary 2} If conditions $(L)$ and $(F)$ hold 
then uniformly in $r, 0<r<\mu, $ 
$$
\P (S\le r) = e ^ {-Q (\ga)} \4 (\tau\sqrt {2\pi}) ^ {-1}
\left (1 + o\4 (1)\right), \ p \nearrow 1,
\tag15$$
where $m(\ga) = r$.
\endproclaim

The result below follows from theorem 2 in the case of finite $n$ (see (1)).
Let now $ S=S_n = \sum_{j=1}^n X _ j$ and
$$\gathered
L(\ga) = \E e ^ {-\ga X}, \quad m(\ga) = -\left (\log {L(\ga)} \right)', 
\quad \si ^ 2 (\ga) = \left (\log {L(\ga)} \right) ",\\
s(\ga) = -\ga\4 m (\ga) -\log {L(\ga)},\ \ga\ge 0.
\endgathered$$

\proclaim {Corollary 3} If condition $(L)$  holds 
then uniformly in $r, 0<r<\mu<\E X, $ 
$$
\P (S\le r\,n) = e^{-n s(\ga)} \4 (\ga\si(\ga)\sqrt {2\pi n}) ^ {-1}
\left (1 + o\4 (1)\right), \ n\to\infty,
\tag16$$
where $m(\ga) = r$.
\endproclaim

The similar result under assumption that $V(r)$ is regular varying at zero,
was obtained by  T. Hoglund,"A unified formulation of the central limit 
theorem for small and large deviations from the mean", 
Z.Wahrscheinlichkeitstheor.verw. Geb., v. 49 (1979), pp. 105 - 117.

\newpage

\head \bf 4. Lemmas.
\endhead

The following lemmas play crucial role in the proofs.

\proclaim {Lemma 1}  
For all $ r, \ 0 < r < \4 \E S $,
$$e^ {-Q (\ga)} \ge \P (S\le r) \ge 
\tfrac {1} {2} \4 e ^ {-2a\4 (1 + \sqrt {1 + 2Q (\ga) /a})} \ e^{-Q(\ga)},
\tag17$$
where $ a = \sup _ {u> 0} \frac {u ^ 2\4\si ^ 2 (u)} {Q (u)} $ and $ \ga $ is 
the unique solution of the equation $ m (\ga) = r $.
\endproclaim

Let random variable $S(h), \, h\ge 0, $ have so called conjugated distribution
$$\P(S(h)\le r)= \int_0^r e ^ {-hy} d\P (S\le y) /\La (h), \ r\ge 0. $$ 
Note (see (3)) that $ m(h) = \E S (h), \quad \si ^ 2 (h) = \V S (h). $

Put 
$$\gathered
g_h (t) = \E\exp {\left (it\4\frac {S (h) -m (h)} {\si (h)} \right)}, 
\ \tau = h\si (h) \ ( h> 0), \\
\dl _ \ep (h) = \int\limits _ 0 ^ {1/\ep} |g_h(t)-e^{-t^2/2}|\, dt\ (\ep > 0). 
\endgathered $$

\proclaim {Lemma 2} For all positive $ r, \ h, \ s $ and $ \ep $
$$\P (r-s < S\le r) = \La (h) \4 e ^ {h\4r} 
\4\frac {1-e ^ {-h \, s}} {\tau\sqrt {2\pi}}\, 
\left(e^{-\be^2/2}+\theta\,  (\be \, e ^ {-\be ^ 2/2} /\tau + 1/\tau ^ 2 + 
\rho) \right). $$
Here
$$\be = \frac {r-m (h)} {\si (h)}, \ \rho=\rho _ \ep (h, s) = \dl _ \ep (h) + 
( 1 + \dl _ \ep (h)) (1 + \tfrac {1} {h \, s}) \, \tau \ep, \
|\theta|\le c, $$ 
where $ c $ is an absolute constant.

In particular, if $0<r<\E X$ and $ m (h) = r $ then 
$$\P (r-s < S\le r) = e^{-Q (h)} \4\frac {1-e ^ {-h \, s}} {\tau\sqrt {2\pi}} \,
\left (1 + \theta \, (1/\tau ^ 2 + \rho _ \ep (h, s) \right).\tag18 $$
\endproclaim

Pay attention that lemmas 1 and 2 do not require the fulfilment 
of conditions $(L)$ or $ (F) $.


Let random variable $ X (h), \ h\ge 0 $, have distribution 
$ e^{-hr} \4 V (dr) /L (h) $.

\proclaim {Lemma 3}  

1) If condition $ (F) $ holds then for all $ 0 < h\le 1 $ 
$$\align 
c_9\le h\E X (h) /G _ 1 (h) \le c _ {10}, \tag 19a \\
c_9\le h ^ 2\bold {Var} X (h) /G (h) \le c _ {10}, \tag 19b \\
h\E X ^ 3 (h) /\bold {Var} X (h) \le c _ {11},\tag 19c
\endalign $$
where $ G (h) = \E\bigl (1\land (h \, X) ^ 2\bigr) $, 
$ G _ 1 (h) = \E (1\land h \, X) $.

2) If condition $(L)$ holds then for all $ h\ge 1 $ 
$$\align 
c _ 9\le h\E X (h)\le c _ {10}, \tag 20a \\
c _ 9\le h ^ 2\bold {Var} X (h) \le c _ {10}, \tag 20b \\
h ^ 3\E X ^ 3 (h) \le c _ {11} .\tag 20c
\endalign $$
\endproclaim

\enddocument
