\vsize = 24cm %\hsize=16true cm \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}\,} \define\ovln#1{\,{\overline{\!#1}}} \define\2{\hskip.5pt} \define\3{\hskip-7pt} \define\4{\hskip1pt} \define\5{\hskip1.3pt} \define\6{\vphantom{\dsize0}} \define\bgl#1{\bigl#1\4} \define\bgr#1{\4\bigr#1} \define\bgm{\5\big|\5} \define\kfrac#1#2{{#1}\big/{#2}} \define\pfrac#1#2{{#1}/{#2}} \define\sfrac#1#2{{}^{#1}\hskip-.5pt\!\big/\!\hskip.5pt{}_{#2}} \define\half{\sfrac12} \define\ffrac#1#2{\raise.5pt\hbox{\eightpoint$\dfrac{\,#1\,}{\,#2\,}$}} \define\Bgr#1{\4\raise.5pt\hbox{\eightpoint$\Bigr#1$}} \define\Bgl#1{\raise.5pt\hbox{\eightpoint$\Bigl#1$}\4} \define\bggr#1{\4\raise.75pt\hbox{\eightpoint$\biggr#1$}} \define\bggl#1{\raise.75pt\hbox{\eightpoint$\biggl#1$}\4} \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 $ $$\gather |df(x)|\le C\,V(x)\,x^{-2}\,dx,\ 0(1+\al)/(1\land\be)$ and $n_0\ge {2 \lor p / (p-1)} $ . Then for all positive $ r, \, s $ and all $ \ga\ge \dl/\la _ {n _ 0} $ $$ \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$$ \endproclaim As one more example we consider small deviation probability of sum $ S=S(p) =(1-p) \sum _ {j\ge 1} p^{j-1}\4 X _ j$ as $p \nearrow 1$. Let constant $\mu<\E X$. \proclaim {Corollary 2} If conditions $(L)$ and $(F)$ hold then uniformly in $r, 0 0} \frac {u ^ 2\4\si ^ 2 (u)} {Q (u)} $ and $ \ga $ is the unique solution of the equation $ m (\ga) = r $. \endproclaim Let random variable $S(h), \, h\ge 0, $ have so called conjugated distribution $$\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