SMALL DEVIATION PROBABILITIES

OF SOME RANDOM SUMS

L. V. Rozovsky

1. Introduction

Denote S = ∑_{j≥1} λ_{j} X_{j}, where λ_{1} ≥ λ_{2} ≥ ⋯ are non-negative numbers and {X_{i}} are independent copies of positive random variable X with distribution function V (x). We assume that

(1)  1 ≤ n = #{j|λ_{j} > 0} ≤ ∞

and that series S converges a.s. if n = ∞.

Our main goal is to examine a behavior of the probabilities P (r −s < S ≤ 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}, c_{2}, … denote positive constants depending only on V and parameters from conditions below, connected with V.

We assume that there exist constants b ∈ (0, 1), c_{1}, c_{2} > 1 and r_{0} > 0 such that for each r ≤ r_{0}

(L)  c_{1} V (br) ≤ V (r) ≤ c_{2} V (br).

Note that condition (L) implies

(2)  c_{3}r^{α} ≤ V (r) ≤ c_{4}r^{β}, r ≤ r_{0},

with some positive constants α, β.

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

For γ ≥ 0 define

(3)

Λ(γ) = E e^{−γS}, m(γ) = − (log Λ(γ))′ , σ^{2}(γ) = (log Λ(γ)) ”,
Q(γ) = −γ m(γ) − log Λ(γ).

Theorem. Let condition (L) hold and let random variable X have

a) a finite variance,

b) absolute continuous distribution. Also assume that

c) {λ_{j}} is a nonincreasing positive sequence such that ∑_{j≥1} λ_{j} < ∞. Then

P (S ≤ r) = (2π)^{−1/2} (γσ(γ))^{−1} e^{−Q(γ)} (1 + o(1)), r ↘ 0,

where the parameter γ = γ(r) is the solution of the equation m(γ) = r. Moreover, if E X^{3} < ∞, then for α from (2) and for each κ ∈ (0, 2)

(4) P (S ≤ r) = (2π)^{−1/2} (γσ(γ))^{−1} e^{−Q(γ)}

(1 + O((γσ(γ))^{−1} + (γσ(γ))^{−κ/α})) , r ↘ 0.

Note that assumptions a) - c) are essentially used in proofs.

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

Typeset by AMS-TEX

2. Results

Theorem 1. Let condition (L) hold. Then

(5)  e^{−Q(γ)} ≥ P (S ≤ r) ≥ e^{−Q(γ)−c_{5} \sqrt{1+Q(γ)}}, 0 < r < E S,

where γ is the unique solution of the equation m(γ) = r.

From (5) it follows that if Q(γ) tends to infinity then log P (S ≤ r) approximately equals −Q(γ).

To formulate more exact result we need some information on a behaviour of V at infinity. Let us introduce the following condition

(F)  lim sup_{r→∞} \frac{r^{2} P (X ≥ r)}{E X^{2} I[X ≤ r]} < ∞.

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

(6)  r^{−ω} E (1 ∧ r X^{2}) ↗, r > 0, ∃ ω ∈ (0, 1).

Theorem 2. Let conditions (L) and (F) hold. Then for all positive r, s and γ

(7) P (r − s < S ≤ r) = Λ(γ)e^{γ r} \frac{1 − e^{−γ s}}{τ\sqrt{2π}}

(e^{−(r−m(γ))^{2}/2σ^{2}(γ)} + θ (τ^{−1} + (\frac{ln (1 + τ)}{τ^{2}})^{1/α} (1 + (γs)^{−1}))) ,

where τ = γ σ(γ) ( see (3)), α is defined in (2) and |θ| ≤ c_{6}.

In particular, if 0 < r < E S and γ is the solution of the equation m(γ) = r, then

(8) P (r − s < S ≤ r) = e^{−Q(γ)} \frac{1 − e^{−γ s}}{τ\sqrt{2π}}

(1 + θ (τ^{−1} + (\frac{ln (1 + τ)}{τ^{2}})^{1/α} (1 + (γs)^{−1}))) .

Remark. In the case n < ∞ (see (1)) condition (F), generally speaking, can be removed: under conditions of theorem 2, relations (7), γ > δ/λ_{n}, and (8), 0 < r < m(δ/λ_{n}), with |θ| ≤ c(V, δ) hold true for each positive δ 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.

Assume that random variable X has an absolute continuous distribution with a density f such that for some positive C, x_{0}, δ and p > 1

|df(x)| ≤ C V (x) x^{−2} dx, 0 < x ≤ x_{0}, (9)

∫_{x_{0}}^{∞}(x^{−δ}f(x))^{p} dx < ∞. (10)

Theorem 3. Let distribution function V and its density f satisfy conditions (L), (F) and (9), (10), respectively. Assume also that n ≥ n_{0}, where integer n_{0} > (1 + α)/(1 ∧ β) and n_{0} ≥ 2 ∨ p/(p − 1) . Then for all positive r, s and all γ ≥ δ/λ_{n_{0}}

(11) P (r − s < S ≤ r) = Λ(γ)e^{γ r} \frac{1 − e^{−γ s}}{γ σ(γ)\sqrt{2π}} (e^{−(r−m(γ))^{2}/2σ^{2}(γ)} + θ(γ σ(γ))^{−1})

with |θ| ≤ c_{8} (1 + λ_{n_{0}}^{−2} ∑_{j≥1} E (1 ∧ λ_{j} X)^{2}) .

In particular, if 0 < r ≤ m(δ/λ_{n_{0}}) and m(γ) = r, then

P (r − s < S ≤ r) = e^{−Q(γ)} \frac{1 − e^{−γ s}}{τ\sqrt{2π}} (1 + θ(γ σ(γ))^{−1}) .

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) = (σ(γ)\sqrt{2π})^{−1} Λ(γ)e^{γ r} (e^{−(r−m(γ))^{2}/2σ^{2}(γ)} + θ(γ σ(γ))^{−1}) ,

and, if 0 < r ≤ m(δ/λ_{n_{0}}), and m(γ) = r,

q(r) = (σ(γ)\sqrt{2π})^{−1} e^{−Q(γ)} (1 + θ (γ σ(γ))^{−1}) .

Remark. If in conditions of theorem 3 to replace (10) by more restrictive assumption

∫_{x_{0}}^{∞}x^{−δ}f(x)dx < ∞

(and to reject the inequality n_{0} ≥ 2 ∨ p/(p − 1)), then relation (11) holds true with |θ| ≤ c_{8}.

3. Corollaries.

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

Let {λ_{j}} be a nonincreasing sequence of positive numbers such that

(12)  ∑_{j≥1} E (1 ∧ λ_{j} X) < ∞.

Note that condition (12) implies the convergence of series ∑_{j} λ_{j} and, if E X < ∞,

conversely.

It is simple to verify that

(12), λ_{j} = j^{−ω}, ω > 1, ⟺ E X^{1/ω} < ∞,
(12), λ_{j} = q^{j}, 0 < q < 1, ⟺ E log (1 + X) < ∞,

and the last condition follows from (F).

Corollary 1. Let condition (L) hold. Then

− log P (S ≤ r) ∼ Q(γ), r → 0,

where m(γ) = r. If conditions (L) and (F) are satisfied then uniformly in r

(13) P (S ≤ r) = Λ(γ)e^{γ r} (τ\sqrt{2π})^{−1}

(e^{−((r−m(γ))/σ(γ))^{2}/2} + O(1/τ + (1/τ)^{2/α} ln^{1/α} τ)) , γ → ∞.

Thus, if m(γ) = r,

(14)  P (S ≤ r) = e^{−Q(γ)} (τ\sqrt{2π})^{−1} (1 + O(1/τ + (1/τ)^{2/α} ln^{1/α} τ) , r → 0.

As one more example we consider small deviation probability of sum S = S(p) = (1 − p) ∑_{j≥1} p^{j−1} X_{j} as p ↗ 1.

Let constant μ < E X.

Corollary 2. If conditions (L) and (F) hold then uniformly in r, 0 < r < μ,

(15)  P (S ≤ r) = e^{−Q(γ)} (τ\sqrt{2π})^{−1} (1 + o(1)) , p ↗ 1,

where m(γ) = r.

The result below follows from theorem 2 in the case of finite n (see (1)). Let now S = S_{n} = ∑_{j=1}^{n} X_{j} and

L(γ) = E e^{−γX}, m(γ) = − (log L(γ))′ , σ^{2}(γ) = (log L(γ)) ”,
s(γ) = −γ m(γ) − log L(γ), γ ≥ 0.

Corollary 3. If condition (L) holds then uniformly in r, 0 < r < μ < E X,

(16)  P (S ≤ r n) = e^{−ns(γ)} (γσ(γ)\sqrt{2πn})^{−1} (1 + o(1)) , n → ∞,

where m(γ) = r.

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.

4. Lemmas.

The following lemmas play crucial role in the proofs.

Lemma 1. For all r, 0 < r < E S,

(17)  e^{−Q(γ)} ≥ P (S ≤ r) ≥ \frac{1}{2} e^{−2a (1+\sqrt{1+2Q(γ)/a})} e^{−Q(γ)},

where a = sup_{u>0} \frac{u^{2} σ^{2}(u)}{Q(u)} and γ is the unique solution of the equation m(γ) = r.

Let random variable S(h), h ≥ 0, have so called conjugated distribution

P (S(h) ≤ r) = ∫_{0}^{r} e^{−hy}d P (S ≤ y)/Λ(h), r ≥ 0.

Note (see (3)) that m(h) = E S(h), σ^{2}(h) = Var S(h).

Put

\begin{matrix}g_{h}(t) = E exp (it \frac{S(h) − m(h)}{σ(h)}), τ = hσ(h) (h > 0), \\ δ_{ε}(h) = ∫_{0}^{1/ε}|g_{h}(t) − e^{−t^{2}/2}| dt (ε > 0). \end{matrix}

Lemma 2. For all positive r, h, s and ε

P (r − s < S ≤ r) = Λ(h)e^{h r} \frac{1 − e^{−h s}}{τ\sqrt{2π}} (e^{−β^{2}/2} + θ (β e^{−β^{2}/2}/τ + 1/τ^{2} + ρ)) .

Here

β = \frac{r − m(h)}{σ(h)}, ρ = ρ_{ε}(h, s) = δ_{ε}(h) + (1 + δ_{ε}(h))(1 + \frac{1}{h s}) τε, |θ| ≤ c,

where c is an absolute constant.

In particular, if 0 < r < E X and m(h) = r then

(18)  P (r − s < S ≤ r) = e^{−Q(h)} \frac{1 − e^{−h s}}{τ\sqrt{2π}} (1 + θ (1/τ^{2} + ρ_{ε}(h, s)) .

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

Let random variable X(h), h ≥ 0, have distribution e^{−hr} V (dr)/L(h).

Lemma 3.

1) If condition (F) holds then for all 0 < h ≤ 1

c_{9} ≤ h E X(h)/G_{1}(h) ≤ c_{10}, (19a)

c_{9} ≤ h^{2}VarX(h)/G(h) ≤ c_{10}, (19b)

h E X^{3}(h)/VarX(h) ≤ c_{11}, (19c)

where G(h) = E (1 ∧ (h X)^{2}), G_{1}(h) = E (1 ∧ h X).

2) If condition (L) holds then for all h ≥ 1

c_{9} ≤ h E X(h) ≤ c_{10}, (20a)

c_{9} ≤ h^{2}VarX(h) ≤ c_{10}, (20b)

h^{3} E X^{3}(h) ≤ c_{11}. (20c)