Contemporary Mathematics Volume 00, XXXX

A Note on Sums of Independent Random Variables

Paweł Hitczenko and Stephen Montgomery-Smith

Abstract. In this note a two sided bound on the tail probability of sums of independent, and either symmetric or nonnegative, random variables is obtained. We utilize a recent result by Latała on bounds on moments of such sums. We also give a new proof of Latała’s result for nonnegative random variables, and improve one of the constants in his inequality.

1. Introduction

Recently Latała (1997) obtained the following remarkable result: for a sequence of random variables (X_{n}) and 1 ≤ p < ∞ define the following Orlicz norm

(1.1)  |||(X_{k})|||_{p} = inf{λ > 0 : ∏_{n} 𝔼|1 + X_{n}/λ|^{p} ≤ e^{p}}.

Latała proved that

(1.2)  \frac{e − 1}{2e^{2}}|||(X_{k})|||_{p} ≤ (𝔼| ∑ X_{k}|^{p})^{1/p} ≤ e|||(X_{k})|||_{p},

provided (X_{n}) are either symmetric or positive, and in the first case p ≥ 2, and in the second case p ≥ 1. The main novelty here is the fact that, contrary to the classical inequalities, the constants here are independent of p. Certain particular cases of Latała’s result had been known earlier (see e.g. Hitczenko (1993), Gluskin and Kwapień (1995) or Hitczenko, Montgomery-Smith and Oleszkiewicz (1997)), but they can be easily deduced from Latała’s inequality.

Of course, the ultimate goal is to obtain bounds on the tail probabilities for sums of random variables. Latała’s result prompted us to investigate that problem. This program has been completed; our methods, which are based on estimates for the decreasing rearrangement of a random variable, work in a rather general setting. As a result we were able to obtain extensions of Latała’s result in various directions. The details of that approach will be presented elsewhere. The goal of this note is quite different; we will present a very simple argument that allows

\begin{matrix}c \\ ◯ \end{matrix}XXXX American Mathematical Society

1991 Mathematics Subject Classification. Primary 60G50, 60E15; Secondary 46E30.

Key words and phrases. sums of independent random variables, tail distributions.

The first author was partially supported by NSF grant DMS 9401345

The second author was partially supported by NSF grant DMS 9424396, and by the University of Missouri Research Board

one to deduce tail bounds from Latała’s result. As a matter of fact, this approach formally does not really depend on Latała’s result, but it requires a knowledge of his bounds on moments in order to be employed successfully. We will also present a short proof (based on decoupling techniques) of Latała’s result for non-negative random variables. Our proof gives a slightly better constant on the left-hand side of (1.2).

Our notation is standard; for a sequence (z_{k}) we let z_{n}^{∗} = max_{1≤k≤n} |z_{k}|. The letters

c and C denote absolute constants whose values may change from one use to the next. We will write S = ∑_{k=1}^{∞} X_{k}, and S_{n} = ∑_{k=1}^{n} X_{k}, and ∥S∥_{p} = (𝔼|S|^{p})^{1/p}.

2. Tail estimates via moment estimates

In this section we will to obtain two-sided estimates for tails of sums of independent random variables. For the sake of brevity we will concentrate on symmetric random variables, although it will be clear that our arguments work for nonnegative random variables as well. In certain special cases tail inequalities have been obtained from moment inequalities (see Gluskin and Kwapień (1995), Hitczenko and Kwapień (1994) or Hitczenko, Montgomery-Smith and Oleszkiewicz (1997)). Also, in the case of multiples of Rademacher random variables, two-sided estimates have been obtained by Montgomery and Odlyzko (1988), and Montgomery-Smith (1990).

Theorem 2.1. There exist positive constants c, C, α and δ such that for all sequences of independent symmetric random variables (X_{n}), and for all t such that

t ≥ \frac{1}{2}∥∑ X_{i}I(|X_{i}| ≤ t)∥_{2} = \frac{1}{2} (∑_{i=1}^{n} ∥X_{i}I(|X_{i}| ≤ t)∥_{2}^{2})^{1/2} ,

the following holds: Let p_{t} be the least p such that

∥∑ X_{i}I(|X_{i}| ≤ t)∥_{p} ≥ 2t.

Then we have the inequalities

(2.1)  ℙ(|S_{n}| > t) ≥ c{ℙ(X_{n}^{∗} > t) + exp(−αp_{t})},

and

(2.2)  ℙ(|S_{n}| > 4t) ≤ C{ℙ(X_{n}^{∗} > t) + exp(−δp_{t})}.

If t ≤ \frac{1}{2}∥∑ X_{i}I(|X_{i}| ≤ t)∥_{2}, then

ℙ(|S_{n}| > t) ≥ c.

Proof. For a given t, let Y_{i} = X_{i}I(|X_{i}| ≤ t), and let s_{n} = ∑_{j=1}^{n} Y_{j}. Notice that ∥s_{n}∥_{p} is a continuous, increasing function of p, and that ∥s_{n}∥_{2} ≤ 2t. Hence either 2 ≤ p_{t} < ∞ and ∥s_{n}∥_{p_{t}} = 2t, or p_{t} = ∞ and ∥s_{n}∥_{∞} ≤ 2t.

Let us start by proving (2.1). It follows from Levy’s inequality and contraction principle (see e.g. Kwapień Woyczyński (1992, Propositions 1.1.2 and 1.2.1)) that

ℙ(X_{n}^{∗} > t) ≤ 2ℙ(|S_{n}| > t),

and,

ℙ(|s_{n}| > t) ≤ 2ℙ(|S_{n}| > t).

Hence

(2.3)  ℙ(|S_{n}| > t) ≥ \frac{1}{4}(ℙ(X_{n}^{∗} > t) + ℙ(|s_{n}| > t)).

Now we can see that if p_{t} = ∞, then the inequality is established. In the case that p_{t} < ∞, we need to obtain a lower estimate for the tail probability of a maximum of partial sums of uniformly bounded symmetric random variables (Y_{i}). But for such random variables, the following inequality is true (cf. Hitczenko (1994)): for all q ≥ p ≥ 1, we have

(2.4)  \begin{matrix}∥s_{n}∥_{q} ≤ & C \frac{q}{p}\{∥s_{n}∥_{p} + ∥Y_{n}^{∗}∥_{q}\} \\ ≤ & C \frac{q}{p}\{∥s_{n}∥_{p} + t\}. \end{matrix}

We also use the Paley-Zygmund inequality that states that for any non-negative random variable Z, and 0 < λ < 1,

ℙ(Z > λEZ) ≥ (1 − λ)^{2} \frac{(EZ)^{2}}{EZ^{2}} .

Since t = \frac{1}{2}∥s_{n}∥_{p_{t}} we have that

\begin{aligned}ℙ(|s_{n}| > t) & ≥ ℙ(|s_{n}|^{p_{t}} > 2^{−p_{t}}∥s_{n}∥_{p_{t}}^{p_{t}}) \\ & ≥ (1 − 2^{−p_{t}})^{2} \frac{∥s_{n}∥_{p_{t}}^{2p_{t}}}{∥s_{n}∥_{2p_{t}}^{2p_{t}}} . \end{aligned}

It follows from (2.4) that the denominator is no more than

C^{2p_{t}}\{∥s_{n}∥_{p_{t}} + t\}^{2p_{t}} ≤ (\frac{3}{2}C)^{2p_{t}}∥s_{n}∥_{p_{t}}^{2p_{t}}.

Therefore, we get the estimate

ℙ(|s_{n}| ≥ t) ≥ (1 − 2^{−p_{t}})^{2}(\frac{3}{2}C)^{−p_{t}} ≥ exp(−αp_{t}),

which, together with (2.3) gives (2.1).

Inequality (2.2) is an easy consequence of Chebyshev’s inequality. If p_{t} < ∞, then

\begin{matrix}ℙ(|S_{n}| > 4t) ≤ & ℙ(X_{n}^{∗} > t) + ℙ(|s_{n}| > 4t) ≤ ℙ(X_{n}^{∗} > t) + \frac{E|s_{n}|^{p_{t}}}{(4t)^{p_{t}}} \\ ≤ & ℙ(X_{n}^{∗} > t) + 2^{−p_{t}} = ℙ(X_{n}^{∗} > t) + exp(−δp_{t}). \end{matrix}

If p_{t} = ∞, we use the same ideas, noticing that ℙ(|s_{n}| > 2t) = 0.

Finally, if t ≤ \frac{1}{2}∥s_{n}∥_{2}, we apply the contraction principle, the Paley-Zygmund inequality, and (2.4), to get

2ℙ(|S_{n}| > t) ≥ ℙ(|s_{n}|^{2} > \frac{1}{4}E|s_{n}|^{2})

≥ \frac{9∥s_{n}∥_{2}^{4}}{16∥s_{n}∥_{4}^{4}}

≥ \frac{9∥s_{n}∥_{2}^{4}}{16C^{4}(∥s_{n}∥_{2} + t)^{4}}

which is bounded below by a universal constant. □

Remark. The above theorem allows us to approximate tails of the sums of independent random variables in terms of tails of the individual summands. This follows from the fact that in view of Latała’s result p_{t} can be approximated using only information about marginal distributions, and from the well known inequality

(2.5)  \frac{∑ ℙ(|X_{i}| > u)}{1 + ∑ ℙ(|X_{i}| > u)} ≤ ℙ(X_{n}^{∗} > u) ≤ 2\frac{∑ ℙ(|X_{i}| > u)}{1 + ∑ ℙ(|X_{i}| > u)},

which gives tails of X_{n}^{∗} in terms of tails of individual summands.

3. Another proof of Latała’s result for nonnegative rv’s

Here we intend to give another proof of Latała’s formula concerning ∥S∥_{p} for nonnegative random variables.

Theorem 3.1. Let (X_{n}) be a sequence of positive independent random variables. Then for all p ≥ 1 we have that

(3.1)  κ|||(X_{n})|||_{p} ≤ ∥S∥_{p} ≤ (e^{p} − 1)^{1/p}|||(X_{n})|||_{p},

where |||(X_{n})|||_{p} is given by (1.1), and κ is the positive number for which f(κ) = e, where

f(x) = ∑_{k=0}^{∞} \frac{(2k + 1)^{k}}{k!} x^{k}.

Proof. First note that if |||(X_{n})|||_{p} ≤ 1, then since 1+∑_{n} X_{n} ≤ ∏_{n}(1+X_{n}), we have that ∥S∥_{p}^{p} ≤ e^{p} − 1. This proves the second inequality in (3.1) To prove the first, we use certain results concerning decoupling. These ideas appear often in the literature (usually in the context of mean-zero or symmetric random variables, see e.g. Kwapień and Woyczyński (1992).) However, since we will need control of constants, we cite the following, which is a special case of de la Peña, Montgomery-Smith and Szulga (1994, Theorem 2.1.)

Lemma 3.2. Let (X_{n}) be a sequence of real valued independent random variables. Let (X_{n}^{(l)}) be independent copies of (X_{n}) for 1 ≤ l ≤ k. Furthermore, let f_{i_{1},...,i_{k}} be elements of a Banach space such that f_{i_{1},...,i_{k}} = 0 unless the i_{1}, …, i_{k} are distinct. Then for any 1 ≤ p ≤ ∞, we have that

∥ ∑_{i_{1},...,i_{k}} f_{i_{1},...,i_{k}}X_{i_{1}} ⋯ X_{i_{k}} ∥_{p} ≤ (2k + 1)^{k}∥ ∑_{i_{1},...,i_{k}} f_{i_{1},...,i_{k}}X_{i_{1}}^{(1)} ⋯ X_{i_{k}}^{(k)}∥_{p}.

Now let us finish the proof of Theorem 3.1. Note that

∏_{n} 𝔼|1 + X_{n}|^{p} = ∥∏_{n}(1 + X_{n})∥_{p}^{p},

and so by Minkowski’s inequality we have that

∥∏_{n}(1 + X_{n})∥_{p} ≤ 1 + ∑_{k=1}^{∞} ∥ ∑_{i_{1}<···<i_{k}} X_{i_{1}} ⋯ X_{i_{k}} ∥_{p}.

But if k ≥ 1

∥ ∑_{i_{1}<···<i_{k}} X_{i_{1}} ⋯ X_{i_{k}} ∥_{p} = \frac{1}{k!}∥ ∑_{distinct^{i_{1},...,i_{k}}} X_{i_{1}} ⋯ X_{i_{k}} ∥_{p}

≤ \frac{(2k + 1)^{k}}{k!} ∥ ∑_{distinct^{i_{1},...,i_{k}}} X_{i_{1}}^{(1)} ⋯ X_{i_{k}}^{(k)}∥_{p}

(where (X_{n}^{(l)}) are independent copies of (X_{n}) for 1 ≤ l < ∞)

≤ \frac{(2k + 1)^{k}}{k!} ∥ ∑_{i_{1},...,i_{k}} X_{i_{1}}^{(1)} ⋯ X_{i_{k}}^{(k)}∥_{p}

= \frac{(2k + 1)^{k}}{k!} ∥S∥_{p}^{k}.

Hence

∥∏_{n}(1 + X_{n})∥_{p} ≤ f(∥S∥_{p}),

So, if ∥S∥_{p} ≤ κ, then

∥∏_{n}(1 + X_{n})∥_{p} ≤ e,

that is,

|||(X_{n})|||_{p} ≤ 1.
□

Remark. Our constant in the second inequality of (3.1) is essentially the same as Latała’s constant. But in the first inequality our constant, which may numerically be shown to be about 0.1549, is slightly better than Latała’s constant, which is about 0.1162.

References

[1] V. de la Peña, S.J. Montgomery-Smith and J. Szulga, Contraction and decoupling inequalities for multilinear forms and U-statistics, Ann. of Probab. 22 (1994), 1745–1765.

[2] E.D. Gluskin and S. Kwapień, Tail and moment estimates for sums of independent random variables with logarithmically concave tails, Studia Math. 114 (1995), 303 - 309.

[3] P. Hitczenko, Domination inequality for martingale transforms of Rademacher sequence, Israel J. Math. 84 (1993), 161–178.

[4] P. Hitczenko, On a domination of sums of random variables by sums of conditionally independent ones, Ann. Probab. 22 (1994), 453–468.

[5] P. Hitczenko and S. Kwapień, On the Rademacher series, Probability in Banach Spaces, Nine, Sandbjerg, Denmark (J. Hoffmann-Jørgensen, J. Kuelbs, M.B. Marcus, ed.), Birkhäuser, Boston, 1994, pp. 31–36.

[6] P. Hitczenko, S.J. Montgomery-Smith and K. Oleszkiewicz, Moment inequalities for sums of certain independent symmetric random variables, Studia Math 123 (1997), 15–42.

[7] S. Kwapień and W.A. Woyczyński, Random Series and Stochastic Integrals. Single and Multiple, Birkhäuser, Boston, 1992.

[8] R. Latała, Estimation of moments of sums of independent random variables, Ann. Probab. 25 (1997), 1502–1513.

[9] H.L. Montgomery and A.M. Odlyzko, Large deviations of sums of independent random variables, Acta Arithmetica 49 (1988), 427–434.

[10] S.J. Montgomery-Smith, The distribution of Rademacher sum, Proc. Amer. Math. Soc. 109 (1990), 517–522.

Department of Mathematics, North Carolina State University, Raleigh, NC 27695–8205

E-mail address: pawel@math.ncsu.edu
URL: http://math.ncsu.edu/~pawel/

Department of Mathematics, University of Missouri–Columbia, Columbia, MO 65211

E-mail address: stephen@math.missouri.edu
URL: http://math.missouri.edu/~stephen