Measuring the magnitude of sums of independent random variables

Paweł Hitczenko^{∗} Department of Mathematics and Computer Science Drexel University Philadelphia, PA 19104 phitczen@mcs.drexel.edu http://www.mcs.drexel.edu/˜ phitczen

Stephen Montgomery-Smith^{†} Department of Mathematics University of Missouri-Columbia Columbia, MO 65211 stephen@math.missouri.edu http://www.math.missouri.edu/˜ stephen

Abstract

This paper considers how to measure the magnitude of the sum of independent random variables in several ways. We give a formula for the tail distribution for sequences that satisfy the so called Lévy property. We then give a connection between the tail distribution and the pth moment, and between the pth moment and the rearrangement invariant norms.

Keywords: sum of independent random variables, tail distributions, decreasing rearrangement, pth moment, rearrangement invariant space, disjoint sum, maximal function, Hoffmann-Jørgensen/Klass-Nowicki Inequality, Lévy Property.

A.M.S. Classification (1991): Primary 60G50, 60E15, 46E30; Secondary 46B09.

∗The first named author was partially supported by NSF grant DMS 9401345.

†The second named author was partially supported by NSF grants DMS 9424396 and DMS 9870026, and by the University of Missouri Research Board.

1 Introduction

This paper is about the following type of problem: given independent (not necessarily identically distributed) random variables X_{1}, X_{2}, …, X_{N}, find the ‘size’ of |S|, where

S = ∑_{n=1}^{N} X_{n}.

We will examine several ways to measure this size. The first will be through tail distributions, that is, Pr(|S| > t). Finding an exact solution to this problem would be a dream of probabilists, so we have to temper our desires in some manner. In fact, this problem goes back to the foundations of probability in the following form: if the sequence (X_{n}) consists of random variables that are mean zero, identically distributed and have finite variance, find the asymptotic value of Pr(|S| > \sqrt{N}t) as N → ∞. This is answered, of course, by the Central Limit Theorem, which tells us that the answer is the Gaussian distribution. There has been a tremendous amount of work on generalizing this. We refer the reader to almost any advanced work on probability.

Our approach is different. Instead of seeking asymptotic solutions, we will look for approximate solutions. That is, we seek a function f(t), computed from (X_{n}), such that there is a positive constant c with

c^{−1}f(ct) ≤ Pr(|S| > t) ≤ c f(c^{−1}t).

The second measurement of the size of |S| will be through the pth moments, ∥S∥_{p} = (𝔼|S|^{p})^{1/p}. Again, we shall be searching for approximate solutions, that is, finding a quantity A such that there is a positive constant c so that

c^{−1}A ≤ ∥S∥_{p} ≤ c A.

While this may seem like quite a different problem, in fact, as we will show, there is a precise connection between the two, in that obtaining an approximate formula for ∥S∥_{p} with constants that are uniform as p → ∞ is equivalent to obtaining an approximate formula for the tail distribution.

The third way that we shall look at is to find the size of |S| in a rearrangement invariant space. This line of research was began by Carothers and Dilworth (1988) who obtained results for Lorentz spaces, and was completed by Johnson and Schechtman (1989). Our results will give a comparison of the size of |S| in the rearrangement invariant space with ∥S∥_{p}, obtaining a greater control on the sizes of the constants involved than the previous works.

Many of the results of this paper will be true for all sums of independent random variables, even those that are vector valued, with the following proviso. Instead of considering the sum S = ∑_{n} X_{n}, we will consider the maximal function U = sup_{n} |∑_{k=1}^{n} X_{k}|. We will define a property for sequences called the Lévy property, which will imply that U is comparable to S. Sequences with this Lévy property will include positive random variables, symmetric random variables, and identically distributed random variables. The result of this paper that gives the tail distribution for S is only valid for real valued sequences of random variables that satisfy the Lévy property. However the results connecting the L_{p} and the rearrangement invariant norms to the tail distributions of U are valid for all sequences of vector valued independent random variables. (Since this paper was submitted, Mark Rudelson pointed out to us that some of the inequalities can be extended from U to S by a simple symmetrization argument. We give details at the end of each relevant section.)

Let us first give the historical context for these results, considering first the problem of approximate formulae for the tail distribution. Perhaps the earliest works are the Paley-Zygmund inequality (see for example Kahane (1968, Theorem 3, Chapter 2)), and Kolmogorov’s reverse maximal inequality (see for example Shiryaev (1980, Chapter 4, section 2.)) Both give (under an extra assumption) a lower bound on the probability that a sum of independent, mean zero random variables exceeds a fraction of its standard deviation and both may be regarded as a sort of converse to the Chebyshev’s inequality. Next, in 1929, Kolmogorov, proved a two-sided exponential inequality for sums of independent, meanzero, uniformly bounded, random variables (see for example Stout (1974, Theorem 5.2.2) or Ledoux and Talagrand (1991, Lemma 8.1)). All of these results require some restriction on the nature of the sequence (X_{n}), and on the size of the level t.

Hahn and Klass (1997) obtained very good bounds on one sided tail probabilities for sums of independent, identically distributed, real valued random variables. Their result had no restrictions on the nature of the random variable, or on the size of the level t. In effect, their result worked by removing the very large parts of the random variables, and then using an exponential estimate on the rest. We will take a similar approach in this paper.

Let us next look at the pth moments. Khintchine (1923) gave an inequality for Rademacher (Bernoulli) sums. This very important formula has found extensive applications in analysis and probability. Khintchine’s result was extended to any sequence of positive or mean zero random variables by the celebrated result of Rosenthal (1970). The order of the best constants as p → ∞ was obtained by Johnson, Schechtman and Zinn (1983), and Pinelis (1994) refined this still further. Now even more precise results are known, and we refer the reader to Figiel, Hitczenko, Johnson, Schechtman and Zinn (1997) (see also Ibragimov and Sharakhmetov (1997)). However, the problem with all these results is that the constants were not uniformly bounded as p → ∞.

Khintchine’s inequality was generalized independently by Montgomery and Odlyzko (1988) and Montgomery-Smith (1990). They were able to give approximate bounds on the tail probability for Rademacher sums, with no restriction on the level t. Hitczenko (1993) obtained an approximate formula for the L_{p} norm of Rademacher sums, where the constants were uniformly bounded as p → ∞. (A more precise version of this last result was obtained in Hitczenko-Kwapień (1994) and it was used to give a simple proof of the lower bound in Kolmogorov’s exponential inequality.)

Continuing in the direction of Montgomery and Odlyzko, Montgomery-Smith and Hitczenko, Gluskin and Kwapień (1995) extended tail and moment estimates from Rademacher sums to weighted sums of random variables with logarithmically concave tails (that is, P (|X| ≥ t) = exp(−ϕ(t)), where ϕ : [0, ∞) → [0, ∞) is convex). After that, Hitczenko, Montgomery-Smith, and Oleszkiewicz (1997) treated the case of logarithmically convex tails (that is, the ϕ above is concave rather than convex). It should be emphasized that in the last paper, the result of Hahn and Klass (1997) played a critical role.

The breakthrough came with the paper of Latała (1997), who solved the problem of finding upper and lower bounds for general sums of positive or symmetric random variables, with uniform constants as p → ∞. His method made beautiful use of special properties of the function t ↦ t^{p}. In a short note, Hitczenko and Montgomery-Smith (1999) showed how to use Latała’s result to derive upper and lower bounds on tail probabilities. Latała’s result is the primary motivation for this paper.

The main tool we will use is the Hoffmann-Jørgensen Inequality. In fact, we will use a stronger form of this inequality, due to Klass and Nowicki (1998). The principle in many of our proofs is the following idea. Given a sequence of random variables (X_{n}), we choose an appropriate level s > 0. Each random variable X_{n} is split into the sum X_{n}^{(≤s)} + X_{n}^{(>s)}, where X_{n}^{(≤s)} = X_{n}I_{|X_{n}|≤s}, and X_{n}^{(>s)} = X_{n}I_{|X_{n}|>s}. It turns out that the quantity (X_{n}^{(>s)}) can either be disregarded, or it can be considered as a sequence of disjoint random variables. (By “disjoint” we mean that the random variables are disjointly supported as functions on the underlying probability space.) As for the quantity ∑_{n} X_{n}^{(≤s)}, it will turn out that the level s allows one to apply the Hoffmann-Jørgensen/Klass-Nowicki Inequality so that it may be compared with quantities that we understand rather better.

Let us give an outline of this paper. In Section 2, we will give definitions. This will include the notion of decreasing rearrangement, that is, the inverse to the distribution function. Many results of this paper will be written in terms of the decreasing rearrangement. Section 3 is devoted to the Klass-Nowicki Inequality. Since our result is slightly stronger than that currently in the literature, we will include a full proof. In Section 4, we will introduce and discuss the Lévy property. This will include a “reduced comparison principle” for sequences with this property. Section 5 contains the formula for the tail distribution of sums of real valued random variables. Then in Section 6, we demonstrate the connection between L_{p}-norms of such sums and their tail distributions. In Section 7 we will discuss sums of independent random variables in rearrangement invariant spaces.

2 Notation and definitions

Throughout this paper, a random variable will be a measurable function from a probability space to some Banach space (often the real line). The norm in the implicit Banach space will always be denoted by | · |.

Suppose that f : [0, ∞) → [0, ∞] is a non-increasing function. Define the left continuous inverse to be

f^{−1}(x−) = sup{y : f(y) ≥ x},

and the right continuous inverse to be

f^{−1}(x+) = sup{y : f(y) > x}.

In describing the tail distribution of a random variable X, instead of considering the function t ↦ Pr(|X| > t), we will consider its right continuous inverse, which we will denote by X^{∗}(t). In fact, this quantity appears very much in the literature, and is more commonly referred to as the decreasing rearrangement (or more correctly the non-increasing rearrangement) of |X|. Notice that if one considers X^{∗} to be a random variable on the probability space [0, 1] (with Lebesgue measure), then X^{∗} has exactly the same law as |X|. We might also consider the left continuous inverse t ↦ X^{∗}(t−). Notice that X^{∗}(t) ≤ x ≤ X^{∗}(t−) if and only if Pr(|X| > x) ≤ t ≤ Pr(|X| ≥ x).

If A and B are two quantities (that may depend upon certain parameters), we will write A ≈ B to mean that there exist positive constants c_{1} and c_{2} such that c_{1}^{−1}A ≤ B ≤ c_{2}A. We will call c_{1} and c_{2} the constants of approximation. If f(t) and g(t) are two (usually non-increasing) functions on [0, ∞), we will write f(t) ≈_{t} g(t) if there exist positive constants

c_{1}, c_{2}, c_{3} and c_{4} such that c_{1}^{−1}f(c_{2}t) ≤ g(t) ≤ c_{3}f(c_{4}^{−1}t) for all t ≥ 0. Again, we will call c_{1}, c_{2}, c_{3} and c_{4} the constants of approximation.

Suppose that X and Y are random variables. Then the statement Pr(|X| > t) ≈_{t} Pr(|Y | > t) is the same as the statement X^{∗}(t) ≈_{t} Y^{∗}(t). Since X^{∗}(t) = 0 for t ≥ 1 the latter statement is equivalent to the existence of positive constants c_{1}, c_{2}, c_{3}, c_{4} and c_{5} such that c_{1}^{−1}X^{∗}(c_{2}t) ≤ Y^{∗}(t) ≤ c_{3}X^{∗}(c_{4}^{−1}t) for t ≤ c_{5}^{−1}.

To avoid bothersome convergence problems, we will always suppose that our sequence of independent random variables (X_{n}) is of finite length. Given a sequence of independent random variables (X_{n}), when no confusion will arise, we will use the following notations. If A is a finite subset of ℕ, we will let S_{A} = ∑_{n∈A} X_{n}, and M_{A} = sup_{n∈A} |X_{n}|. If k is a positive integer, then S_{k} = S_{{1,...,k}} and M_{k} = M_{{1,...,k}}. We will define the maximal function U_{k} = sup_{1≤n≤k} |S_{n}|. Furthermore, S = S_{N}, M = M_{N}, and U = U_{N}, where N is the length of the sequence (X_{n}).

If s is a real number, we will write X_{n}^{(>s)} = X_{n}I_{|X_{n}|>s} and X_{n}^{(≤s)} = X_{n}I_{|X_{n}|≤s} = X_{n}−X_{n}^{(>s)}. For A ⊂ ℕ, we will write S_{A}^{(≤s)} = ∑_{n∈A} X_{n}^{(≤s)}. Similarly we define S_{A}^{(>s)}, S_{k}^{(≤s)}, etc.

Another quantity that we shall care about is the decreasing rearrangement of the disjoint sum of random variables. This notion was used by Johnson, Maurey, Schechtman and Tzafriri (1979), Carothers and Dilworth (1988), and Johnson and Schechtman (1989), all in the context of sums of independent random variables. The disjoint sum of the sequence (X_{n}) is the measurable function on the measure space Ω × ℕ that takes (ω, n) to X_{n}(ω). We shall denote the decreasing rearrangement of the disjoint sum by ℓ̃: [0, ∞) → [0, ∞], that is, ℓ̃(t) is the least number such that

∑_{n} Pr(|X_{n}| > ℓ̃(t)) ≤ t.

Define ℓ(t) to be ℓ̃(t) if 0 ≤ t ≤ 1, and 0 otherwise. Since ℓ(t) is only non-zero when 0 ≤ t ≤ 1, we will think of ℓ as being a random variable on the probability space [0, 1] with Lebesgue measure. The quantity ℓ is effectively M in disguise. This next result (and its proof) essentially appears in Giné and Zinn (1983).

Proposition 2.1 If 0 < t < 1, then

ℓ(2t) ≤ ℓ(t/(1 − t)) ≤ M^{∗}(t) ≤ ℓ(t).

Proof: The first inequality follows easily once one notices that both sides of this inequality are zero if t > 1/2.

To get the second inequality, note that, by an easy argument, if α_{1}, α_{2}, ⋯ ≥ 0 with ∑_{n} α_{n} ≤ 1, then

1 − ∑_{n} α_{n} ≤ ∏_{n}(1 − α_{n}) ≤ 1 − \frac{∑_{n} α_{n}}{1 + ∑_{n} α_{n}} .

So, if Pr(ℓ > x) = ∑_{n} Pr(|X_{n}| > x) ≤ 1, then

Pr(M > x) = 1 − ∏_{n}(1 − Pr(|X_{n}| > x)),

and hence

\frac{Pr(ℓ > x)}{1 + Pr(ℓ > x)} ≤ Pr(M > x) ≤ Pr(ℓ > x).

Taking inverses, the result follows. □

3 The Klass-Nowicki Inequality

This section is devoted to the following result — the Klass-Nowicki Inequality.

Theorem 3.1 Let (X_{n}) be a sequence of Banach valued independent random variables. Then for all positive integers K we have

Pr(U > 2Kt + (K − 1)s) ≤ \frac{1}{K!} (\frac{Pr(U > t)}{1 − Pr(U > t)})^{K} + Pr(M > s),

whenever Pr(U > t) < 1.

The original inequality of this form was for Rademacher (or Bernoulli) sums and K = 2, and was due to Kahane (1968). This was extended by Hoffmann-Jørgensen (1974) to general sums, at least for positive or symmetric random variables, for the case K = 2. Indeed, if one wants Theorem 3.1 for K > 2, but without the K! factor, this may be obtained by iterating the Hoffmann-Jørgensen Inequality, as was done by Johnson and Schechtman (1989, Lemmas 6 and 7). (Both Kahane and Hoffmann-Jørgensen obtained slightly different constants than those we have presented. Also, in neither case did a factor like (1 − Pr(U > t)) appear in their formulae.)

Klass and Nowicki (1998) were able to obtain Theorem 3.1, at least in the case when the random variables are positive or symmetric. (However their constants are better than ours.) Removing the positive or symmetric condition is really not so hard, but because it does not appear in the literature in this manner, we will give a complete proof of Theorem 3.1.

We also note that this inequality has some comparison with a result that appears in Ledoux and Talagrand (1991, Theorem 6.17.)

Proof: Let N be the length of the sequence (X_{n}). During this proof, let us write (m, n] for the set of integers greater than m and not greater than n.

We start with the observation

Pr(U > 2Kt + (K − 1)s) ≤ Pr(U > 2Kt + (K − 1)s and M ≤ s) + Pr(M > s).

Now, if we have that both U > 2Kt + (K − 1)s and M ≤ s, then we ensure the existence of an increasing sequence of non-negative integers m_{0}, …, m_{K}, bounded by N, and defined as follows. Set m_{0} = 0. If we have picked m_{l−1}, let m_{l} be the smallest positive integer greater than m_{l−1} such that |S_{(m_{l−1},m_{l}]}| > 2t. For l = 1, it is clear that such an integer exists. Let us explain why the integer m_{l} ≤ N exists if 2 ≤ l ≤ K.

For 1 ≤ l′ ≤ l − 1, and m_{l′−1} < k ≤ m_{l′} − 1, we have that |S_{(m_{l′−1},k]}| ≤ 2t and |X_{m_{l′}}| ≤ s. Hence for 1 ≤ l′ ≤ l − 1

|S_{m_{l′}}| = |∑_{j=1}^{l′}(S_{(m_{j−1},m_{j}−1]} + X_{m_{j}})| ≤ 2l′t + l′s,

and for m_{l′−1} < k ≤ m_{l′} − 1

|S_{k}| = |(∑_{j=1}^{l′−1}(S_{(m_{j−1},m_{j}−1]} + X_{m_{j}})) + S_{(m_{l′−1},k]}| ≤ 2l′t + (l′ − 1)s.

But we know that there exists a number m such that |S_{m}| > 2Kt + (K − 1)s. Hence, we must have that m > m_{l−1}, and that |S_{(m_{l−1},m]}| > 2Kt + (K − 1)s − 2(l − 1)t − (l − 1)s ≥ 2t.

Therefore

Pr(U > 2Kt + (K − 1)s and M ≤ s) ≤ ∑_{1≤m_{1}<···<m_{K}≤N} p_{0,m_{1}}p_{m_{1},m_{2}} ⋯ p_{m_{K−1},m_{K}},

where

p_{m,n} = Pr(|S_{(m,k]}| ≤ 2t for m ≤ k < n, and |S_{(m,n]}| > 2t).

Now let us show the following inequality:

∑_{k=m+1}^{n} p_{m,k} ≤ \frac{1}{1 − Pr(U > t)} ∑_{k=m+1}^{n} p̃_{k},

where

p̃_{n} = Pr(|S_{k}| ≤ t for 1 ≤ k < n, and |S_{n}| > t).

Using independence, we have that

∑_{k=m+1}^{n} p_{m,k} = Pr( sup_{m<k≤n} |S_{k} − S_{m}| > 2t)
= Pr( sup_{m<k≤n} |S_{k} − S_{m}| > 2t|U_{m} ≤ t)
≤ Pr( sup_{m<k≤n} |S_{k}| > t|U_{m} ≤ t)
= \frac{Pr(sup_{m<k≤n} |S_{k}| > t and sup_{1≤k≤m} |S_{k}| ≤ t)}{Pr(U_{m} ≤ t)}
≤ \frac{1}{1 −Pr(U > t)} ∑_{k=m+1}^{n} p̃_{k},

as required.

Now we rearrange the sum as follows:

∑_{1≤m_{1}<···<m_{K}≤N} p_{0,m_{1}}p_{m_{1},m_{2}} ⋯ p_{m_{K−1},m_{K}}

= ∑_{1≤m_{1}<···<m_{K−1}≤N} p_{0,m_{1}}p_{m_{1},m_{2}} ⋯ p_{m_{K−2},m_{K−1}} ∑_{m_{K}=m_{K−1}+1}^{N} p_{m_{K−1},m_{K}}

≤ \frac{1}{1 − Pr(U > t)} ∑_{1≤m_{1}<···<m_{K−1}≤N} p_{0,m_{1}}p_{m_{1},m_{2}} ⋯ p_{m_{K−2},m_{K−1}} ∑_{m_{K}=m_{K−1}+1}^{N} p̃_{m_{K}}.

Now we rearrange this last quantity to get

∑_{1≤m_{1}<···<m_{K}≤N} p_{0,m_{1}}p_{m_{1},m_{2}} ⋯ p_{m_{K−1},m_{K}}

≤ \frac{1}{1 − Pr(U > t)} ∑_{1≤m_{1}<···<m_{K−2}<m_{K}≤N} p_{0,m_{1}}p_{m_{1},m_{2}} ⋯ p_{m_{K−3},m_{K−2}} p̃_{m_{K}} ×

× ∑_{m_{K−1}=m_{K−2}+1}^{m_{K}} p_{m_{K−2},m_{K−1}}

≤ \frac{1}{(1 − Pr(U > t))^{2}} ∑_{1≤m_{1}<···<m_{K−2}<m_{K}≤N} p_{0,m_{1}}p_{m_{1},m_{2}} ⋯ p_{m_{K−3},m_{K−2}} p̃_{m_{K}} ×

× ∑_{m_{K−1}=m_{K−2}+1}^{m_{K}} p̃_{m_{K−1}}.

Repeating this argument (K − 2) more times, we eventually see that

∑_{1≤m_{<}···<m_{K}≤N} p_{0,m_{1}}p_{m_{1},m_{2}} ⋯ p_{m_{K−1},m_{K}}

≤ \frac{1}{(1 − Pr(U > t))^{K}} ∑_{1≤m_{1}<···<m_{K}≤N} p̃_{m_{1}} p̃_{m_{2}} ⋯ p̃_{m_{K}}.

Now, since K distinct numbers may be rearranged in K! different ways, we have that

∑_{1≤m_{1}<···<m_{K}≤N} p̃_{m_{1}} p̃_{m_{2}} ⋯ p̃_{m_{K}}

= \frac{1}{K!} ∑_{1≤m_{1},m_{2},...,m_{K}≤N m_{1},m_{2},...,m_{K} distinct} p̃_{m_{1}} p̃_{m_{2}} ⋯ p̃_{m_{K}}

≤ \frac{1}{K!} ∑_{1≤m_{1},m_{2},...,m_{K}≤N} p̃_{m_{1}} p̃_{m_{2}} ⋯ p̃_{m_{K}}

= \frac{1}{K!} (∑_{k=1}^{N} p̃_{k})^{K} .

Since

∑_{k=1}^{N} p̃_{k} = Pr(U > t),

we obtain the result. □

Let us now understand what this result means in terms of the decreasing rearrangement.

Corollary 3.2 There exists a universal positive constant c_{1} such that for any sequence of Banach valued independent random variables (X_{n}), and for 0 < t ≤ s ≤ 1/2 we have

U^{∗}(t) ≤ c_{1} \frac{log(1/t)}{max\{log(1/s), log log(4/t)\}}(U^{∗}(s) + M^{∗}(t/2)).

Proof: Notice that if f, g : [0, ∞) → [0, ∞] are non-increasing functions, then (max{f, g})^{−1} = max{f^{−1}, g^{−1}}, and if f ≤ g, then f^{−1} ≤ g^{−1}, where here f^{−1} denotes either the left or right continuous inverse of f. Since A + B ≤ 2 max{A, B} for any two positive numbers A and B, from Theorem 3.1, and setting s = t, we have that if Pr(U > t) ≤ 1/2, then for all positive integers K

Pr(U > (3K − 1)t) ≤ 2 max \{\frac{1}{K!}(2 Pr(U > t))^{K}, Pr(M > t)\} .

Taking inverses, we see that if (K!t/2)^{1/K} ≤ 1/2, then

\frac{1}{3K − 1}U^{∗}(t) ≤ max \{U^{∗} (\frac{1}{2} (\frac{K!t}{2})^{1/K}) , M^{∗} (\frac{t}{2})\} .

Now, using the fact that max{A, B} ≤ A + B for any positive numbers A and B, and by choosing K to be the smallest integer such that s ≤ (K!t/2)^{1/K}, and by some elementary but tedious algebra, the result follows. □

Since this paper was submitted, Mark Rudelson pointed out to us a couple of ways that Theorem 3.1 can be improved. First, we may obtain a result closer to that of Ledoux and Talagrand (1991, Theorem 6.17. Let (|X_{n}|^{∗}) be the order statistics of (|X_{n}|), that is, the values of (|X_{n}|) rearranged in decreasing order. Then exactly the same proofs gives the following strengthening: for all positive integers K

Pr(U > 2Kt + (K − 1)s) ≤ \frac{1}{K!} (\frac{Pr(U > t)}{1 − Pr(U > t)})^{K} + Pr (∑_{n=1}^{K} |X_{n}|^{∗} > (K − 1)s) ,

whenever Pr(U > t) < 1.

Secondly, a similar result is also true if we replace U by |S|. This is certainly the case if the sequence (X_{n}) consists of symmetric random variables, since they satisfy the Lévy property. Now let (X̄_{n}) be an independent copy of (X_{n}), and let X̃_{n} = X_{n} − X̄_{n}. Let S̄ and S̃ respectively denote the sums formed from these two sequences of random variables. Thus we have the result for |S̃|, since it is a sum of symmetric random variables. But

Pr(|S̃| > ct) ≥ Pr(|S| > (c + 1)t and |S̄| ≤ t) = Pr(|S| > (c + 1)t) Pr(|S| ≤ t).

Arguing in this way, we quickly see that there are constants c_{1}, c_{2}, and c_{3} such that

Pr(S > c_{1}K(s + t)) ≤ \frac{Pr(S > t)^{K}}{K!(1 − c_{2} Pr(S > c_{3}t))^{K+1}} + Pr(M > s),

whenever Pr(S > c_{3}t) < c_{2}^{−1}.

Thus a version of Corollary 3.2 is also true when U is replaced by |S|.

4 The Lévy Property

Let (X_{n}) be a sequence of independent random variables. We will say that (X_{n}) satisfies the Lévy property with constants c_{1} and c_{2} if whenever A ⊆ B ⊆ ℕ, with A and B finite, then for t > 0

Pr(|S_{A}| > c_{1}t) ≤ c_{2} Pr(|S_{B}| > t).

The casual reader should beware that this property has nothing to do with Lévy processes.

The sequence (X_{n}) has the strong Lévy property with constants c_{1} and c_{2} if for all s > 0 the sequence (X_{n}^{(≤s)}) has the Lévy property with constants c_{1} and c_{2}.

Here are examples of sequences with the strong Lévy property. (It may be easily seen that in all these cases it is sufficient to show that they have the Lévy property.)

(i) Positive sequences, with constants 1 and 1.

(ii) Sequences of symmetric random variables with constants 1 and 2. This “reflection property” plays a major role in results attributed to Lévy, hence the name of the property.

(iii) Sequences of identically distributed random variables. This was shown independently by Montgomery-Smith (1993) with constants 10 and 3, and by Latała (1993) with constants 5 and 4, or 7 and 2.

We see that sequences with the Lévy property satisfy a maximal inequality.

Proposition 4.1 Let (X_{n}) be a sequence of independent random satisfying the Lévy property with constants c_{1} and c_{2}. Then for all t > 0

Pr(U > 3c_{1}t) ≤ 3c_{2} Pr(|S| > t).

Thus M^{∗}(t) ≤ 6c_{1}S^{∗}(t/3c_{2}).

Proof: The first statement is an immediate corollary of the following result known as Lévy-Ottaviani inequality:

Pr(U_{N} > 3t) ≤ 3 sup_{1≤k≤N} Pr(|S_{k}| > t).

(Billingsley (1995, Theorem 22.5, p. 288) attributes this result to Etemadi (1985) who proved it with constants 4 in both places, but the same proof gives constants 3; see, for example, Billingsley. However the first named author learned this result from Kwapień in 1980.)

The second statement follows from the first, since M ≤ 2U. □

We end with a lemma that lists some elementary properties. Part (i) of the lemma might be thought of as a kind of reduced comparison principle.

Lemma 4.2 Let (X_{n}) be a sequence of random variables satisfying the strong Lévy property.

(i) There exist positive constants c_{1} and c_{2}, depending only upon the Lévy constants of (X_{n}), such that if s ≤ 1/2 and 0 ≤ t ≤ 1, then

(S^{(≤M^{∗}(s))})^{∗}(t) ≤ c_{1}S^{∗}(c_{2}^{−1}t).

(ii) There exist positive constants c_{1} and c_{2}, depending only upon the strong Lévy constants of (X_{n}), such that if r ≤ s ≤ 1/2, and if 0 ≤ t ≤ 1, then (S^{(≤M^{∗}(s))})^{∗}(t) ≤ c_{1}(S^{(≤M^{∗}(r))})^{∗}(c_{2}^{−1}t).

(iii) If 0 ≤ s ≤ t ≤ 1, then S^{∗}(t) ≤ (S^{(≤M^{∗}(s))})^{∗}(t − s), and (S^{(≤M^{∗}(s))})^{∗}(t) ≤ S^{∗}(t − s). In particular, S^{∗}(t) ≤ (S^{(≤M^{∗}(t/2))})^{∗}(t/2), and (S^{(≤M^{∗}(t/2))})^{∗}(t) ≤ S^{∗}(t/2).

(iv) For α, β > 0, we have that

(S^{(≤M^{∗}(t))})^{∗}(t) ≈_{t} (S^{(≤M^{∗}(αt))})^{∗}(βt)

where the constants of approximation depend only upon α, β and the strong Lévy constants of (X_{n}).

(v) We have that

S^{∗}(t) ≈_{t} (S^{(≤M^{∗}(t))})^{∗}(t) ≈_{t} (S^{(≤ℓ(t))})^{∗}(t),

where the constants of approximation depend only upon the strong Lévy constants of (X_{n}).

Proof: Let us start with part (i). For each set A ⊆ ℕ, define the event

E_{A} = {|X_{n}| ≤ M^{∗}(s) if and only if n ∈ A}.

Note that the whole probability space is the disjoint union of these events. Also

{|S^{(≤M^{∗}(s))}| > x} ∩ E_{A} = {|S_{A}| > x} ∩ E_{A}.

Furthermore, by independence, we see that

Pr(|S_{A}| > x and E_{A})
= Pr(|S_{A}| > x and |X_{n}| ≤ M^{∗}(s) for n ∈ A) Pr(|X_{n}| > M^{∗}(s) for n ∉ A).

Hence

Pr(|S^{(≤M^{∗}(s))}| > x)

= ∑_{A⊆ℕ}Pr(|S_{A}| > x and |X_{n}| ≤ M^{∗}(s) for n ∈ A) Pr(|X_{n}| > M^{∗}(s) for n ∉ A)

≤ 2 ∑_{A⊆ℕ}Pr(|S_{A}| > x) Pr(|X_{n}| ≤ M^{∗}(s) for n ∈ A) Pr(|X_{n}| > M^{∗}(s) for n ∉ A)

≤ c_{2} Pr(|S| > c_{1}^{−1}x),

where in the first inequality we have used the fact that

Pr(|X_{n}| ≤ M^{∗}(s) for n ∈ A) ≥ Pr(M ≤ M^{∗}(s)) ≥ 1 − s ≥ 1/2.

Part (ii) follows by applying part (i) to S^{(≤M^{∗}(r))}.

Part (iii) follows from the observation that

Pr(S ≠ S^{(≤M^{∗}(s))}) ≤ Pr(M > M^{∗}(s)) ≤ s.

Hence, if Pr(S > α) ≥ t, then Pr(S^{(≤M^{∗}(s))} > α) ≥ t − s, and conversely, if Pr(S^{(≤M^{∗}(s))} > α) > t then Pr(S > α) ≥ t − s.

To show part (iv), we may suppose without loss of generality that α = 1 and β > 1. Clearly S^{(≤M^{∗}(t))}(t) ≥ S^{(≤M^{∗}(t))}(βt), so we need only show an opposite inequality. From part (ii), there are positive constants c_{1} and c_{2}, depending only upon the strong Lévy constants of (X_{n}), such that for 0 ≤ t ≤ 1/2

S^{(≤M^{∗}(t))}(t) ≤ c_{1}S^{(≤M^{∗}(c_{2}^{−1}β^{−1})t)}(c_{2}^{−1}t) ≤ c_{1}S^{(≤M^{∗}(c_{3}^{−1}t)}(c_{3}^{−1}βt),

where c_{3} = c_{2}β.

Part (v) follows easily by combining part (iii), part (iv), and Proposition 2.1. □

5 Tail distributions

In this section, we will state and prove the formula for the tail distribution of the sum of independent, real valued, random variables that satisfy the Lévy Property.

If one restricts the formula to the case of sums of independent, identically distributed random variables, one obtains a formula very similar to the main result of Hahn and Klass (1997). The main differences are that their inequality involves one sided inequalities, and also that their inequality is more precise.

This formula also has a strong resemblance to the result of Latała. As we shall show in Section 6, computing the L_{p} norm of U is effectively equivalent to computing U^{∗}(e^{−p}). Then if one notices that (1 + x)^{p} is very close to e^{xp} for small positive x, one can see that this result and the result of Latała are very closely related. Presumably one could derive Latała’s result by combining Theorem 5.1 with Theorem 6.1. However the technical difficulties are quite tricky, and since Latała’s proof is elegant, we will not carry out this program here.

Theorem 5.1 Let (X_{n}) be a sequence of real valued independent random variables satisfying the strong Lévy property. Define the functions F_{1}(t) and F_{2}(t) to be 0 if t > 1, and if 0 ≤ t ≤ 1,

F_{1}(t) = inf {λ > 0 : ∏_{n} 𝔼(t^{X_{n}^{(≤ℓ(t))}/λ}) ≤ t^{−1} and ∏_{n} 𝔼(t^{−X_{n}^{(≤ℓ(t))}/λ}) ≤ t^{−1}} ,

F_{2}(t) = inf {λ > 0 : ∏_{n} 𝔼(t^{X_{n}^{(≤M^{∗}(t))}/λ}) ≤ t^{−1} and ∏_{n} 𝔼(t^{−X_{n}^{(≤M^{∗}(t))}/λ}) ≤ t^{−1}} .

Then

S^{∗}(t) ≈_{t} F_{1}(t) ≈_{t} F_{2}(t),

where the constants of approximation depend only upon the strong Lévy constants of (X_{n}).

Let us start with gaining some understanding of Orlicz spaces. There is a huge literature on Orlicz spaces, see for example Lindenstrauss and Tzafriri (1977). Suppose that Φ : [0, ∞) → [0, ∞] is an increasing function (usually convex with Φ(0) = 0). Then the Orlicz norm of a random variable X is defined according to the formula

∥X∥_{Φ} = inf{λ > 0 : 𝔼Φ(|X|/λ) ≤ 1}.

We will be concerned with the special functions

Φ_{t}(x) = \frac{t^{−x} − 1}{t^{−1} − 1}.

The following is a special case of results that appear in Montgomery-Smith (1992).

Lemma 5.2 For any random variable X, and for t ≤ 1/4, we have that

∥X∥_{Φ_{t}} ≈ sup_{0≤x≤1} \frac{log(t)}{log(xt)}X^{∗}(x),

with constants of approximation bounded by 2.

Proof: Suppose first that ∥X∥_{Φ_{t}} ≤ 1. Then 𝔼Φ_{t}(X) ≤ 1, which implies that

xt^{−X^{∗}(x)} ≤ ∫_{0}^{1} t^{−X^{∗}(y)} dy ≤ 𝔼(t^{−|X|}) ≤ t^{−1},

that is, X^{∗}(x) ≤ log(xt)/ log(t).

Conversely, suppose that X^{∗}(x) ≤ log(xt)/ log(t) for 0 ≤ x ≤ 1. Then

EΦ_{t}(X/2) ≤ ∫_{0}^{1} Φ_{t} (\frac{log(xt)}{2 log(t)}) dx = \frac{2t^{−1/2} − 1}{t^{−1} − 1} ≤ 1.

□

Proof of Theorem 5.1: Let us start with the proof that S^{∗}(t) ≈_{t} F_{1}(t). Since the random

variables X_{n}^{(≤ℓ(t))} are independent, we have that

F_{1}(t) = inf {λ > 0 : 𝔼(t^{S^{(≤ℓ(t))}/λ}) ≤ t^{−1} and 𝔼(t^{−S^{(≤ℓ(t))}/λ}) ≤ t^{−1}} .

Now we notice that for any random variable Y, and 0 ≤ t ≤ 1, we have that

\frac{1}{2}𝔼(t^{−|Y |}) ≤ max\{𝔼(t^{Y}), 𝔼(t^{−Y})\} ≤ 𝔼(t^{−|Y |}).

Hence

F_{1}(t) ≤ inf {λ > 0 : 𝔼(t^{−|S^{(≤ℓ(t))}/λ|}) ≤ t^{−1}} = ∥S^{(≤ℓ(t))}∥_{Φ_{t}},

and

F_{1}(t) ≥ inf {λ > 0 : 𝔼(t^{−|S^{(≤ℓ(t))}/λ|}) ≤ 2t^{−1}} = ∥S^{(≤ℓ(t))}∥_{Ψ_{t}},

where Ψ_{t}(x) = \frac{t^{−x} − 1}{2t^{−1} − 1}. However, we quickly see that for x ≥ 0 that if t ≤ 1/2 then

Ψ_{t}(x) ≥ \frac{1}{3}Φ_{t}(x) ≥ Φ_{t}(x/3), since Φ_{t} is a convex function. Hence

F_{1}(t) ≈ ∥S^{(≤ℓ(t))}∥_{Φ_{t}}

with constants of approximation bounded by 3.

Next, we apply Lemma 5.2, and we see that

F_{1}(t) ≈ sup_{0≤x≤1} \frac{log(t)}{log(xt)}(S^{(≤ℓ(t))})^{∗}(x).

Taking x = t, we see that the right hand side is bounded below by \frac{1}{2}(S^{(≤ℓ(t))})^{∗}(t). Also, if t ≤ x ≤ 1, then

\frac{log(t)}{log(xt)}(S^{(≤ℓ(t))})^{∗}(x) ≤ (S^{(≤ℓ(t))})^{∗}(t).

Further, by Corollary 3.2 combined with Proposition 4.1, there exist constants c_{1} and c_{2}, depending only on the Lévy constants of (X_{n}), such that if 0 ≤ x ≤ t ≤ c_{1}^{−1}, then

\frac{log(t)}{log(xt)}(S^{(≤ℓ(t))})^{∗}(x) ≤ c_{2} \frac{log(x)}{log(xt)}((S^{(≤ℓ(t))})^{∗}(t) + (M^{(ℓ(t))})^{∗}(x/2))
≤ c_{2}((S^{(≤ℓ(t))})^{∗}(t) + ℓ(t)).

Now, applying Proposition 2.1, Proposition 4.1, and Lemma 4.2 part (v), we finally obtain the desired result.

To show that S^{∗}(t) ≈_{t} F_{2}(t) is an almost identical proof. □

6 L_{p} norms

The main result of this section establishes the relationship between the L_{p} norm of sums of random variables and their tail distributions.

Theorem 6.1 Given p_{0} > 0, if p ≥ p_{0}, and (X_{n}) is a sequence of Banach valued independent random variables, then

∥U∥_{p} ≈ U^{∗}(e^{−p}/4) + ∥ℓ∥_{p} ≈ (U^{(≤ℓ(e^{−p}/8))})^{∗}(e^{−p}/4) + ∥ℓ∥_{p},

where the constants of approximation depend only upon p_{0}.

We should note that we are not able to get universal control over the constants as p_{0} → 0, as is shown by simple examples once one understands that ∥Y ∥_{p} converges to the geometric mean of |Y | as p → 0.

Combining this with Corollary 3.2, we immediately obtain the following result that compares ∥S∥_{q} to ∥S∥_{p}. This result extends results of Talagrand, (see Ledoux and Talagrand (1991, Theorem 6.20), Kwapień and Woyczyński (1992, Proposition 1.4.2 and comments following it; see also Hitczenko (1994, Proposition 4.1)) and Johnson, Schechtman and Zinn (1983). If this result is specialized to symmetric or positive real valued random variables, then by considering the cases p = 2 or p = 1, it implies the inequality of Rosenthal (1970), including the result of Johnson, Schechtman and Zinn (1983) that gives correct order of the constants as p → ∞. Note that ∥ℓ∥_{q} ≤ 2^{1/q}∥M∥_{q} by Proposition 2.1.

Theorem 6.2 Let (X_{n}) be a sequence of Banach valued independent random variables and let p_{0} > 0. Then there exist positive constants c_{1}, c_{2} and c_{3}, depending only upon p_{0}, such that for q ≥ p ≥ p_{0} we have

∥U∥_{q} ≤ c_{1} \frac{q}{max\{p, log(e + q)\}}(∥U∥_{p} + M^{∗}(c_{2}^{−1}e^{−q})) + c_{1}∥M∥_{q}
≤ c_{3} \frac{q}{max\{p, log(e + q)\}}(∥U∥_{p} + ∥M∥_{q}).

Let us proceed with the proofs. First we need a lemma that allows us to deal with the “large” parts of U, so that they might be effectively considered as a sum of disjoint random variables.

Lemma 6.3 Let (X_{n}) be a sequence of Banach valued independent random variables, and let 0 < r < 1. Then we may express U^{(>ℓ(r))} = ∑_{k=1}^{∞} V_{k}, where the random variables V_{k} are disjoint, and V_{k}^{∗}(t) ≤ kℓ(t(k − 1)!/r^{k−1}).

Proof: In proving this result, we may suppose without loss of generality that X_{n} = X_{n}^{(>ℓ(r))}, that is, we may suppose that ∑_{n} Pr(X_{n} ≠ 0) ≤ r.

If A is a finite subset of ℕ, define the event

E_{A} = {X_{n} ≠ 0 if and only if n ∈ A}.

For each positive integer k, let E_{k} = ⋃\begin{matrix}A⊆ℕ \\ |A|=k \end{matrix} E_{A}. Set V_{k} = UI_{E_{k}}. Notice that if |A| = k, then

Pr(UI_{E_{A}} > x) ≤ ∑_{n∈A} Pr(|X_{n}| > x/k and E_{A})
= ∑_{n∈A} Pr(|X_{n}| > x/k) ∏_{m∈A\{n}} Pr(X_{m} ≠ 0).

Hence,

Pr(V_{k} > x) = ∑_{\begin{matrix}A⊆ℕ \\ |A|=k \end{matrix}} Pr(UI_{E_{A}} > x)

≤ ∑_{i_{1}<···<i_{k}} ∑_{j=1}^{k} Pr(|X_{i_{j}}| > x/k) ∏_{\begin{matrix}l=1 \\ l≠j \end{matrix}}^{k} Pr(X_{i_{l}} ≠ 0)

≤ \frac{1}{k!} ∑_{i_{1}} ⋯ ∑_{i_{k}} ∑_{j=1}^{k} Pr(|X_{i_{j}}| > x/k) ∏_{\begin{matrix}l=1 \\ l≠j \end{matrix}}^{k} Pr(X_{i_{l}} ≠ 0)

= \frac{k}{k!} ∑_{i_{1}} ⋯ ∑_{i_{k}} Pr(|X_{i_{1}}| > x/k) ∏_{l=2}^{k} Pr(X_{i_{l}} ≠ 0)

= \frac{k}{k!} (∑_{n} Pr(|X_{n}| > x/k)) (∑_{n} Pr(X_{n} ≠ 0))^{k−1}

≤ \frac{r^{k−1}}{(k − 1)!} Pr(ℓ > x/k).
□

Corollary 6.4 Let (X_{n}) be a sequence of Banach valued independent random variables, let 0 < r < 1, and let 0 < p < ∞. Then

∥U^{(>ℓ(r))}∥_{p} ≤ 2e^{2^{p}r/p}∥ℓ∥_{p}.

Proof: Apply Lemma 6.3 to obtain the V_{k}. Using the fact that k ≤ 2^{k}, we obtain that

∥V_{k}∥_{p}^{p} ≤ k^{p} \frac{r^{k−1}}{(k − 1)!}∥ℓ∥_{p}^{p} ≤ 2^{p} \frac{(2^{p}r)^{k−1}}{(k − 1)!} ∥ℓ∥_{p}^{p}.

Thus

∥U^{(>ℓ(r))}∥_{p}^{p} = ∑_{k=1}^{∞} ∥V_{k}∥_{p}^{p} ≤ 2^{p}∥ℓ∥_{p}^{p} ∑_{k=1}^{∞} \frac{(2^{p}r)^{k−1}}{(k − 1)!} = 2^{p}e^{2^{p}r}∥ℓ∥_{p}^{p}.

□

Proof of Theorem 6.1: Applying Proposition 2.1, we see that

∥U∥_{p} ≥ \frac{1}{2}∥M∥_{p} ≥ 2^{−1−1/p}∥ℓ∥_{p}.

Also, we have that

∥U∥_{p}^{p} = ∫_{0}^{1}(U^{∗}(t))^{p} dt ≥ 8^{−1}e^{−p}(U^{∗}(e^{−p}/8))^{p},

that is, ∥U∥_{p} ≥ 8^{−1/p}e^{−1}U^{∗}(e^{−p}/8) ≥ 8^{−1/p}e^{−1}U^{∗}(e^{−p}/4). Hence we have shown that there exists a constant c_{1}, depending only upon p_{0}, such that

∥U∥_{p} ≥ c_{1}^{−1}(U^{∗}(e^{−p}/4) + ∥ℓ∥_{p}).

Furthermore, by Proposition 2.1,

Pr(U ≠ U^{(≤ℓ(e^{−p}/8))}) ≤ Pr(M > ℓ(e^{−p}/8)) ≤ Pr(M > M^{∗}(e^{−p}/8)) ≤ e^{−p}/8.

Hence U^{∗}(e^{−p}/8) ≥ (U^{(≤ℓ(e^{−p}/8))})^{∗}(e^{−p}/4), and so we have shown that there is a constant c_{2} > 0, depending only upon p_{0}, such that

∥U∥_{p} ≥ c_{2}^{−1}((U^{(≤ℓ(e^{−p}/8))})^{∗}(e^{−p}/4) + ∥ℓ∥_{p}).

Now let us derive the converse inequalities. Corollary 3.2 tells us that for t ≤ e^{−p}/2 that

(U^{(≤ℓ(e^{−p}/8))})^{∗}(t) ≤ c_{2} \frac{log(1/t)}{p + log(2)}((U^{(≤ℓ(e^{−p}/8))})^{∗}(e^{−p}/2) + ℓ(e^{−p}/8)).

Thus

∥U^{(≤ℓ(e^{−p}/8))}∥_{p}^{p} ≤ ∫_{0}^{e^{−p}/2}((U^{(≤ℓ(e^{−p}/8))})^{∗}(t))^{p} dt + (1 − e^{−p}/2)(U^{(≤ℓ(e^{−p}/8))})^{∗}(e^{−p}/2)
≤ \frac{1}{(p + log(2))^{p}} ∫_{0}^{1}(log(1/t))^{p} dt ((U^{(≤ℓ(e^{−p}/8))})^{∗}(e^{−p}/2) + ℓ(e^{−p}/8))^{p}
+(U^{(≤ℓ(e^{−p}/8))})^{∗}(e^{−p}/2)
≤ c_{3}^{p}((U^{(≤ℓ(e^{−p}/8))})^{∗}(e^{−p}/2) + ℓ(e^{−p}/8))^{p},

where c_{3} > 0 depends only upon p_{0}. Furthermore,

ℓ(e^{−p}/8)^{p} ≤ 8e^{p} ∫_{0}^{1} ℓ(t)^{p} dt = 8e^{p}∥ℓ∥_{p}^{p}.

Hence, applying Corollary 6.4, and the (quasi-)triangle inequality for L_{p}, we deduce that there exists a constant c_{4}, depending only upon p_{0}, such that

∥U∥_{p} ≤ c_{4}((U^{(≤ℓ(e^{−p}/8))})^{∗}(e^{−p}/2) + ∥ℓ∥_{p}).

Finally the result follows by noticing that

(U^{(≤ℓ(e^{−p}/8))})^{∗}(e^{−p}/2) ≤ (U^{(≤ℓ(e^{−p}/8))})^{∗}(e^{−p}/4),

and also, by an argument similar to one presented above, that

(U^{(≤ℓ(e^{−p}/8))})^{∗}(e^{−p}/2) ≤ U^{∗}(3e^{−p}/8) ≤ U^{∗}(e^{−p}/4).
□

Finally we remark that from the results mentioned at the end of Section 3 we can obtain one sided versions of Theorem 6.1 with |S| in place of U, for example, given p ≥ p_{0},

∥S∥_{p} ≤ cS^{∗}(e^{−p}/c) + c∥ℓ∥_{p}.

where the constants depend only upon p_{0}.

Obviously if the sequence of random variables satisfy the Lévy property, then we can obtain the two sided inequality, but otherwise the other side of the inequality need not hold, as is shown by the example X_{1} = 1, X_{2} = −1, X_{n} = 0 (n ≥ 2).

7 Rearrangement invariant spaces

Rearrangement invariant spaces are studied in much of the literature, see for example Lindenstrauss and Tzafriri (1977). However, we will work with a definition that is a little less restrictive. A rearrangement invariant space on the random variables is a quasi-normed Banach space L of random variables such that 1 ∈ L, and if X^{∗} ≤ Y^{∗} and Y ∈ L, then X ∈ L and ∥X∥_{L} ≤ ∥Y ∥_{L}. Obviously the spaces L_{p} for 0 < p ≤ ∞ are rearrangement invariant spaces.

Given a rearrangement invariant space L, we define the quasi-constant of L to be the least constant K > 0 such that ∥X + Y ∥_{L} ≤ K(∥X∥_{L} + ∥Y ∥_{L}) for all X, Y ∈ L. Notice that if X^{∗}(2t) ≤ Y^{∗}(t), and Y ∈ L, then X may be written as the sum of two disjoint random variables Y_{1} and Y_{2} with Y_{1}^{∗}(t), Y_{2}^{∗}(t) ≤ Y^{∗}(t), and hence ∥X∥_{L} ≤ 2K∥Y ∥_{L}.

Given two rearrangement invariant spaces L and M, we will say that L embeds into M if there is a positive constant c such that if X ∈ L, then X ∈ M and ∥X∥_{M} ≤ c∥X∥_{L}. We will call the least such c the embedding constant of L into M.

Theorem 7.1 Let p_{0} > 0, and let L be a rearrangement invariant space such that L embeds into L_{p}, and L_{q} embeds into L, where q ≥ p ≥ p_{0}. Then there is a positive constant c, depending only upon the quasi-constant of L, the embedding constants, p_{0} and q/p, such that for any sequence of Banach valued independent random variables (X_{n})

c^{−1}(∥U∥_{p} + ∥ℓ∥_{L}) ≤ ∥U∥_{L} ≤ c(∥U∥_{p} + ∥ℓ∥_{L}).

Proof: Let us first obtain the left hand side inequality. It follows by hypothesis that ∥U∥_{L} ≥ c_{1}^{−1}∥U∥_{p}, where c_{1} is the embedding constant of L into L_{p}. Furthermore, U ≥ \frac{1}{2}M, and by Proposition 2.1, ℓ(t) ≤ M^{∗}(2t). Hence ∥U∥_{L} ≥ (4K)^{−1}∥ℓ∥_{L}, where K is the quasiconstant of L.

Now let us obtain the right hand inequality. By Corollary 3.2, we have that there is a universal positive c_{2} for 0 ≤ t ≤ 1

U^{∗}(t)I_{0≤t≤2^{−2q/p}} ≤ c_{2} \frac{2q}{p} (U^{∗}(t^{p/2q}) + M^{∗}(t/2)).

Now U^{∗}(t)I_{0≤t≤2^{−2q/p}} ≥ U^{∗}(2^{2q/p}t), and hence

∥U∥_{L} ≤ (2K)^{⌈2q/p⌉}c_{2}K \frac{2q}{p} (∥t ↦ U^{∗}(t^{p/2q})∥_{L} + ∥M∥_{L}).

To finish the proof, suppose that ∥U∥_{p} = λ. Then it is easily seen that U^{∗}(t) ≤ λt^{−1/p}. Thus, if c_{3} is the embedding constant of L_{q} into L, then

∥t ↦ U^{∗}(t^{p/2q})∥_{L} ≤ c_{3}∥t ↦ U^{∗}(t^{p/2q})∥_{q}
= c_{3} (∫_{0}^{1}(U^{∗}(t^{p/2q}))^{q} dt)^{1/q}

≤ c_{3}λ (∫_{0}^{1} t^{−1/2} dt)^{1/q}
= 2^{1/q}c_{3}λ.
□

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