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Let { ξ 1 , ξ 2 , … } be a sequence of independent but not necessarily identically distributed non-negative random variables a number of which has distribution functions with dominatingly varying tails. Let η be a counting random variable independent of { ξ 1 , ξ 2 , … } . We consider conditions for random variables { ξ 1 , ξ 2 , … } and η under which distribution of the random sum ξ 1 + ξ 2 + ⋯ + ξ η preserves dominatingly varying tail.

The slash distribution of the sum of independent logistic random variables is introduced. For the distribution of the sum of two logistic random variables, the characteristic function and several distributional identities are obtained. The self-decomposability of this distribution is proved.

In the paper it is shown that the well-known asymptotics for small deviation probabilities of weighed sum of independent positive random variables with a common distribution still holds true if this distribution decreases at zero not faster than some power.

Let ℚp, where p > 2, be a field of p-adic numbers. We consider two independent identically distributed random variables with values in ℚp and distribution μ with a continuous density. We prove that the sum and the squared difference of these random variables are independent if and only if μ is an idempotent distribution, i.e., a shift of the Haar distribution of a compact subgroup of the additive group of the field ℚp.

□ We calculate the asymptotics of the moments as well as the limiting distribution (after the appropriate normalization) of the maximum of independent, not identically distributed, geometric random variables. In many cases, the limit distribution turns out to be the standard Gumbel. The motivation comes from a variant of the genomic evolutionary model proposed by Wilf and Ewens[ 15 ] as an answer to the criticism of the Darwinian theory of evolution stating that the time required for the appropriate mutations is huge. A byproduct of our analysis is the asymptotics of the moments as well as the limiting distribution (after the appropriate normalization) of the maximum of independent, not identically distributed, exponential random variables.

In this chapter, the behavior of the sum of independent random variables is first investigated. Then the limiting behavior of the mean of independent and identically distributed (i.i.d.) samples when the number of samples tends to infinity is discussed.

In this paper, we generalize some results of [V. Bentkus, A new method for approximation in probability and operator theories, Lith. Math. J., 43(4):367–388, 2003] for independent identically distributed summands to to the case of independent non-identically distributed real summands. We derive the Edgeworth expansion with the first term only. Proofs are given following [V. Bentkus, A new method for approximation in probability and operator theories, Lith. Math. J., 43(4):367–388, 2003].

In a 1957 paper F. Spitzer published, among other things, one estimate of the distribution function of sums of independent symmetric random variables with a common absolutely continuous distribution. This estimation was constructed with the help of one identity for the distribution of a sum of the above-mentioned random variables, the proof of which was not given. In this paper we prove the Spitzer identity for any independent random variables, and by using it we construct an estimate of the distribution of a sum of independent identically distributed random variables. The Spitzer estimate can be derived as a particular case of the proposed estimation.

The family consisting of the distributions of products of two independent beta variables is extended to include cases where some of the parameters are not positive but negative or complex. This “beta product” distribution is expressible as a Meijer G function. An example (from risk theory) where such a distribution arises is given: an infinite sum of products of independent random variables is shown to have a distribution that is the product convolution of a complex-parameter beta product and an independent exponential. The distribution of the infinite sum is a new explicit solution of the stochastic equation X = (in law) B(X + C). Characterizations of some G distributions are also proved.

Let fXn; n ‚ 1g be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), deflned on a probability space (›;A;P) and let fNn; n ‚ 1g be a sequence of positive integer-valued r.vs., deflned on the same probability space (›;A;P): Furthermore, we assume that the r.vs. Nn; n ‚ 1

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution. In distribution, the compound Poisson random variable is equal to the sum of a Poisson-distributed number of positive random variables, which are independent and identically distributed and have a common specified density function. Based on the representation of the stochastic integral, we prove that the transition distribution of the tempered stable Ornstein–Uhlenbeck process is self-decomposable and that the transition density is a C∞-function.

This article considers large deviation results for sums of independent non identically distributed random variables, generalizing the result of Petrov (1968) by using a weaker and more natural condition on bounds of the cumulant generating functions of the sequence of random variables.

A classical theorem on the growth of partial sums of independent identically distributed random variables with infinite expectations due to Feller [Feller, W., 1946. A limit theorem for random variables with infinite moments. Amer. J. Math. 66 (2), 257–262] is reinforced and generalized to the case of pairwise independent random variables.

A new estimate of the probability density function (PDF) of the sum of a random number of independent and identically distributed (IID) random variables Xi (i=1, 2, …, N) is shown. The sum PDF is represented as a sum of normal PDFs weighted according to the N PDF. The analytical model is verified by numerical simulations. The comparison is made by the Chi-Square Goodness-of-Fit test.

Let {X1,X2, . . .} be a sequence of independent and identically distributed positive random variables of Pareto-type and let {N(t); t >_ 0} be a counting process independent of the Xi?s. For any fixed t> _ 0, define: TN(t) := X2 1 + X2 2 + ? ? ? + X2N (t) (X1 + X2 + ? ? ? + XN(t))2 if N(t) >_ 1 and TN(t) := 0 otherwise. We derive limits in distribution for TN(t) under some convergence conditions on the counting process. This is even achieved when both the numerator and the denominator defining TN(t) exhibit an erratic behavior (EX1 = ?) or when only the numerator has an erratic behavior (EX1 < ? and EX2 1 = ?). Armed with these results, we obtain asymptotic properties of two popular risk measures, namely the sample coefficient of variation and the sample dispersion. References.

Let S = w 1 S 1 + w 2 S 2 + ⋯ + w N S N . Here S j is a sum of identically distributed random variables with weight w j > 0. We consider the cases where S j is a sum of independent random variables, the sum of independent lattice variables, or has the Markov binomial distribution. Apart from the general case, we investigate the case of symmetric random variables. Distribution of S is approximated by a compound Poisson distribution, by a second-order asymptotic expansion, and by a signed exponential measure. Lower bounds for the accuracy of approximations in uniform metric are established.

In an insurance context, the discounted sum of losses within a finite or infinite time period can be described as a randomly weighted sum of a sequence of independent random variables. These independent random variables represent the amounts of losses in successive development years, while the weights represent the stochastic discount factors. In this paper, we investigate the problem of approximating the tail probability of this weighted sum in the case when the losses have Pareto-like distributions and the discount factors are mutually dependent. We also give some simulation results. © 2005 Taylor & Francis Group, LLC. ispartof: Scandinavian actuarial journal vol:2005 issue:6 pages:446-461 status: published

Suppose be a sequence of independent and identically distributed random variables from Type I extreme value distribution. Let X L (1), X L (2), … be the lower record values of this distribution. We assume the first observation as the first lower record, X L (1). It is known that X L ( m ) is identically distributed as X L ( m +1) + W/m, where W has the exponential distribution with cumulative distribution function F(x) = 1−e −x , x>0. In this article we will characterize the Type I extreme value distribution using this distribution property.

We investigate necessary and sufficient conditions under which one estimate of exponential type is valid for large deviation probabilities of sums of independent identically distributed random variables. Bibliography: 3 titles.

It is commonly accepted that Borel’s theorem for Bernoulli trials was the first strong law of large numbers.

The structure of the nonuniform estimate of convergence rate in the local central limit theorem for the densities of sums of independent identically distributed random variables is made more accurate. The absolute constants are written out explicitly.

We study the mean survival probability (n) at time n on a random one-dimensional chain with perfect absorbers at 0 and L. The transition probabilities gi at the lattice sites i, are independent identically distributed random variables having the distribution p(gi) = 1 for 0 gi1. We prove the asymptotic inequality, C1(n)n2/(logn)L-3 C2 where C1 and C2 are finite positive constants which depend on the lattice size L, but not on n. We confirm this result by simulations for lattice sizes up to L = 17.

Let ( X ij ) be a double sequence of independent, identically distributed random variables, with mean zero and variance one, whose moment generating function is finite in a neighbourhood of the origin. Let S i ( t ) be the partial sum process constructed from X i . We consider the sets F n = { γ −1 n S i ( n .): i ⩽ a n }, where a n is a nondecreasing sequence of integers and γ n is a suitable normalizing sequence. We prove a strong approximation result that in particular implies a Strassen-type law if a n grows slower than exponential. If a n grows at an exponential rate, we prove another Strassen-type result.

Let S = X[subscript 1]+···+X[subscript n] be a sum of n independent integer random variables X[subscript i], where each X[subscript i] is supported on {0, 1, ..., k - 1} but otherwise may have an arbitrary distribution (in particular the Xi's need not be identically distributed). How many samples are required to learn the distribution S to high accuracy? In this paper we show that the answer is completely independent of n, and moreover we give a computationally efficient algorithm which achieves this low sample complexity. More precisely, our algorithm learns any such S to ε-accuracy (with respect to the total variation distance between distributions) using poly(k, 1/ε) samples, independent of n. Its running time is poly(k, 1/ε) in the standard word RAM model. Thus we give a broad generalization of the main result of [DDS12b] which gave a similar learning result for the special case k = 2 (when the distribution S is a Poisson Binomial Distribution). Prior to this work, no nontrivial results were known for learning these distributions even in the case k = 3. A key difficulty is that, in contrast to the case of k = 2, sums of independent {0, 1, 2}-valued random variables may behave very differently from (discretized) normal distributions, and in fact may be rather complicated - they are not log-concave, they can be Θ(n)-modal, there is no relationship between Kolmogorov distance and total variation distance for the class, etc. Nevertheless, the heart of our learning result is a new limit theorem which characterizes what the sum of an arbitrary number of arbitrary independent {0, 1, ... , k-1}-valued random variables may look like. Previous limit theorems in this setting made strong assumptions on the “shift invariance” of the random variables Xi in order to force a discretized normal limit. We believe that our new limit theorem, as the first result for truly arbitrary sums of independent {0, 1, ... - k-1}-valued random variables, is of independent interest.

Let X 1 ,X 2 ,…,X n be compatible independent random variables of continuous type, X (r) be the r th order statistic and I ( r ) : I ( r ) = i if X ( r ) = X i . It is shown that I (r) and X (r) are independent if and only if identically distributed.

We consider the random sum Sn= X1 + X2+ ... + XNwhere X1, X2,...are independent and identically distributed random variables, AT is a geometric random variable, and Nis in general dependent on X.A common mistake in the literature is corrected.

For two types of non-parametric hypotheses optimum tests are derived against certain classes of alternatives. The two kinds of hypotheses are related and may be illustrated by the following example: (1) The joint distribution of the variables Xi, • • •, Xm, Yi, • • •, Yn is invariant under all permutations of the variables; (2) the variables are independently and identically distributed. It is shown that the theory of optimum tests for hypotheses of the first kind is the same as that of optimum similar tests for hypotheses of the second kind. Most powerful tests are obtained against arbitrary simple alternatives, and in a number of important cases most stringent tests are derived against certain composite alternatives. For the example (1), if the distributions are restricted to probability densities, Pitman’s test based on ȳ — x is most powerful against the alternatives that the X’s and F’s are independently normally distributed with common variance, and that E(Xi) = ξ, E(Yi) =η where η > ξ If η — ξ may be positive or negative the test based on | ȳ — x| is most stringent. The definitions are sufficiently general that the theory applies to both continuous and discrete problems, and that tied observations present no difficulties. It is shown that continuous and discrete problems may be combined. Pitman’s test for example, when applied to certain discrete problems, coincides with Fisher’s exact test, and when m = n the test based on | ȳ — x | is most stringent for hypothesis (1) against a broad class of alternatives which includes both discrete and absolutely continuous distributions.

A classic result in probability theory states that two independent real-valued random variables having independent sum and difference are either constant or normally distributed with the same variance. In this article conditions are round on independent random variables X and Y taking values in the group of real numbers modulo 2π so that X +Y and X − Y are independent. When X and Y are identically distributed, the small number of possible distributions for which X and Y have the desired property is known. In the general case there is a richer family of possible distributions for X and Y.

We give a short survey on the characterizations of probability distributions that satisfy the property that the sum of a random number of independent identically distributed (i.i.d.) random variables is of the same type as one random variable from that sum, together with some related new results.

In this short note, we revisit the work of T. Tao and V. Vu on large non-hermitian random matrices with independent and identically distributed entries with mean zero and unit variance. We prove under weaker assumptions that the limit spectral distribution of sum and product of non-hermitian random matrices is universal. As a byproduct, we show that the generalized eigenvalues distribution of two independent matrices converges almost surely to the uniform measure on the Riemann sphere., 8 pages, statement of main theorem slightly improved

The domain of a space is studied in which independent observations of random vectors are concentrated if the dimension of the space and the number of observations approach infinity, and the components of the observable vectors are independent and identically distributed with a stable distribution with a parameter 0< α < 2.

The exponential distribution is characterized by means of the distribution of a weighted sum of independent, identically distributed random variables. The conditions considered turn out to be sufficient in the case of two random variables only.

Let C be a two-dimensional copula, i.e. a two-way distribution function C(x, y) = P(U < x, V < y) with the unidimensional random variables U and V uniformly distributed in the interval (0,1) (see, e.g., [3]). Let us study what happens when the r.v.’s U + V and U - V are independent.

A 0–1probability space is a probability space (Ω, 2Ω,P), where the sample space Ω⊂-{0, 1} n for somen. A probability space isk-wise independent if, whenY i is defined to be theith coordinate or the randomn-vector, then any subset ofk of theY i 's is (mutually) independent, and it is said to be a probability spacefor p1,p2, ...,p n ifP[Y i =1]=p i .

If one of two independent random variables (possibly both) is uniformly distributed, then so is their sum modulo m; we show that the converse is true if m is a prime number and the relevant probabilities are rational numbers.

Let { X n } \{X_n\} be a sequence of independent, identically distributed (i.i.d.) random variables with common mean μ ≠ 0 \mu \ne 0 and variance σ 2 > 0 \sigma ^2 > 0 . Let { S n } \{S_n\} be a sequence of nonnegative integer-valued random variables such that for each n n the random variables S n S_n , X 1 X_1 , X 2 X_2 , … are independent. Then ( X 1 + ⋯ + X S n − n μ ) / n σ 2 → L (X_1 + \dots + X_{S_n} - n \mu ) / \sqrt {n\sigma ^2} \stackrel {\mathcal {L}}{\rightarrow } {} (some) Z Z if and only if ( S n − n ) / n → L (S_n - n)/\sqrt {n} \stackrel {\mathcal {L}}{\rightarrow } {} (some) U U , in which case the distribution of Z Z is that of X + Y X+Y , where X X and Y Y are independent random variables, X X being N ( 0 , 1 ) \mathcal {N}(0, 1) and Y Y having the same distribution as μ U / σ \mu U/\sigma .

In this note, we make some remarks on the construction of sequences of independent identically distributed random variables and of Markov chains concretely on a probability space (Ω,A,P). We also show that there are non non-trivial martingales of exponential type.

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