Recursive estimation of distributional fix-points

2000 ◽  
Vol 37 (1) ◽  
pp. 73-87
Author(s):  
Paul Embrechts ◽  
Harro Walk
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Stochastic Approximation ◽  
Stochastic Models ◽  
Queueing Theory ◽  
Renewal Theory ◽  
Random Variable ◽  
Recursive Estimation ◽  
Real Random Variable ◽  
Index Space

In various stochastic models the random equation of implicit renewal theory appears where the real random variable S and the stochastic process Ψ with index space and state space R are independent. By use of stochastic approximation the distribution function of S is recursively estimated on the basis of independent or ergodic copies of Ψ. Under integrability assumptions almost sure L1-convergence is proved. The choice of gains in the recursion is discussed. Applications are given to insurance mathematics (perpetuities) and queueing theory (stationary waiting and queueing times).

Recursive estimation of distributional fix-points

2000 ◽  
Vol 37 (01) ◽  
pp. 73-87
Author(s):  
Paul Embrechts ◽  
Harro Walk
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Stochastic Approximation ◽  
Stochastic Models ◽  
Queueing Theory ◽  
Renewal Theory ◽  
Random Variable ◽  
Recursive Estimation ◽  
Real Random Variable ◽  
Index Space

In various stochastic models the random equation of implicit renewal theory appears where the real random variable S and the stochastic process Ψ with index space and state space R are independent. By use of stochastic approximation the distribution function of S is recursively estimated on the basis of independent or ergodic copies of Ψ. Under integrability assumptions almost sure L 1-convergence is proved. The choice of gains in the recursion is discussed. Applications are given to insurance mathematics (perpetuities) and queueing theory (stationary waiting and queueing times).

A thermal energy storage process with controlled input

1982 ◽  
Vol 14 (02) ◽  
pp. 257-271 ◽  
Author(s):  
D. J. Daley ◽  
J. Haslett
Keyword(s):  
Stochastic Process ◽  
Energy Storage ◽  
Thermal Energy ◽  
Thermal Energy Storage ◽  
Queueing Theory ◽  
Random Variable ◽  
Stationary Sequence ◽  
Storage Process ◽  
Storage Model ◽  
Special Cases

The stochastic process {Xn } satisfying Xn +1 = max{Yn +1 + αβ Xn , βXn } where {Yn } is a stationary sequence of non-negative random variables and , 0<β <1, can be regarded as a simple thermal energy storage model with controlled input. Attention is mostly confined to the study of μ = EX where the random variable X has the stationary distribution for {Xn }. Even for special cases such as i.i.d. Yn or α = 0, little explicit information appears to be available on the distribution of X or μ . Accordingly, bounding techniques that have been exploited in queueing theory are used to study μ . The various bounds are illustrated numerically in a range of special cases.

A characterization of stable processes

1969 ◽  
Vol 6 (02) ◽  
pp. 409-418 ◽  
Author(s):  
Eugene Lukacs
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Interior Point ◽  
Random Variable ◽  
Stable Processes ◽  
Process With Independent Increments ◽  
Independent Increments

Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t 1 , t 2 ɛ T, t 1 〈 t2; the random variable X(t 2) – X(t 1) is called the increment of the process X(t) over the interval [t 1, t 2]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plimτ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, limτ→0 P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].

scholarly journals A generalized unimodality

1970 ◽  
Vol 7 (1) ◽  
pp. 21-34 ◽  
Author(s):  
Richard A. Olshen ◽  
Leonard J. Savage
Keyword(s):  
Distribution Function ◽  
Random Variable ◽  
Stieltjes Transform ◽  
Real Random Variable ◽  
Unimodal Distribution ◽  
Image Position

Khintchine (1938) showed that a real random variable Z has a unimodal distribution with mode at 0 iff Z ~ U X (that is, Z is distributed like U X), where U is uniform on [0, 1] and U and X are independent. Isii ((1958), page 173) defines a modified Stieltjes transform of a distribution function F for w complex thus: Apparently unaware of Khintchine's work, he proved (pages 179-180) that F is unimodal with mode at 0 iff there is a distribution function Φ for which . The equivalence of Khintchine's and Isii's results is made vivid by a proof (due to L. A. Shepp) in the next section.

Smallest-fit selection of random sizes under a sum constraint: weak convergence and moment comparisons

1999 ◽  
Vol 31 (1) ◽  
pp. 178-198 ◽  
Author(s):  
Frans A. Boshuizen ◽  
Robert P. Kertz
Keyword(s):  
Distribution Function ◽  
Point Processes ◽  
Continuous Mapping ◽  
Random Variables ◽  
Random Measure ◽  
Renewal Theory ◽  
Random Variable ◽  
Brownian Bridges ◽  
On Line ◽  
Image Position

In this paper, in work strongly related with that of Coffman et al. [5], Bruss and Robertson [2], and Rhee and Talagrand [15], we focus our interest on an asymptotic distributional comparison between numbers of ‘smallest’ i.i.d. random variables selected by either on-line or off-line policies. Let X1,X2,… be a sequence of i.i.d. random variables with distribution function F(x), and let X1,n,…,Xn,n be the sequence of order statistics of X1,…,Xn. For a sequence (cn)n≥1 of positive constants, the smallest fit off-line counting random variable is defined by Ne(cn) := max {j ≤ n : X1,n + … + Xj,n ≤ cn}. The asymptotic joint distributional comparison is given between the off-line count Ne(cn) and on-line counts Nnτ for ‘good’ sequential (on-line) policies τ satisfying the sum constraint ∑j≥1XτjI(τj≤n) ≤ cn. Specifically, for such policies τ, under appropriate conditions on the distribution function F(x) and the constants (cn)n≥1, we find sequences of positive constants (Bn)n≥1, (Δn)n≥1 and (Δ'n)n≥1 such that for some non-degenerate random variables W and W'. The major tools used in the paper are convergence of point processes to Poisson random measure and continuous mapping theorems, strong approximation results of the normalized empirical process by Brownian bridges, and some renewal theory.

On the non-closure under convolution of the subexponential family

1989 ◽  
Vol 26 (01) ◽  
pp. 58-66 ◽  
Author(s):  
J. R. Leslie
Keyword(s):  
Queueing Theory ◽  
Branching Processes ◽  
Renewal Theory ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Family Of Distributions

A distribution F of a non-negative random variable belongs to the subexponential family of distributions S if 1 – F (2)(x) ~ 2(1 – F(x)) as x →∞. This family is of considerable interest in branching processes, queueing theory, transient renewal theory and infinite divisibility theory. Much is known about the kind of distributions that belong to S but the question of whether S is closed under convolution has remained unresolved for some time. This paper contains an example which demonstrates that S is not in fact closed.

On a three-state sojourn time problem

1971 ◽  
Vol 8 (04) ◽  
pp. 716-723 ◽  
Author(s):  
A. E. Gibson ◽  
B. W. Conolly
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Sojourn Time ◽  
Random Variable ◽  
The Real ◽  
Time Problem ◽  
Arbitrary Space ◽  
Image Position

Consider the real-valued stochastic process {S(t), 0 ≦ t < ∞} which assumes values in an arbitrary space X. For a given subset T ⊂ X we define which represents the length in time of a visit to state T. We shall restrict ourselves to processes such that τT is a random variable having a differentiable distribution function which is independent of the time t 0 at which the visit to state T begins.

scholarly journals Modelling Real World Using Stochastic Processes and Filtration

2016 ◽  
Vol 24 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Peter Jaeger
Keyword(s):  
Stochastic Process ◽  
Real World ◽  
Random Variable ◽  
Call Option ◽  
Relevant Market ◽  
Real Random Variable ◽  
Time Points ◽  
Arbitrage Opportunity ◽  
Definition Of ◽  
The Given

Summary First we give an implementation in Mizar [2] basic important definitions of stochastic finance, i.e. filtration ([9], pp. 183 and 185), adapted stochastic process ([9], p. 185) and predictable stochastic process ([6], p. 224). Second we give some concrete formalization and verification to real world examples. In article [8] we started to define random variables for a similar presentation to the book [6]. Here we continue this study. Next we define the stochastic process. For further definitions based on stochastic process we implement the definition of filtration. To get a better understanding we give a real world example and connect the statements to the theorems. Other similar examples are given in [10], pp. 143-159 and in [12], pp. 110-124. First we introduce sets which give informations referring to today (Ωnow, Def.6), tomorrow (Ωfut1 , Def.7) and the day after tomorrow (Ωfut2 , Def.8). We give an overview for some events in the σ-algebras Ωnow, Ωfut1 and Ωfut2, see theorems (22) and (23). The given events are necessary for creating our next functions. The implementations take the form of: Ωnow ⊂ Ωfut1 ⊂ Ωfut2 see theorem (24). This tells us growing informations from now to the future 1=now, 2=tomorrow, 3=the day after tomorrow. We install functions f : {1, 2, 3, 4} → ℝ as following: f1 : x → 100, ∀x ∈ dom f, see theorem (36), f2 : x → 80, for x = 1 or x = 2 and f2 : x → 120, for x = 3 or x = 4, see theorem (37), f3 : x → 60, for x = 1, f3 : x → 80, for x = 2 and f3 : x → 100, for x = 3, f3 : x → 120, for x = 4 see theorem (38). These functions are real random variable: f1 over Ωnow, f2 over Ωfut1, f3 over Ωfut2, see theorems (46), (43) and (40). We can prove that these functions can be used for giving an example for an adapted stochastic process. See theorem (49). We want to give an interpretation to these functions: suppose you have an equity A which has now (= w1) the value 100. Tomorrow A changes depending which scenario occurs − e.g. another marketing strategy. In scenario 1 (= w11) it has the value 80, in scenario 2 (= w12) it has the value 120. The day after tomorrow A changes again. In scenario 1 (= w111) it has the value 60, in scenario 2 (= w112) the value 80, in scenario 3 (= w121) the value 100 and in scenario 4 (= w122) it has the value 120. For a visualization refer to the tree: The sets w1,w11,w12,w111,w112,w121,w122 which are subsets of {1, 2, 3, 4}, see (22), tell us which market scenario occurs. The functions tell us the values to the relevant market scenario: For a better understanding of the definition of the random variable and the relation to the functions refer to [7], p. 20. For the proof of certain sets as σ-fields refer to [7], pp. 10-11 and [9], pp. 1-2. This article is the next step to the arbitrage opportunity. If you use for example a simple probability measure, refer, for example to literature [3], pp. 28-34, [6], p. 6 and p. 232 you can calculate whether an arbitrage exists or not. Note, that the example given in literature [3] needs 8 instead of 4 informations as in our model. If we want to code the first 3 given time points into our model we would have the following graph, see theorems (47), (44) and (41): The function for the “Call-Option” is given in literature [3], p. 28. The function is realized in Def.5. As a background, more examples for using the definition of filtration are given in [9], pp. 185-188.

Approximating a Cumulative Distribution Function by Generalized Hyperexponential Distributions

1997 ◽  
Vol 11 (1) ◽  
pp. 11-18 ◽  
Author(s):  
Jihong Ou ◽  
Jingwen Li ◽  
Süleyman Özekici
Keyword(s):  
Distribution Function ◽  
Performance Measures ◽  
Stochastic Models ◽  
Queueing Theory ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Idle Period ◽  
Period Distribution ◽  
Different Types ◽  
Recent Developments

Recent developments in stochastic modeling show that enormous analytical advantages can be gained if a general cumulative distribution function (c.d.f.) can be approximated by generalized hyperexponential distributions. In this paper, we introduce a procedure to explicitly construct such approximations of an arbitrary c.d.f. Although our approach can be used in different types of stochastic models, the main motivation comes from queueing theory in obtaining approximations of the idle-period distribution and other performance measures in GI/G/1 queues.

Smallest-fit selection of random sizes under a sum constraint: weak convergence and moment comparisons

1999 ◽  
Vol 31 (01) ◽  
pp. 178-198 ◽  
Author(s):  
Frans A. Boshuizen ◽  
Robert P. Kertz
Keyword(s):  
Distribution Function ◽  
Point Processes ◽  
Continuous Mapping ◽  
Random Variables ◽  
Random Measure ◽  
Renewal Theory ◽  
Random Variable ◽  
Brownian Bridges ◽  
On Line ◽  
Image Position

In this paper, in work strongly related with that of Coffman et al. [5], Bruss and Robertson [2], and Rhee and Talagrand [15], we focus our interest on an asymptotic distributional comparison between numbers of ‘smallest’ i.i.d. random variables selected by either on-line or off-line policies. Let X 1,X 2,… be a sequence of i.i.d. random variables with distribution function F(x), and let X 1,n ,…,X n,n be the sequence of order statistics of X 1,…,X n . For a sequence (c n ) n≥1 of positive constants, the smallest fit off-line counting random variable is defined by N e (c n ) := max {j ≤ n : X 1,n + … + X j,n ≤ c n }. The asymptotic joint distributional comparison is given between the off-line count N e (c n ) and on-line counts N n τ for ‘good’ sequential (on-line) policies τ satisfying the sum constraint ∑ j≥1 X τ j I (τ j ≤n) ≤ c n . Specifically, for such policies τ, under appropriate conditions on the distribution function F(x) and the constants (c n ) n≥1, we find sequences of positive constants (B n ) n≥1, (Δ n ) n≥1 and (Δ' n ) n≥1 such that for some non-degenerate random variables W and W'. The major tools used in the paper are convergence of point processes to Poisson random measure and continuous mapping theorems, strong approximation results of the normalized empirical process by Brownian bridges, and some renewal theory.

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