Kambiz Farahmand

The behaviour of some UK equity indices: An application of Hurst and BDS tests1

The characterisation of equity market return series as random in nature has been questioned in recent times by the application of new statistical tools. This study uses recent advances in chaos theory to examine the behaviour of the London Financial Times Stock Exchange (FTSE) All Share, 100, 250 and 350 equity indices. The results reject the hypothesis that the index series examined in this study are random, independent and identically distributed. The results show that the FTSE stock index returns series is not truly random since some cycles or patterns show up more frequently than would be expected in a true random series. A Generalized Autoregressive Conditional Heteroskedasticity (GARCH(1,1)) process appears to explain the behaviour of the index series. The results may have implications for derivative instruments on the indices as well as for weak form market efficiency.

Real zeros of random algebraic polynomials

There are many known asymptotic estimates of the expected number of real zeros of algebraic polynomials with independent random coefficients of equal means. The present paper considers the case when the means of the coefficients are not all necessarily equal. The expected number of crossings of two algebraic polynomials with unequal degree flow from the results

Partial sums of functions of bounded turning

We determine conditions under which the partial sums of the Libera integral operator of functions of bounded turning are also of bounded turning.

Single line queue with recurrent repeated demands

We analyze a model queueing system in which customers cannot be in continuous contact with the server, but must call in to request service. If the server is free, the customer enters service immediately, but if the server is occupied, the unsatisfied customer must break contact and reapply for service later. There are two types of customer present who may reapply. First transit customers who arrive from outside according to a Poisson process and if they find the server busy they join a source of unsatisfied customers, called the orbit, who according to an exponential distribution reapply for service till they find the server free and leave the system on completion of service. Secondly there are a number of recurrent customers present who reapply for service according to a different exponential distribution and immediately go back in to the orbit after each completion of service. We assume a general service time distribution and calculate several characterstic quantities of the system for both the constant rate of reapplying for service and for the case when customers are discouraged and reduce their rate of demand as more customers join the orbit.

Zeros of a random algebraic polynomial with coefficient means in geometric progression

This paper provides the mathematical expectation for the number of real zeros of an algebraic polynomial with non-identical random coefficients. We assume that the coefficients {aj}n−1j=0 of the polynomial T(x)=a0+a1x+a2x2+⋯+an−1xn−1 are normally distributed, with mean E(aj)=μj+1, where μ≠0, and constant non-zero variance. It is shown that the behaviour of the random polynomial is independent of the variance on the interval (−1,1); it differs, however, for the cases of |μ| 1. On the intervals (−∞,−1) and (1,∞) we find the expected number of real zeros is governed by an interesting relationship between the means of the coefficients and their common variance. Our result is consistent with those of previous works for identically distributed coefficients, in that the expected number of real zeros for μ≠0 is half of that for μ=0.

Random Trigonometric Polynomials with Nonidentically Distributed Coefficients

This paper provides asymptotic estimates for the expected number of real zeros of two different forms of random trigonometric polynomials, where the coefficients of polynomials are normally distributed random variables with different means and variances. For the polynomials in the form of 𝑎0

Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

The expected number of real zeros of an algebraic polynomial ð ‘Ž ð ‘œ + ð ‘Ž 1 ð ‘¥ + ð ‘Ž 2 ð ‘¥ 2 + ⋯ + ð ‘Ž ð ‘› ð ‘¥ ð ‘› with random coefficient ð ‘Ž ð ‘— , ð ‘— = 0 , 1 , 2 , … , ð ‘› is known. The distribution of the coefficients is often assumed to be identical albeit allowed to have different classes of distributions. For the nonidentical case, there has been much interest where the variance of the ð ‘— th coefficient is v a r ( ð ‘Ž ð ‘— i€· ) = ð ‘› ð ‘— i€¸ . It is shown that this class of polynomials has significantly more zeros than the classical algebraic polynomials with identical coefficients. However, in the case of nonidentically distributed coefficients it is analytically necessary to assume that the means of coefficients are zero. In this work we study a case when the moments of the coefficients have both binomial and geometric progression elements. That is we assume ð ¸ ( ð ‘Ž ð ‘— i€· ) = ð ‘› ð ‘— i€¸ 𠜇 ð ‘— + 1 and v a r ( ð ‘Ž ð ‘— i€· ) = ð ‘› ð ‘— i€¸ 𠜎 2 ð ‘— . We show how the above expected number of real zeros is dependent on values of 𠜎 2 and 𠜇 in various cases.

ON RANDOM ORTHOGONAL POLYNOMIALS

Let T)(x), T{(x),...,T(x) be a sequence of normalized Legendre polynomials orthogonal with respect to the interval (- 1, 1). The asymptotic estimate of the expected number of real zeros of the random polynomial 9oT)(x) + 9T’(x) +... + 9nTn*(x) where 9j, J O, 1,..., n are independent identically and normally distributed random variables is known. In this paper, we first present the asymptotic value for the above expected number when coefficients are dependent random variables. Further, for the case of independent coefficients, we define the expected number of zero up-crossings with slope greater than u or zero down-crossings with slope less than -u. Promoted by the graphical interpretation, we define these crossings as u-sharp. For the above polynomial, we provide the expected number of such crossings.

Sharp crossings of a non-stationary stochastic process and its application to random polynomials

In this paper we provide the expected number of zero up-crossings with slope greater than udown-crossing with slope less than —uof a Gaussian process ξ(t) Where uis any positive constant. Promoted by graphical interpretation, we define hese crossings as u—sharp. Then the expected number of such crossings of a random lgebraic polynomial of the form with normally distributed coefficients follows from this result. It is Shown that for any bounded uthe expected number of u-sharp crossings is asymptotically equal to 0-sharp crossings while for u→ ∞ as n→ ∞ such that (u 2/3/n)→0 the expected number in he interval (-1,1) asymptotically remains as (1/π) log nand, outside this interval, asymtotically reduces to

Real Zeros of Algebraic Polynomials with Dependent Random Coefficients

The expected number of real zeros of polynomials a 0 + a 1 x + a 2 x 2 +…+a n−1 x n−1 with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π)log n. For the dependent cases studied so far it is shown that this asymptotic value remains O(log n). In this article, we show that when cov(a i , a j ) = 1 − |i − j|/n, for i = 0,…, n − 1 and j = 0,…, n − 1, the above expected number of real zeros reduces significantly to O(log n)1/2.