Lattice distribution

A discrete probability distribution concentrated on a set of points of the form $ a + nh $, where $ h > 0 $, $ a $ is a real number and $ n = 0, \pm 1, \pm 2 , . . . $. The number $ h $ is called the step of the lattice distribution, and if for no $ a _ {1} $ and $ h _ {1} > h $ the distribution is concentrated on a set of the form $ a _ {1} + nh _ {1} $, $ n = 0, \pm 1 , \pm 2 \dots $ then $ h $ is called the maximal step. An arithmetic distribution is a particular case ( $ a= 0 $) of a lattice distribution.

For a probability distribution with characteristic function $ f( t) $ to be a lattice distribution it is necessary and sufficient that there exists a real number $ t _ {0} \neq 0 $ such that $ | f( t _ {0} ) | = 1 $; in this case $ h $ is the maximal step if and only if $ | f( t) | < 1 $ for $ 0 < t < 2 \pi /h $ and $ | f( 2 \pi /h) | = 1 $. The characteristic function of a lattice distribution is periodic.

The inversion formula for a lattice distribution has the form

$$ p _ {n} = \frac{h}{2 \pi } \int\limits _ {| t | < \pi /h } e ^ {- it( a+ nh) } f( t) dt , $$

where $ p _ {n} $ is the probability that the lattice distribution ascribes to the point $ a + nh $ and $ f( t) $ is the corresponding characteristic function. The following equality also holds:

$$ \sum _ {n = - \infty } ^ \infty p _ {n} ^ {2} = \ \frac{h}{2 \pi } \int\limits _ {| t | < \pi /h } | f( t) | ^ {2} dt. $$

The convolution of two lattice distributions with steps $ h _ {1} $ and $ h _ {2} $ and with finite supports is a lattice distribution if and only if $ h _ {1} /h _ {2} $ is a rational number.

In the study of the limit behaviour of sums of independent random variables, the basic result of the central limit theorem on convergence towards the normal distribution is considerably complemented by local theorems for lattice distributions. The simplest example of a local theorem for lattice distributions is the Laplace theorem, which can be generalized as follows: Let $ X _ {1} , X _ {2} \dots $ be a sequence of independent identically-distributed random variables with $ {\mathsf E} X _ {1} = m $, $ {\mathsf D} X _ {1} = \sigma ^ {2} $, and let $ S _ {k} = X _ {1} + \dots + X _ {k} $ while $ X _ {1} $ takes values of the form $ a + nh $, $ h > 0 $. Put

$$ P _ {k} ( n) = {\mathsf P} \{ S _ {k} = ka + nh \} ; $$

for the asymptotic relation

$$ \left | \frac{\sigma \sqrt k }{h} P _ {k} ( n) - \frac{1}{\sqrt {2 \pi } } \mathop{\rm exp} \left \{ - \frac{1}{2} \left ( \frac{ka + nh - km }{\sigma \sqrt h } \right ) ^ {2} \right \} \ \right | \rightarrow 0, $$

as $ k \rightarrow \infty $, to be true uniformly with respect to $ n $, it is necessary and sufficient that the step $ h $ is the maximal step.

References

[1]	B.V. Gnedenko, "The theory of probability", Chelsea, reprint (1962) (Translated from Russian)
[2]	V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)
[3]	Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)

References

[a1]	W. Feller, "An introduction to probability theory and its applications", 1–2 , Wiley (1957–1971)
[a2]	E. Lukacs, "Characteristic functions" , Griffin (1970)
[a3]	N.L. Johnson, S. Kotz, "Distributions in statistics: discrete distributions" , Mifflin (1969)