## On Asymptotics Of Certain Recurrences Arising In Universal Coding (1997)

Venue: | Problems of Information Transmission |

Citations: | 14 - 4 self |

### BibTeX

@ARTICLE{Szpankowski97onasymptotics,

author = {Wojciech Szpankowski},

title = {On Asymptotics Of Certain Recurrences Arising In Universal Coding},

journal = {Problems of Information Transmission},

year = {1997},

volume = {34},

pages = {142--146}

}

### Years of Citing Articles

### OpenURL

### Abstract

Ramanujan's Q-function and the so called "tree function" T (z) defined implicitly by the equation T (z) = ze T (z) found applications in hashing, the birthday paradox problem, random mappings, caching, memory conflicts, and so forth. Recently, several novel applications of these functions to information theory problems such as linear coding and universal portfolios were brought to light. In this paper, we study them in the context of another information theory problem, namely: universal coding which was recently investigated by Shtarkov et al. [Prob. Inf. Trans., 31, 1995]. We provide asymptotic expansions of certain recurrences studied there which describe the optimal redundancy of universal codes. Our methodology falls under the so called analytical information theory that was recently applied successfully to a variety of information theory problems. Key Words: Source coding, multi-alphabet universal coding, redundancy, minimum description length, analytical information theory, si...

### Citations

2197 |
The art of computer programming
- Knuth
- 1973
(Show Context)
Citation Context ...3 +O(1=n), as easy to check by a direct computation. 4 3 Analysis In this section we prove Theorems 1 and 2 using certain tools of analytical analysis of algorithms, notably generating functions (cf. =-=[6, 12]-=-) and singularity analysis (cf. [2]). We shall follow our approach from [11] (cf. also [6, 12]). We start with D n (m) given by (5). Let us introduce a new sequence b D n (m) defined as b D n (m) = n ... |

317 |
Singularity analysis of generating functions
- Flajolet, Odlyzko
- 1990
(Show Context)
Citation Context ... computation. 4 3 Analysis In this section we prove Theorems 1 and 2 using certain tools of analytical analysis of algorithms, notably generating functions (cf. [6, 12]) and singularity analysis (cf. =-=[2]-=-). We shall follow our approach from [11] (cf. also [6, 12]). We start with D n (m) given by (5). Let us introduce a new sequence b D n (m) defined as b D n (m) = n X k=0 / n k ! ` k n ' k ` 1 \Gamma ... |

125 |
Universal Sequential Coding of Single Messages
- Shtarâ€˜kov
- 1987
(Show Context)
Citation Context ...ry for coding theory. In a sense, this work is a continuation of our earlier paper [11] that deals with certain sums arising in linear codes. Here, we deal with (multi-alphabet) universal coding (cf. =-=[9]-=-). A direct motivation for this research is the recent paper of Shtarkov et al. [10] (see also Shtarkov [9]) that designs universal codes in a multi-alphabet environment (e.g., think of constructing t... |

85 | Universal portfolios with side information
- Cover, Ordentlich
- 1996
(Show Context)
Citation Context ... given sequence (the so called additive term), and m in (2) is an additional parameter. In this paper we deal with recurrences of type (2) while in [11] we were more in the framework of (1) (cf. also =-=[1]-=-). 2 Main Results In this section, we formulate precisely the problem and present our main results. Throughout, we shall try to use the notation from [10]. 2 Let x n denote a sequence built over a fin... |

64 | Asymptotic behavior of the Lempelâ€“Ziv parsing scheme and digital search trees
- Jackuet, Szpankowski
- 1995
(Show Context)
Citation Context ... years there has been a resurgence of interest and a few successful applications of analytical methods to problems on words (strings, sequences, codes) that are at the core of information theory (cf. =-=[4, 7, 11]-=-). In this paper, we shall continue developing further tools of analytical information theory for coding theory. In a sense, this work is a continuation of our earlier paper [11] that deals with certa... |

17 |
On the Average Redundancy Rate of the Lempel-Ziv
- Louchard, Szpankowski
- 1997
(Show Context)
Citation Context ... years there has been a resurgence of interest and a few successful applications of analytical methods to problems on words (strings, sequences, codes) that are at the core of information theory (cf. =-=[4, 7, 11]-=-). In this paper, we shall continue developing further tools of analytical information theory for coding theory. In a sense, this work is a continuation of our earlier paper [11] that deals with certa... |

13 |
Multi-alphabet universal coding of memoryless sources
- Shtarkov, Tjalkens, et al.
- 1995
(Show Context)
Citation Context ...[11] that deals with certain sums arising in linear codes. Here, we deal with (multi-alphabet) universal coding (cf. [9]). A direct motivation for this research is the recent paper of Shtarkov et al. =-=[10]-=- (see also Shtarkov [9]) that designs universal codes in a multi-alphabet environment (e.g., think of constructing the best possible code for texts written in English, French and Polish). The authors ... |

12 |
Examples of automatic asymptotic expansions
- Salvy
- 1991
(Show Context)
Citation Context ...2n ` 1 + 1 12n + O(n \Gamma2 ) ' : Plugging this into (13), one can derive the asymptotic expansion for D n (m). Actually, we use directly Maple (more precisely, equivalent program thanks to B. Salvy =-=[8]-=-) to extract coefficients of (B(z) \Gamma 1) m . In fact, equivalent directly gives e \Gamman [z n ](B(z) \Gamma 1) m = n m 2 \Gamma1 2 m 2 \Gamma( m 2 ) \Gamma n m 2 \Gamma 3 2 2 m 2 \Gamma 1 2 / 2m ... |

10 | On Asymptotics of Certain Sums Arising in Coding Theory
- Szpankowski
- 1995
(Show Context)
Citation Context ...on we prove Theorems 1 and 2 using certain tools of analytical analysis of algorithms, notably generating functions (cf. [6, 12]) and singularity analysis (cf. [2]). We shall follow our approach from =-=[11]-=- (cf. also [6, 12]). We start with D n (m) given by (5). Let us introduce a new sequence b D n (m) defined as b D n (m) = n X k=0 / n k ! ` k n ' k ` 1 \Gamma k n ' n\Gammak D n\Gammak (m \Gamma 1) = ... |

8 | Asymptotic Redundancies for Universal Quantum Coding
- Krattenthaler, Slater
- 2000
(Show Context)
Citation Context ... years there has been a resurgence of interest and a few successful applications of analytical methods to problems on words (strings, sequences, codes) that are at the core of information theory (cf. =-=[5, 6, 9, 10, 14]-=-). In this paper, we shall continue developing further tools of analytical information theory for coding theory. In a sense, this work is a continuation of our earlier paper [14] that deals with certa... |

6 | Techniques of the Average Case Analysis of Algorithms
- Szpankowski
- 1998
(Show Context)
Citation Context ...3 +O(1=n), as easy to check by a direct computation. 4 3 Analysis In this section we prove Theorems 1 and 2 using certain tools of analytical analysis of algorithms, notably generating functions (cf. =-=[6, 12]-=-) and singularity analysis (cf. [2]). We shall follow our approach from [11] (cf. also [6, 12]). We start with D n (m) given by (5). Let us introduce a new sequence b D n (m) defined as b D n (m) = n ... |

4 |
Towards Analytical Information Theory: Entropy Computations
- Jacquet, Szpankowski
- 1996
(Show Context)
Citation Context ...t 's W -function. This research was supported in part by NSF Grants NCR-9206315 and NCR-9415491, and in part by NATO Collaborative Grant CGR.950060. 1 1 Introduction Recently, Jacquet and Szpankowski =-=[5]-=- have proposed to call analytical information theory a sub-area of information theory that solves problems of information theory by analytical methods, that is, those in which complex analysis plays a... |

2 |
Analytical Information Theory in Service of Queueing with Aggregated Up/Down Arrivals
- Jacquet
- 1997
(Show Context)
Citation Context ... years there has been a resurgence of interest and a few successful applications of analytical methods to problems on words (strings, sequences, codes) that are at the core of information theory (cf. =-=[5, 6, 9, 10, 14]-=-). In this paper, we shall continue developing further tools of analytical information theory for coding theory. In a sense, this work is a continuation of our earlier paper [14] that deals with certa... |